d03psf integrates a system of linear or nonlinear convection-diffusion equations in one space dimension, with optional source terms and scope for coupled ordinary differential equations (ODEs). The system must be posed in conservative form. This routine also includes the option of automatic adaptive spatial remeshing. Convection terms are discretized using a sophisticated upwind scheme involving a user-supplied numerical flux function based on the solution of a Riemann problem at each mesh point. The method of lines is employed to reduce the partial differential equations (PDEs) to a system of ODEs, and the resulting system is solved using a backward differentiation formula (BDF) method or a Theta method.
$\stackrel{.}{V}$ denotes its derivative with respect to time, and ${U}_{x}$ is the spatial derivative of $U$.
In (2), ${P}_{i,j}$,
${F}_{i}$ and ${C}_{i}$ depend on $x$, $t$, $U$ and $V$; ${D}_{i}$ depends on $x$, $t$, $U$, ${U}_{x}$ and $V$; and ${S}_{i}$ depends on $x$, $t$, $U$, $V$ and linearly on $\stackrel{.}{V}$. Note that ${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${S}_{i}$ must not depend on any space derivatives, and ${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${D}_{i}$ must not depend on any time derivatives. In terms of conservation laws, ${F}_{i}$, $\frac{{C}_{i}\partial {D}_{i}}{\partial x}$ and ${S}_{i}$ are the convective flux, diffusion and source terms respectively.
In (3), $\xi $ represents a vector of ${n}_{\xi}$ spatial coupling points at which the ODEs are coupled to the PDEs. These points may or may not be equal to PDE spatial mesh points. ${U}^{*}$, ${U}_{x}^{*}$ and ${U}_{t}^{*}$ are the functions $U$, ${U}_{x}$ and ${U}_{t}$ evaluated at these coupling points. Each ${R}_{i}$ may depend only linearly on time derivatives. Hence (3) may be written more precisely as
$$R=L-M\stackrel{.}{V}-N{U}_{t}^{*}\text{,}$$
(4)
where $R={[{R}_{1},\dots ,{R}_{{\mathbf{nv}}}]}^{\mathrm{T}}$, $L$ is a vector of length nv, $M$ is an nv by nv matrix, $N$ is an nv by $({n}_{\xi}\times {\mathbf{npde}})$ matrix and the entries in $L$, $M$ and $N$ may depend on $t$, $\xi $, ${U}^{*}$, ${U}_{x}^{*}$ and $V$. In practice you only need to supply a vector of information to define the ODEs and not the matrices $L$, $M$ and $N$. (See Section 5 for the specification of odedef.)
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{npts}}}$ are the leftmost and rightmost points of a user-defined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$ defined initially by you and (possibly) adapted automatically during the integration according to user-specified criteria.
The initial $(t={t}_{0})$ values of the functions $U(x,t)$ and $V\left(t\right)$ must be specified in uvinit. Note that uvinit will be called again following any initial remeshing, and so $U(x,{t}_{0})$ should be specified for all values of $x$ in the interval $a\le x\le b$, and not just the initial mesh points.
The PDEs are approximated by a system of ODEs in time for the values of ${U}_{i}$ at mesh points using a spatial discretization method similar to the central-difference scheme used in d03pcf/d03pca,d03phf/d03phaandd03ppf/d03ppa, but with the flux ${F}_{i}$ replaced by a numerical flux, which is a representation of the flux taking into account the direction of the flow of information at that point (i.e., the direction of the characteristics). Simple central differencing of the numerical flux then becomes a sophisticated upwind scheme in which the correct direction of upwinding is automatically achieved.
The numerical flux, ${\hat{F}}_{i}$ say, must be calculated by you in terms of the left and right values of the solution vector $U$ (denoted by ${U}_{L}$ and ${U}_{R}$ respectively), at each mid-point of the mesh ${x}_{\mathit{j}-\frac{1}{2}}=({x}_{\mathit{j}-1}+{x}_{\mathit{j}})/2$, for $\mathit{j}=2,3,\dots ,{\mathbf{npts}}$. The left and right values are calculated by d03psf from two adjacent mesh points using a standard upwind technique combined with a Van Leer slope-limiter (see LeVeque (1990)). The physically correct value for ${\hat{F}}_{i}$ is derived from the solution of the Riemann problem given by
where $y=x-{x}_{j-\frac{1}{2}}$, i.e., $y=0$ corresponds to $x={x}_{j-\frac{1}{2}}$, with discontinuous initial values $U={U}_{L}$ for $y<0$ and $U={U}_{R}$ for $y>0$, using an approximate Riemann solver. This applies for either of the systems (1) or (2); the numerical flux is independent of the functions ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$. A description of several approximate Riemann solvers can be found in LeVeque (1990) and Berzins et al. (1989). Roe's scheme (see Roe (1981)) is perhaps the easiest to understand and use, and a brief summary follows. Consider the system of PDEs ${U}_{t}+{F}_{x}=0$ or equivalently ${U}_{t}+A{U}_{x}=0$. Provided the system is linear in $U$, i.e., the Jacobian matrix $A$ does not depend on $U$, the numerical flux $\hat{F}$ is given by
where ${F}_{L}$ (${F}_{R}$) is the flux $F$ calculated at the left (right) value of $U$, denoted by ${U}_{L}$ (${U}_{R}$); the ${\lambda}_{k}$ are the eigenvalues of $A$; the ${e}_{k}$ are the right eigenvectors of $A$; and the ${\alpha}_{k}$ are defined by
Examples are given in the documents for d03pffandd03plf.
If the system is nonlinear, Roe's scheme requires that a linearized Jacobian is found (see Roe (1981)).
