naginterfaces.library.pde.dim1_​parab_​remesh_​keller

naginterfaces.library.pde.dim1_parab_remesh_keller(npde, ts, tout, pdedef, bndary, uvinit, u, x, nleft, nv, xi, rtol, atol, itol, norm, laopt, algopt, remesh, xfix, nrmesh, dxmesh, trmesh, ipminf, comm, itask, itrace, ind, odedef=None, xratio=1.5, con=None, monitf=None, lrsave_estim=0, lisave_estim=0, data=None, io_manager=None, spiked_sorder='C')

dim1_parab_remesh_keller integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs), and automatic adaptive spatial remeshing. The spatial discretization is performed using the Keller box scheme (see Keller (1970)) and the method of lines is employed to reduce the PDEs to a system of ODEs. The resulting system is solved using a Backward Differentiation Formula (BDF) method or a Theta method (switching between Newton’s method and functional iteration).

For full information please refer to the NAG Library document for d03pr

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d03/d03prf.html

Parameters

npdeint

The number of PDEs to be solved.

tsfloat

The initial value of the independent variable $t$.

toutfloat

The final value of $t$ to which the integration is to be carried out.

pdedefcallable (res, ires) = pdedef(t, x, u, ut, ux, v, vdot, ires, data=None)

$pdedef$ must evaluate the functions $G_i$ which define the system of PDEs. $pdedef$ is called approximately midway between each pair of mesh points in turn by dim1_parab_remesh_keller.

Parameters

tfloat

The current value of the independent variable $t$.

xfloat

The current value of the space variable $x$.

ufloat, ndarray, shape $\left(\text{npde}\right)$

$u[\text{i}-1]$ contains the value of the component $U_{\text{i}}(x,t)$, for $\text{i}=1,2,\dots ,npde$.

utfloat, ndarray, shape $\left(\text{npde}\right)$

$ut[\text{i}-1]$ contains the value of the component $\frac{\partial U_{\text{i}}(x,t)}{\partial t}$, for $\text{i}=1,2,\dots ,npde$.

uxfloat, ndarray, shape $\left(\text{npde}\right)$

$ux[\text{i}-1]$ contains the value of the component $\frac{\partial U_{\text{i}}(x,t)}{\partial x}$, for $\text{i}=1,2,\dots ,npde$.

vfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $v[\text{i}-1]$ contains the value of the component $V_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

vdotfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $vdot[\text{i}-1]$ contains the value of component ${\dot{V}}_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

iresint

The form of $G_i$ that must be returned in the array $res$.

$ires=-1$

Equation (9) must be used.

$ires=1$

Equation (0) must be used.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

resfloat, array-like, shape $\left(\text{npde}\right)$

$res[\text{i}-1]$ must contain the $\text{i}$th component of $G$, for $\text{i}=1,2,\dots ,npde$, where $G$ is defined as

$$G_{\text{i}}=\sum _{\text{j}=1}^{npde}{P}_{\text{i},\text{j}}\frac{\partial U_{\text{j}}}{\partial t}+\sum _{\text{j}=1}^{nv}{Q}_{\text{i},\text{j}}{\dot{V}}_{\text{j}}\text{,}$$

i.e., only terms depending explicitly on time derivatives, or

$$G_{\text{i}}=\sum _{\text{j}=1}^{npde}{P}_{\text{i},\text{j}}\frac{\partial U_{\text{j}}}{\partial t}+\sum _{\text{j}=1}^{nv}{Q}_{\text{i},\text{j}}{\dot{V}}_{\text{j}}+S_{\text{i}}\text{,}$$

i.e., all terms in equation (3).

The definition of $G$ is determined by the input value of $ires$.

iresint

Should usually remain unchanged. However, you may set $ires$ to force the integration function to take certain actions, as described below:

$ires=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $errno$ = 6.

$ires=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires=3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires=3$, dim1_parab_remesh_keller returns to the calling function with the error indicator set to $errno$ = 4.

bndarycallable (res, ires) = bndary(t, ibnd, nobc, u, ut, v, vdot, ires, data=None)

$bndary$ must evaluate the functions $G_i^L$ and $G_i^R$ which describe the boundary conditions, as given in (5) and (6).

Parameters

tfloat

The current value of the independent variable $t$.

ibndint

Specifies which boundary conditions are to be evaluated.