The functions ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ (but not${F}_{i}$) must be specified in pdedef. The numerical flux ${\hat{F}}_{i}$ must be supplied in numflx. For problems in the form (2),
the actual argument d03plp
may be used for pdedef.
d03plp is included in the NAG Library and
sets the matrix with entries ${P}_{i,j}$ to the identity matrix, and the functions ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ to zero.
For second-order problems, i.e., diffusion terms are present, a boundary condition is required for each PDE at both boundaries for the problem to be well-posed. If there are no diffusion terms present, then the continuous PDE problem generally requires exactly one boundary condition for each PDE, that is npde boundary conditions in total. However, in common with most discretization schemes for first-order problems, a numerical boundary condition is required at the other boundary for each PDE. In order to be consistent with the characteristic directions of the PDE system, the numerical boundary conditions must be derived from the solution inside the domain in some manner (see below). You must supply both types of boundary conditions, i.e., a total of npde conditions at each boundary point.
The position of each boundary condition should be chosen with care. In simple terms, if information is flowing into the domain then a physical boundary condition is required at that boundary, and a numerical boundary condition is required at the other boundary. In many cases the boundary conditions are simple, e.g., for the linear advection equation. In general you should calculate the characteristics of the PDE system and specify a physical boundary condition for each of the characteristic variables associated with incoming characteristics, and a numerical boundary condition for each outgoing characteristic.
A common way of providing numerical boundary conditions is to extrapolate the characteristic variables from the inside of the domain (note that when using banded matrix algebra the fixed bandwidth means that only linear extrapolation is allowed, i.e., using information at just two interior points adjacent to the boundary). For problems in which the solution is known to be uniform (in space) towards a boundary during the period of integration then extrapolation is unnecessary; the numerical boundary condition can be supplied as the known solution at the boundary. Another method of supplying numerical boundary conditions involves the solution of the characteristic equations associated with the outgoing characteristics. Examples of both methods can be found in the documents for d03pffandd03plf.
The boundary conditions must be specified in bndary in the form
Note that spatial derivatives at the boundary are not passed explicitly to bndary, but they can be calculated using values of $U$ at and adjacent to the boundaries if required. However, it should be noted that instabilities may occur if such one-sided differencing opposes the characteristic direction at the boundary.
The algebraic-differential equation system which is defined by the functions ${R}_{i}$ must be specified in odedef. You must also specify the coupling points $\xi $ (if any) in the array xi.
In total there are ${\mathbf{npde}}\times {\mathbf{npts}}+{\mathbf{nv}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF or Theta method, optionally switching between Newton's method and functional iteration (see Berzins et al. (1989) and the references therein).
The adaptive space remeshing can be used to generate meshes that automatically follow the changing time-dependent nature of the solution, generally resulting in a more efficient and accurate solution using fewer mesh points than may be necessary with a fixed uniform or non-uniform mesh. Problems with travelling wavefronts or variable-width boundary layers for example will benefit from using a moving adaptive mesh. The discrete time-step method used here (developed by Furzeland (1984)) automatically creates a new mesh based on the current solution profile at certain time-steps, and the solution is then interpolated onto the new mesh and the integration continues.
The method requires you to supply a monitf which specifies in an analytical or numerical form the particular aspect of the solution behaviour you wish to track. This so-called monitor function is used by the routine to choose a mesh which equally distributes the integral of the monitor function over the domain. A typical choice of monitor function is the second space derivative of the solution value at each point (or some combination of the second space derivatives if there is more than one solution component), which results in refinement in regions where the solution gradient is changing most rapidly.
You must specify the frequency of mesh updates together with certain other criteria such as adjacent mesh ratios. Remeshing can be expensive and you are encouraged to experiment with the different options in order to achieve an efficient solution which adequately tracks the desired features of the solution.
Note that unless the monitor function for the initial solution values is zero at all user-specified initial mesh points, a new initial mesh is calculated and adopted according to the user-specified remeshing criteria. uvinit will then be called again to determine the initial solution values at the new mesh points (there is no interpolation at this stage) and the integration proceeds.
The problem is subject to the following restrictions:
(i)In (1), ${\stackrel{.}{V}}_{\mathit{j}}\left(t\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in the functions
${S}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, with a similar restriction for ${G}_{i}^{L}$ and ${G}_{i}^{R}$;
(ii)${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${S}_{i}$ must not depend on any space derivatives; and ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${F}_{i}$ must not depend on any time derivatives;
(iii)${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction;
(iv)The evaluation of the terms ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ is done by calling the pdedef at a point approximately midway between each pair of mesh points in turn. Any discontinuities in these functions must, therefore, be at one or more of the fixed mesh points specified by xfix;
(v)At least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the PDE problem.
Berzins M, Dew P M and Furzeland R M (1989) Developing software for time-dependent problems using the method of lines and differential-algebraic integrators Appl. Numer. Math.5 375–397
Furzeland R M (1984) The construction of adaptive space meshes TNER.85.022 Thornton Research Centre, Chester
Hirsch C (1990) Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows John Wiley
LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag
Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw.20 63–99
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys.43 357–372
5Arguments
1: $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs to be solved.
Constraint:
${\mathbf{npde}}\ge 1$.
2: $\mathbf{ts}$ – Real (Kind=nag_wp)Input/Output
On entry: the initial value of the independent variable $t$.
On exit: the value of $t$ corresponding to the solution values in u. Normally ${\mathbf{ts}}={\mathbf{tout}}$.
Constraint:
${\mathbf{ts}}<{\mathbf{tout}}$.
3: $\mathbf{tout}$ – Real (Kind=nag_wp)Input
On entry: the final value of $t$ to which the integration is to be carried out.
4: $\mathbf{pdedef}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
pdedef must evaluate the functions ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ which partially define the system of PDEs. ${P}_{i,j}$ and ${C}_{i}$ may depend on $x$, $t$, $U$ and $V$; ${D}_{i}$ may depend on $x$, $t$, $U$, ${U}_{x}$ and $V$; and ${S}_{i}$ may depend on $x$, $t$, $U$, $V$ and linearly on $\stackrel{.}{V}$. pdedef is called approximately midway between each pair of mesh points in turn by d03psf. The actual argument d03plp may be used for pdedef for problems in the form (2). (d03plp is included in the NAG Library.)
On entry: the current value of the independent variable $t$.
3: $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the current value of the space variable $x$.
4: $\mathbf{u}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}\left(\mathit{i}\right)$ contains the value of the component ${U}_{\mathit{i}}(x,t)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
5: $\mathbf{ux}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ux}}\left(\mathit{i}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
6: $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
7: $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
8: $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
Note:${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in
${S}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
9: $\mathbf{p}({\mathbf{npde}},{\mathbf{npde}})$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{p}}(\mathit{i},\mathit{j})$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}(x,t,U,V)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
10: $\mathbf{c}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{c}}\left(\mathit{i}\right)$ must be set to the value of ${C}_{\mathit{i}}(x,t,U,V)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
11: $\mathbf{d}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{d}}\left(\mathit{i}\right)$ must be set to the value of ${D}_{\mathit{i}}(x,t,U,{U}_{x},V)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
12: $\mathbf{s}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{s}}\left(\mathit{i}\right)$ must be set to the value of ${S}_{\mathit{i}}(x,t,U,V,\stackrel{.}{V})$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
13: $\mathbf{ires}$ – IntegerInput/Output
On entry: set to $\mathrm{-1}$ or $1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03psf returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
5: $\mathbf{numflx}$ – Subroutine, supplied by the user.External Procedure
numflx must supply the numerical flux for each PDE given the left and right values of the solution vector ${\mathbf{u}}$. numflx is called approximately midway between each pair of mesh points in turn by d03psf.
On entry: the current value of the independent variable $t$.
3: $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the current value of the space variable $x$.
4: $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
5: $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
6: $\mathbf{uleft}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{uleft}}\left(\mathit{i}\right)$ contains the left value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
7: $\mathbf{uright}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{uright}}\left(\mathit{i}\right)$ contains the right value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
8: $\mathbf{flux}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{flux}}\left(\mathit{i}\right)$ must be set to the numerical flux ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
9: $\mathbf{ires}$ – IntegerInput/Output
On entry: set to $\mathrm{-1}$ or $1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03psf returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
numflx must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:numflx should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
6: $\mathbf{bndary}$ – Subroutine, supplied by the user.External Procedure
bndary must evaluate the functions ${G}_{i}^{L}$ and ${G}_{i}^{R}$ which describe the physical and numerical boundary conditions, as given by (8) and (9).
On entry: the number of mesh points in the interval $[a,b]$.
3: $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: the current value of the independent variable $t$.
4: $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left(1\right)$ corresponds to the left-hand boundary, $a$, and ${\mathbf{x}}\left({\mathbf{npts}}\right)$ corresponds to the right-hand boundary, $b$.
5: $\mathbf{u}({\mathbf{npde}},{\mathbf{npts}})$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}(\mathit{i},\mathit{j})$ contains the value of the component ${U}_{\mathit{i}}(x,t)$ at $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
Note: if banded matrix algebra is to be used then the functions ${G}_{\mathit{i}}^{L}$ and ${G}_{\mathit{i}}^{R}$ may depend on the value of ${U}_{\mathit{i}}(x,t)$ at the boundary point and the two adjacent points only.
6: $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
7: $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
8: $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
Note:${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in
${G}_{\mathit{j}}^{L}$ and ${G}_{\mathit{j}}^{R}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
9: $\mathbf{ibnd}$ – IntegerInput
On entry: specifies which boundary conditions are to be evaluated.
${\mathbf{ibnd}}=0$
bndary must evaluate the left-hand boundary condition at $x=a$.
${\mathbf{ibnd}}\ne 0$
bndary must evaluate the right-hand boundary condition at $x=b$.
10: $\mathbf{g}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{g}}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of either ${G}_{\mathit{i}}^{L}$ or ${G}_{\mathit{i}}^{R}$ in (8) and (9), depending on the value of ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
11: $\mathbf{ires}$ – IntegerInput/Output
On entry: set to $\mathrm{-1}$ or $1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03psf returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
7: $\mathbf{uvinit}$ – Subroutine, supplied by the user.External Procedure
uvinit must supply the initial $(t={t}_{0})$ values of $U(x,t)$ and $V\left(t\right)$ for all values of $x$ in the interval $a\le x\le b$.
On entry: the number of mesh points in the interval [$a,b$].
3: $\mathbf{nxi}$ – IntegerInput
On entry: the number of ODE/PDE coupling points.
4: $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the current mesh. ${\mathbf{x}}\left(\mathit{i}\right)$ contains the value of ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5: $\mathbf{xi}\left({\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left(\mathit{i}\right)$ contains the ODE/PDE coupling point, ${\xi}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
6: $\mathbf{u}({\mathbf{npde}},{\mathbf{npts}})$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{u}}(\mathit{i},\mathit{j})$ contains the value of the component ${U}_{\mathit{i}}({x}_{\mathit{j}},{t}_{0})$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
7: $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
8: $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ must contain the value of component ${V}_{\mathit{i}}\left({t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
uvinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
8: $\mathbf{u}\left({\mathbf{neqn}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{ind}}=1$ the value of u must be unchanged from the previous call.