$ibnd=0$

$bndary$ must compute the left-hand boundary condition at $x=a$.

$ibnd\ne 0$

$bndary$ must compute of the right-hand boundary condition at $x=b$.

nobcint

Specifies the number $n_a$ of boundary conditions at the boundary specified by $ibnd$.

ufloat, ndarray, shape $\left(\text{npde}\right)$

$u[\text{i}-1]$ contains the value of the component $U_{\text{i}}(x,t)$ at the boundary specified by $ibnd$, for $\text{i}=1,2,\dots ,npde$.

utfloat, ndarray, shape $\left(\text{npde}\right)$

$ut[\text{i}-1]$ contains the value of the component $\frac{\partial U_{\text{i}}(x,t)}{\partial t}$, for $\text{i}=1,2,\dots ,npde$.

vfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $v[\text{i}-1]$ contains the value of the component $V_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

vdotfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $vdot[\text{i}-1]$ contains the value of component ${\dot{V}}_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

Note: $vdot[\text{i}-1]$, for $\text{i}=1,2,\dots ,nv$, may only appear linearly as in (1) and (2).

iresint

The form of $G_i^L$ (or $G_i^R$) that must be returned in the array $res$.

$ires=-1$

Equation (1) must be used.

$ires=1$

Equation (2) must be used.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

resfloat, array-like, shape $\left(nobc\right)$

$res[\text{i}-1]$ must contain the $\text{i}$th component of $G^L$ or $G^R$, depending on the value of $ibnd$, for $\text{i}=1,2,\dots ,nobc$, where $G^L$ is defined as

$$G_{\text{i}}^L=\sum _{\text{j}=1}^{npde}{E}_{\text{i},\text{j}}^L\frac{\partial U_{\text{j}}}{\partial t}+\sum _{\text{j}=1}^{nv}{H}_{\text{i},\text{j}}^L{\dot{V}}_{\text{j}}\text{,}$$

i.e., only terms depending explicitly on time derivatives, or

$$G_{\text{i}}^L=\sum _{\text{j}=1}^{npde}{E}_{\text{i},\text{j}}^L\frac{\partial U_{\text{j}}}{\partial t}+\sum _{\text{j}=1}^{nv}{H}_{\text{i},\text{j}}^L{\dot{V}}_{\text{j}}+K_{\text{i}}^L\text{,}$$

i.e., all terms in equation (7), and similarly for $G_{\text{i}}^R$.

The definitions of $G^L$ and $G^R$ are determined by the input value of $ires$.

iresint

Should usually remain unchanged. However, you may set $ires$ to force the integration function to take certain actions as described below:

$ires=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $errno$ = 6.

$ires=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires=3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires=3$, dim1_parab_remesh_keller returns to the calling function with the error indicator set to $errno$ = 4.

uvinitcallable (u, v) = uvinit(npde, x, xi, nv, data=None)

$uvinit$ 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 $[a,b]$.

Parameters

npdeint

The number of PDEs in the system.

xfloat, ndarray, shape $\left(\text{npts}\right)$

The current mesh. $x[\text{i}-1]$ contains the value of $x_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{npts}$.

xifloat, ndarray, shape $\left(\text{nxi}\right)$

If $\text{nxi}>0$, $xi[\text{i}-1]$ contains the ODE/PDE coupling point, ${\xi }_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{nxi}$.

nvint

The number of coupled ODEs in the system.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

ufloat, array-like, shape $(npde,\text{npts})$

$u[\text{i}-1,\text{j}-1]$ contains the value of the component $U_{\text{i}}(x_{\text{j}},t_0)$, for $\text{j}=1,2,\dots ,\text{npts}$, for $\text{i}=1,2,\dots ,npde$.

vfloat, array-like, shape $\left(nv\right)$

If $nv>0$, $v[\text{i}-1]$ must contain the value of component $V_{\text{i}}\left(t_0\right)$, for $\text{i}=1,2,\dots ,nv$.

ufloat, array-like, shape $\left(\text{neqn}\right)$

If $ind=1$ the value of $u$ must be unchanged from the previous call.

xfloat, array-like, shape $\left(\text{npts}\right)$

The initial mesh points in the space direction. $x\left[0\right]$ must specify the left-hand boundary, $a$, and $x[\text{npts}-1]$ must specify the right-hand boundary, $b$.

nleftint

The number $n_a$ of boundary conditions at the left-hand mesh point $x\left[0\right]$.

nvint

The number of coupled ODE components.

xifloat, array-like, shape $\left(\text{nxi}\right)$

$xi[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{nxi}$, must be set to the ODE/PDE coupling points, ${\xi }_{\text{i}}$.

rtolfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $itol\text{in}(1,2)$: $1$; if $itol\text{in}(3,4)$: $\text{neqn}$; otherwise: $0$.