On exit: the computed solution
${U}_{\mathit{i}}({x}_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and
${V}_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,{\mathbf{nv}}$, evaluated at $t={\mathbf{ts}}$, as follows:
${\mathbf{u}}\left({\mathbf{npde}}\times (\mathit{j}-1)+\mathit{i}\right)$ contain ${U}_{\mathit{i}}({x}_{\mathit{j}},t)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and
${\mathbf{u}}\left({\mathbf{npts}}\times {\mathbf{npde}}+\mathit{i}\right)$ contain ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
9: $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $[a,b]$.
Constraint:
${\mathbf{npts}}\ge 3$.
10: $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the mesh points in the space direction. ${\mathbf{x}}\left(1\right)$ must specify the left-hand boundary, $a$, and ${\mathbf{x}}\left({\mathbf{npts}}\right)$ must specify the right-hand boundary, $b$.
12: $\mathbf{odedef}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
odedef must evaluate the functions $R$, which define the system of ODEs, as given in (4).
If you wish to compute the solution of a system of PDEs only (i.e., ${\mathbf{nv}}=0$), odedef must be the dummy routine d03pek. (d03pek is included in the NAG Library.)
On entry: the current value of the independent variable $t$.
3: $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
4: $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
5: $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
6: $\mathbf{nxi}$ – IntegerInput
On entry: the number of ODE/PDE coupling points.
7: $\mathbf{xi}\left({\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left(\mathit{i}\right)$ contains the ODE/PDE coupling point, ${\xi}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
8: $\mathbf{ucp}({\mathbf{npde}},{\mathbf{nxi}})$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucp}}(\mathit{i},\mathit{j})$ contains the value of ${U}_{\mathit{i}}(x,t)$ at the coupling point $x={\xi}_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
9: $\mathbf{ucpx}({\mathbf{npde}},{\mathbf{nxi}})$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpx}}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial {U}_{\mathit{i}}(x,t)}{\partial x}$ at the coupling point $x={\xi}_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
10: $\mathbf{ucpt}({\mathbf{npde}},{\mathbf{nxi}})$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpt}}(\mathit{i},\mathit{j})$ contains the value of $\frac{\partial {U}_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi}_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
11: $\mathbf{r}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{r}}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $R$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, where $R$ is defined as
$$R=L-M\stackrel{.}{V}-N{U}_{t}^{*}\text{,}$$
(10)
or
$$R=-M\stackrel{.}{V}-N{U}_{t}^{*}\text{.}$$
(11)
The definition of $R$ is determined by the input value of ires.
12: $\mathbf{ires}$ – IntegerInput/Output
On entry: the form of $R$ that must be returned in the array r.
On exit: should usually remain unchanged. However, you may reset ires to force the integration routine to take certain actions, as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03psf returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
13: $\mathbf{nxi}$ – IntegerInput
On entry: the number of ODE/PDE coupling points.
Constraints:
if ${\mathbf{nv}}=0$, ${\mathbf{nxi}}=0$;
if ${\mathbf{nv}}>0$, ${\mathbf{nxi}}\ge 0$.
14: $\mathbf{xi}\left({\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$, must be set to the ODE/PDE coupling points.
16: $\mathbf{rtol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array rtol
must be at least
$1$ if ${\mathbf{itol}}=1$ or $2$ and at least ${\mathbf{neqn}}$ if ${\mathbf{itol}}=3$ or $4$.
On entry: the relative local error tolerance.
Constraint:
${\mathbf{rtol}}\left(i\right)\ge 0.0$ for all relevant $i$.
17: $\mathbf{atol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array atol
must be at least
$1$ if ${\mathbf{itol}}=1$ or $3$ and at least ${\mathbf{neqn}}$ if ${\mathbf{itol}}=2$ or $4$.
On entry: the absolute local error tolerance.
Constraint:
${\mathbf{atol}}\left(i\right)\ge 0.0$ for all relevant $i$.
Note: corresponding elements of rtol and atol cannot both be $0.0$.
18: $\mathbf{itol}$ – IntegerInput
On entry: a value to indicate the form of the local error test.
If ${e}_{\mathit{i}}$ is the estimated local error for ${\mathbf{u}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqn}}$, and $\Vert \text{\hspace{1em}}\Vert $, denotes the norm, the error test to be satisfied is $\Vert {e}_{\mathit{i}}\Vert <1.0$. itol indicates to d03psf whether to interpret either or both of rtol and atol as a vector or scalar in the formation of the weights ${w}_{\mathit{i}}$ used in the calculation of the norm (see the description of norm):
See the description of itol for the formulation of the weight vector $w$.
Constraint:
${\mathbf{norm}}=\text{'1'}$ or $\text{'2'}$.
20: $\mathbf{laopt}$ – Character(1)Input
On entry: the type of matrix algebra required.
${\mathbf{laopt}}=\text{'F'}$
Full matrix methods to be used.
${\mathbf{laopt}}=\text{'B'}$
Banded matrix methods to be used.
${\mathbf{laopt}}=\text{'S'}$
Sparse matrix methods to be used.
Constraint:
${\mathbf{laopt}}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.
Note: you are recommended to use the banded option when no coupled ODEs are present (${\mathbf{nv}}=0$). Also, the banded option should not be used if the boundary conditions involve solution components at points other than the boundary and the immediately adjacent two points.
21: $\mathbf{algopt}\left(30\right)$ – Real (Kind=nag_wp) arrayInput
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, ${\mathbf{algopt}}\left(1\right)$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:
${\mathbf{algopt}}\left(1\right)$
Selects the ODE integration method to be used. If ${\mathbf{algopt}}\left(1\right)=1.0$, a BDF method is used and if ${\mathbf{algopt}}\left(1\right)=2.0$, a Theta method is used. The default is ${\mathbf{algopt}}\left(1\right)=1.0$.
If ${\mathbf{algopt}}\left(1\right)=2.0$, then
${\mathbf{algopt}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,4$, are not used.
${\mathbf{algopt}}\left(2\right)$
Specifies the maximum order of the BDF integration formula to be used. ${\mathbf{algopt}}\left(2\right)$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is ${\mathbf{algopt}}\left(2\right)=5.0$.
${\mathbf{algopt}}\left(3\right)$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ${\mathbf{algopt}}\left(3\right)=1.0$ a modified Newton iteration is used and if ${\mathbf{algopt}}\left(3\right)=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is ${\mathbf{algopt}}\left(3\right)=1.0$.
${\mathbf{algopt}}\left(4\right)$
Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as
${P}_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, for some $i$ or when there is no ${\stackrel{.}{V}}_{i}\left(t\right)$ dependence in the coupled ODE system. If ${\mathbf{algopt}}\left(4\right)=1.0$, the Petzold test is used. If ${\mathbf{algopt}}\left(4\right)=2.0$, the Petzold test is not used. The default value is ${\mathbf{algopt}}\left(4\right)=1.0$.
If ${\mathbf{algopt}}\left(1\right)=1.0$,
${\mathbf{algopt}}\left(\mathit{i}\right)$, for $\mathit{i}=5,6,7$, are not used.
${\mathbf{algopt}}\left(5\right)$
Specifies the value of Theta to be used in the Theta integration method. $0.51\le {\mathbf{algopt}}\left(5\right)\le 0.99$. The default value is ${\mathbf{algopt}}\left(5\right)=0.55$.
${\mathbf{algopt}}\left(6\right)$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ${\mathbf{algopt}}\left(6\right)=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left(6\right)=2.0$, a functional iteration method is used. The default value is ${\mathbf{algopt}}\left(6\right)=1.0$.
${\mathbf{algopt}}\left(7\right)$
Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If ${\mathbf{algopt}}\left(7\right)=1.0$, switching is allowed and if ${\mathbf{algopt}}\left(7\right)=2.0$, switching is not allowed. The default value is ${\mathbf{algopt}}\left(7\right)=1.0$.
${\mathbf{algopt}}\left(11\right)$
Specifies a point in the time direction, ${t}_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of ${t}_{\mathrm{crit}}$ is described under the argument itask. If ${\mathbf{algopt}}\left(1\right)\ne 0.0$, a value of $0.0$ for ${\mathbf{algopt}}\left(11\right)$, say, should be specified even if itask subsequently specifies that ${t}_{\mathrm{crit}}$ will not be used.
${\mathbf{algopt}}\left(12\right)$
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left(12\right)$ should be set to $0.0$.
${\mathbf{algopt}}\left(13\right)$
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left(13\right)$ should be set to $0.0$.
${\mathbf{algopt}}\left(14\right)$
Specifies the initial step size to be attempted by the integrator. If ${\mathbf{algopt}}\left(14\right)=0.0$, the initial step size is calculated internally.
${\mathbf{algopt}}\left(15\right)$
Specifies the maximum number of steps to be attempted by the integrator in any one call. If ${\mathbf{algopt}}\left(15\right)=0.0$, no limit is imposed.
${\mathbf{algopt}}\left(23\right)$
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, ${U}_{t}$, $V$ and $\stackrel{.}{V}$. If ${\mathbf{algopt}}\left(23\right)=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left(23\right)=2.0$, functional iteration is used. The default value is ${\mathbf{algopt}}\left(23\right)=1.0$.
${\mathbf{algopt}}\left(29\right)$ and ${\mathbf{algopt}}\left(30\right)$ are used only for the sparse matrix algebra option, i.e., ${\mathbf{laopt}}=\text{'S'}$.
${\mathbf{algopt}}\left(29\right)$
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<{\mathbf{algopt}}\left(29\right)<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ${\mathbf{algopt}}\left(29\right)$ lies outside the range then the default value is used. If the routines regard the Jacobian matrix as numerically singular, increasing ${\mathbf{algopt}}\left(29\right)$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is ${\mathbf{algopt}}\left(29\right)=0.1$.
${\mathbf{algopt}}\left(30\right)$
Used as the relative pivot threshold during subsequent Jacobian decompositions (see ${\mathbf{algopt}}\left(29\right)$) below which an internal error is invoked. ${\mathbf{algopt}}\left(30\right)$ must be greater than zero, otherwise the default value is used. If ${\mathbf{algopt}}\left(30\right)$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian matrix is found to be numerically singular (see ${\mathbf{algopt}}\left(29\right)$). The default value is ${\mathbf{algopt}}\left(30\right)=0.0001$.
22: $\mathbf{remesh}$ – LogicalInput
On entry: indicates whether or not spatial remeshing should be performed.
${\mathbf{remesh}}=\mathrm{.TRUE.}$
Indicates that spatial remeshing should be performed as specified.
${\mathbf{remesh}}=\mathrm{.FALSE.}$
Indicates that spatial remeshing should be suppressed.
Note:remesh should not be changed between consecutive calls to d03psf. Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.
Note: the end points ${\mathbf{x}}\left(1\right)$ and ${\mathbf{x}}\left({\mathbf{npts}}\right)$ are fixed automatically and hence should not be specified as fixed points.
24: $\mathbf{xfix}\left({\mathbf{nxfix}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{xfix}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxfix}}$, must contain the value of the $x$ coordinate at the $\mathit{i}$th fixed mesh point.
Constraints:
${\mathbf{xfix}}\left(\mathit{i}\right)<{\mathbf{xfix}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxfix}}-1$;
each fixed mesh point must coincide with a user-supplied initial mesh point, that is ${\mathbf{xfix}}\left(\mathit{i}\right)={\mathbf{x}}\left(\mathit{j}\right)$ for some $\mathit{j}$, $2\le \mathit{j}\le {\mathbf{npts}}-1$..
Note: the positions of the fixed mesh points in the array ${\mathbf{x}}\left({\mathbf{npts}}\right)$ remain fixed during remeshing, and so the number of mesh points between adjacent fixed points (or between fixed points and end points) does not change. You should take this into account when choosing the initial mesh distribution.
25: $\mathbf{nrmesh}$ – IntegerInput
On entry: specifies the spatial remeshing frequency and criteria for the calculation and adoption of a new mesh.
${\mathbf{nrmesh}}<0$
Indicates that a new mesh is adopted according to the argument dxmesh. The mesh is tested every $\left|{\mathbf{nrmesh}}\right|$ timesteps.
${\mathbf{nrmesh}}=0$
Indicates that remeshing should take place just once at the end of the first time step reached when $t>{\mathbf{trmesh}}$.
${\mathbf{nrmesh}}>0$
Indicates that remeshing will take place every nrmesh time steps, with no testing using dxmesh.
Note:nrmesh may be changed between consecutive calls to d03psf to give greater flexibility over the times of remeshing.
26: $\mathbf{dxmesh}$ – Real (Kind=nag_wp)Input
On entry: determines whether a new mesh is adopted when nrmesh is set less than zero. A possible new mesh is calculated at the end of every $\left|{\mathbf{nrmesh}}\right|$ time steps, but is adopted only if
dxmesh thus imposes a lower limit on the difference between one mesh and the next.
Constraint:
${\mathbf{dxmesh}}\ge 0.0$.
27: $\mathbf{trmesh}$ – Real (Kind=nag_wp)Input
On entry: specifies when remeshing will take place when nrmesh is set to zero. Remeshing will occur just once at the end of the first time step reached when $t$ is greater than trmesh.
Note:trmesh may be changed between consecutive calls to d03psf to force remeshing at several specified times.
28: $\mathbf{ipminf}$ – IntegerInput
On entry: the level of trace information regarding the adaptive remeshing. Details are directed to the current advisory message unit (see x04abf).
${\mathbf{ipminf}}=0$
No trace information.
${\mathbf{ipminf}}=1$
Brief summary of mesh characteristics.
${\mathbf{ipminf}}=2$
More detailed information, including old and new mesh points, mesh sizes and monitor function values.
Constraint:
${\mathbf{ipminf}}=0$, $1$ or $2$.
29: $\mathbf{xratio}$ – Real (Kind=nag_wp)Input
On entry: an input bound on the adjacent mesh ratio (greater than $1.0$ and typically in the range $1.5$ to $3.0$). The remeshing routines will attempt to ensure that
On entry: an input bound on the sub-integral of the monitor function ${F}^{\mathrm{mon}}\left(x\right)$ over each space step. The remeshing routines will attempt to ensure that
(see Furzeland (1984)). con gives you more control over the mesh distribution, e.g., decreasing con allows more clustering. A typical value is $2.0/({\mathbf{npts}}-1)$, but you are encouraged to experiment with different values. Its value is not critical and the mesh should be qualitatively correct for all values in the range given below.
31: $\mathbf{monitf}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
monitf must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.
If you specify ${\mathbf{remesh}}=\mathrm{.FALSE.}$, i.e., no remeshing, monitf will not be called and the dummy routine d03pel may be used for monitf. (d03pel is included in the NAG Library.)
On entry: the current value of the independent variable $t$.
2: $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $[a,b]$.
3: $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system.
4: $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the current mesh. ${\mathbf{x}}\left(\mathit{i}\right)$ contains the value of ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$.
5: $\mathbf{u}({\mathbf{npde}},{\mathbf{npts}})$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}(\mathit{i},\mathit{j})$ contains the value of ${U}_{\mathit{i}}(x,t)$ at $x={\mathbf{x}}\left(\mathit{j}\right)$ and time $t$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
6: $\mathbf{fmon}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{fmon}}\left(i\right)$ must contain the value of the monitor function ${F}^{\mathrm{mon}}\left(x\right)$ at mesh point $x={\mathbf{x}}\left(i\right)$.
monitf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03psf is called. Arguments denoted as Input must not be changed by this procedure.
Note:monitf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03psf. If your code inadvertently does return any NaNs or infinities, d03psf is likely to produce unexpected results.
32: $\mathbf{rsave}\left({\mathbf{lrsave}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
If ${\mathbf{ind}}=0$, rsave need not be set on entry.
If ${\mathbf{ind}}=1$, rsave must be unchanged from the previous call to the routine because it contains required information about the iteration.
33: $\mathbf{lrsave}$ – IntegerInput
On entry: the dimension of the array rsave as declared in the (sub)program from which d03psf is called.
Its size depends on the type of matrix algebra selected.
If ${\mathbf{laopt}}=\text{'F'}$, ${\mathbf{lrsave}}\ge {\mathbf{neqn}}\times {\mathbf{neqn}}+{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\text{'B'}$, ${\mathbf{lrsave}}\ge (3\mathit{mlu}+1)\times {\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\text{'S'}$, ${\mathbf{lrsave}}\ge 4{\mathbf{neqn}}+11{\mathbf{neqn}}/2+1+\mathit{nwkres}+\mathit{lenode}$.
Where $\mathit{mlu}$ is the lower or upper half bandwidths such that
Note: when ${\mathbf{laopt}}=\text{'S'}$, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the routine returns with ${\mathbf{ifail}}={\mathbf{15}}$.
If ${\mathbf{ind}}=1$, isave must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular the following components of the array isave concern the efficiency of the integration:
${\mathbf{isave}}\left(1\right)$
Contains the number of steps taken in time.
${\mathbf{isave}}\left(2\right)$
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves evaluating the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
${\mathbf{isave}}\left(3\right)$
Contains the number of Jacobian evaluations performed by the time integrator.
${\mathbf{isave}}\left(4\right)$
Contains the order of the BDF method last used in the time integration, if applicable. When the Theta method is used, ${\mathbf{isave}}\left(4\right)$ contains no useful information.
${\mathbf{isave}}\left(5\right)$
Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.
35: $\mathbf{lisave}$ – IntegerInput
On entry: the dimension of the array isave as declared in the (sub)program from which d03psf is called. Its size depends on the type of matrix algebra selected:
if ${\mathbf{laopt}}=\text{'F'}$, ${\mathbf{lisave}}\ge 25$;
if ${\mathbf{laopt}}=\text{'B'}$, ${\mathbf{lisave}}\ge {\mathbf{neqn}}+{\mathbf{nxfix}}+25$;
if ${\mathbf{laopt}}=\text{'S'}$, ${\mathbf{lisave}}\ge 25\times {\mathbf{neqn}}+{\mathbf{nxfix}}+25$.
Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the routine returns with ${\mathbf{ifail}}={\mathbf{15}}$.
36: $\mathbf{itask}$ – IntegerInput
On entry: the task to be performed by the ODE integrator.
${\mathbf{itask}}=1$
Normal computation of output values u at $t={\mathbf{tout}}$ (by overshooting and interpolating).
${\mathbf{itask}}=2$
Take one step in the time direction and return.
${\mathbf{itask}}=3$
Stop at first internal integration point at or beyond $t={\mathbf{tout}}$.
${\mathbf{itask}}=4$
Normal computation of output values u at $t={\mathbf{tout}}$ but without overshooting $t={t}_{\mathrm{crit}}$ where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
${\mathbf{itask}}=5$
Take one step in the time direction and return, without passing ${t}_{\mathrm{crit}}$, where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
Constraint:
${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
37: $\mathbf{itrace}$ – IntegerInput
On entry: the level of trace information required from d03psf and the underlying ODE solver. itrace may take the value $\mathrm{-1}$, $0$, $1$, $2$ or $3$.
${\mathbf{itrace}}=\mathrm{-1}$
No output is generated.
${\mathbf{itrace}}=0$
Only warning messages from the PDE solver are printed on the current error message unit (see x04aaf).
${\mathbf{itrace}}>0$
Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
If ${\mathbf{itrace}}<\mathrm{-1}$, $\mathrm{-1}$ is assumed and similarly if ${\mathbf{itrace}}>3$, $3$ is assumed.
The advisory messages are given in greater detail as itrace increases. You are advised to set ${\mathbf{itrace}}=0$, unless you are experienced with Sub-chapter D02M–N.
38: $\mathbf{ind}$ – IntegerInput/Output
On entry: indicates whether this is a continuation call or a new integration.
${\mathbf{ind}}=0$
Starts or restarts the integration in time.
${\mathbf{ind}}=1$
Continues the integration after an earlier exit from the routine. In this case, only the arguments tout, ifail, nrmesh and trmesh may be reset between calls to d03psf.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
39: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one point in xi lies outside $[{\mathbf{x}}\left(1\right),{\mathbf{x}}\left({\mathbf{npts}}\right)]$: ${\mathbf{x}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left({\mathbf{npts}}\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{con}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{con}}\le 10.0/({\mathbf{npts}}-1)$.
On entry, ${\mathbf{con}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{con}}\ge 0.1/({\mathbf{npts}}-1)$.
On entry, ${\mathbf{dxmesh}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{dxmesh}}\ge 0.0$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{xfix}}\left(\mathit{i}+1\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{xfix}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xfix}}\left(\mathit{i}+1\right)>{\mathbf{xfix}}\left(\mathit{i}\right)$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{j}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left(\mathit{j}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{npts}}\right)$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{xi}}\left(\mathit{i}+1\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{xi}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xi}}\left(\mathit{i}+1\right)>{\mathbf{xi}}\left(\mathit{i}\right)$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{atol}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{atol}}\left(\mathit{i}\right)\ge 0.0$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{j}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: corresponding elements ${\mathbf{atol}}\left(\mathit{i}\right)$ and ${\mathbf{rtol}}\left(\mathit{j}\right)$ cannot both be $0.0$.
On entry, $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{rtol}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{rtol}}\left(\mathit{i}\right)\ge 0.0$.
On entry, ${\mathbf{ind}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{ipminf}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ipminf}}=0$, $1$ or $2$.
On entry, ${\mathbf{itask}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
On entry, ${\mathbf{itol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{itol}}=1$, $2$, $3$ or $4$.
On entry, ${\mathbf{laopt}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{laopt}}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.
On entry, ${\mathbf{lisave}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lisave}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{lrsave}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lrsave}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{neqn}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{npde}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nv}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}\times {\mathbf{npts}}+{\mathbf{nv}}$.
On entry, ${\mathbf{norm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{norm}}=\text{'1'}$ or $\text{'2'}$.
On entry, ${\mathbf{npde}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{npts}}\ge 3$.
On entry, ${\mathbf{nv}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nv}}\ge 0$.
On entry, ${\mathbf{nv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nxi}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nxi}}=0$ when ${\mathbf{nv}}=0$.
On entry, ${\mathbf{nv}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nxi}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nxi}}\ge 0$ when ${\mathbf{nv}}>0$.
On entry, ${\mathbf{nxfix}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{npts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nxfix}}\le {\mathbf{npts}}-2$.
On entry, ${\mathbf{nxfix}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nxfix}}\ge 0$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
On entry, the point ${\mathbf{xfix}}\left(\mathit{i}\right)$ does not coincide with any ${\mathbf{x}}\left(\mathit{j}\right)$: $\mathit{i}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{xfix}}\left(\mathit{i}\right)=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{tout}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}-{\mathbf{ts}}$ is too small:
${\mathbf{tout}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{xratio}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{xratio}}>1.0$.
${\mathbf{ifail}}=2$
Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol.
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=3$
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts:
${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t={\mathbf{ts}}$. The problem may have a singularity, or the error requirement may be inappropriate. Incorrect specification of boundary conditions may also result in this error.
${\mathbf{ifail}}=4$
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting ${\mathbf{ires}}=3$ in pdedef, numflx, bndary or odedef.
${\mathbf{ifail}}=5$
Singular Jacobian of ODE system. Check problem formulation.
${\mathbf{ifail}}=6$
In evaluating residual of ODE system, ${\mathbf{ires}}=2$ has been set in user-supplied subroutines pdedef, numflx, bndary or odedef. Integration is successful as far as ts: ${\mathbf{ts}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=7$
atol and rtol were too small to start integration.
Serious error in internal call to an auxiliary. Increase itrace for further details.
${\mathbf{ifail}}=10$
Integration completed, but small changes in atol or rtol are unlikely to result in a changed solution.
The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when ${\mathbf{itask}}\ne 2$ or $5$.)
${\mathbf{ifail}}=11$
Error during Jacobian formulation for ODE system. Increase itrace for further details.
${\mathbf{ifail}}=12$
In solving ODE system, the maximum number of steps ${\mathbf{algopt}}\left(15\right)$ has been exceeded. ${\mathbf{algopt}}\left(15\right)=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=13$
Zero error weights encountered during time integration.
Some error weights ${w}_{i}$ became zero during the time integration (see the description of itol). Pure relative error control (${\mathbf{atol}}\left(i\right)=0.0$) was requested on a variable (the $i$th) which has become zero. The integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=14$
The functions $P$, $D$, or $C$ appear to depend on time derivatives.
${\mathbf{ifail}}=15$
When using the sparse option lisave or lrsave is too small:
${\mathbf{lisave}}=\u27e8\mathit{\text{value}}\u27e9$,
${\mathbf{lrsave}}=\u27e8\mathit{\text{value}}\u27e9$.
fmon is negative at one or more mesh points, or zero mesh spacing has been obtained due to a poor monitor function.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
d03psf controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should, therefore, test the effect of varying the accuracy arguments, atol and rtol.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d03psf is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.
d03psf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03psf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
d03psf is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional second-order diffusion terms. The use of the routine to solve systems which are not naturally in this form is discouraged, and you are advised to use one of the central-difference scheme routines for such problems.
You should be aware of the stability limitations for hyperbolic PDEs. For most problems with small error tolerances the ODE integrator does not attempt unstable time steps, but in some cases a maximum time step should be imposed using ${\mathbf{algopt}}\left(13\right)$. It is worth experimenting with this argument, particularly if the integration appears to progress unrealistically fast (with large time steps). Setting the maximum time step to the minimum mesh size is a safe measure, although in some cases this may be too restrictive.
Problems with source terms should be treated with caution, as it is known that for large source terms stable and reasonable looking solutions can be obtained which are in fact incorrect, exhibiting non-physical speeds of propagation of discontinuities (typically one spatial mesh point per time step). It is essential to employ a very fine mesh for problems with source terms and discontinuities, and to check for non-physical propagation speeds by comparing results for different mesh sizes. Further details and an example can be found in Pennington and Berzins (1994).
The time taken depends on the complexity of the system, the accuracy requested, and the frequency of the mesh updates. For a given system with fixed accuracy and mesh-update frequency it is approximately proportional to neqn.
10Example
For this routine two examples are presented, with a main program and two example problems given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
This example is a simple model of the advection and diffusion of a cloud of material:
for $x\in [0,1]$ and $t\le 0\le 0.3$. In this example the constant wind speed $W=1$ and the diffusion coefficient $C=0.002$.
The cloud does not reach the boundaries during the time of integration, and so the two (physical) boundary conditions are simply $U(0,t)=U(1,t)=0.0$, and the initial condition is
for $0\le x\le 1$ and $t\ge 0$. The discontinuity is modelled by a ramp function of width $0.01$ and gradient $100$, so that the exact solution at any time $t\ge 0$ is
where $\delta =100(0.1-x+t)$. The initial profile is given by the exact solution. The characteristic points into the domain at $x=0$ and out of the domain at $x=1$, and so a physical boundary condition $u(0,t)=1$ is imposed at $x=0$, with a numerical boundary condition at $x=1$ which can be specified as $u(1,t)=0$ since the discontinuity does not reach $x=1$ during the time of integration.
The numerical flux is simply $\hat{F}={U}_{L}$ at all times.
The remeshing monitor function (described below) is chosen to create an increasingly fine mesh towards the discontinuity in order to ensure good resolution of the discontinuity, but without loss of efficiency in the surrounding regions. However, refinement must be limited so that the time step required for stability does not become unrealistically small. The region of refinement must also keep up with the discontinuity as it moves across the domain, and hence it cannot be so small that the discontinuity moves out of the refined region between remeshing.
The above requirements mean that the use of the first or second spatial derivative of $U$ for the monitor function is inappropriate; the large relative size of either derivative in the region of the discontinuity leads to extremely small mesh-spacing in a very limited region, and the solution is then far more expensive than for a very fine fixed mesh.
An alternative monitor function based on a cosine function proves very successful. It is only semi-automatic as it requires some knowledge of the solution (for problems without an exact solution an initial approximate solution can be obtained using a coarse fixed mesh). On each call to monitf the discontinuity is located by finding the maximum spatial derivative of the solution. On the first call the desired width of the region of nonzero monitor function is set (this can be changed at a later time if desired). Then on each call the monitor function is assigned using a cosine function so that it has a value of one at the discontinuity down to zero at the edges of the predetermined region of refinement, and zero outside the region. Thus the monitor function and the subsequent refinement are limited, and the region is large enough to ensure that there is always sufficient refinement at the discontinuity.