The relative local error tolerance.

atolfloat, array-like, shape $(:)$

Note: the required length for this argument is determined as follows: if $itol\text{in}(1,3)$: $1$; if $itol\text{in}(2,4)$: $\text{neqn}$; otherwise: $0$.

The absolute local error tolerance.

itolint

A value to indicate the form of the local error test. $itol$ indicates to dim1_parab_remesh_keller whether to interpret either or both of $rtol$ or $atol$ as a vector or scalar.

normstr, length 1

The type of norm to be used.

$norm=\text{'M'}$

Maximum norm.

$norm=\text{'A'}$

Averaged $L_2$ norm.

If $U_{norm}$ denotes the norm of the vector $u$ of length $\text{neqn}$, then for the averaged $L_2$ norm

$$U_{norm}=\sqrt{\frac{1}{\text{neqn}}\sum _{i=1}^{\text{neqn}}{\left(U\left(i\right)/w_i\right)}^2}\text{,}$$

while for the maximum norm

$$U_{norm}={max}_i\left|u\right[i-1\left]/w_i\right|\text{.}$$

See the description of $itol$ for the formulation of the weight vector $w$.

laoptstr, length 1

The type of matrix algebra required.

$laopt=\text{'F'}$

Full matrix methods to be used.

$laopt=\text{'B'}$

Banded matrix methods to be used.

$laopt=\text{'S'}$

Sparse matrix methods to be used.

algoptfloat, array-like, shape $\left(30\right)$

May be set to control various options available in the integrator. If you wish to employ all the default options, $algopt\left[0\right]$ 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:

$algopt\left[0\right]$

Selects the ODE integration method to be used. If $algopt\left[0\right]=1.0$, a BDF method is used and if $algopt\left[0\right]=2.0$, a Theta method is used. The default value is $algopt\left[0\right]=1.0$.

If $algopt\left[0\right]=2.0$, then $algopt[\text{i}-1]$, for $\text{i}=2,3,\dots ,4$, are not used.

$algopt\left[1\right]$

Specifies the maximum order of the BDF integration formula to be used. $algopt\left[1\right]$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is $algopt\left[1\right]=5.0$.

$algopt\left[2\right]$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $algopt\left[2\right]=1.0$ a modified Newton iteration is used and if $algopt\left[2\right]=2.0$ 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 $algopt\left[2\right]=1.0$.

$algopt\left[3\right]$

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 $P_{i,\text{j}}=0.0$, for $\text{j}=1,2,\dots ,npde$, for some $i$ or when there is no ${\dot{V}}_i\left(t\right)$ dependence in the coupled ODE system. If $algopt\left[3\right]=1.0$, the Petzold test is used. If $algopt\left[3\right]=2.0$, the Petzold test is not used. The default value is $algopt\left[3\right]=1.0$.

If $algopt\left[0\right]=1.0$, $algopt[\text{i}-1]$, for $\text{i}=5,6,\dots ,7$, are not used.

$algopt\left[4\right]$

Specifies the value of Theta to be used in the Theta integration method. $0.51\le algopt\left[4\right]\le 0.99$. The default value is $algopt\left[4\right]=0.55$.

$algopt\left[5\right]$

Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $algopt\left[5\right]=1.0$, a modified Newton iteration is used and if $algopt\left[5\right]=2.0$, a functional iteration method is used. The default value is $algopt\left[5\right]=1.0$.

$algopt\left[6\right]$

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 $algopt\left[6\right]=1.0$, switching is allowed and if $algopt\left[6\right]=2.0$, switching is not allowed. The default value is $algopt\left[6\right]=1.0$.

$algopt\left[10\right]$

Specifies a point in the time direction, $t_{crit}$, beyond which integration must not be attempted. The use of $t_{crit}$ is described under the argument $itask$. If $algopt\left[0\right]\ne 0.0$, a value of $0.0$, for $algopt\left[10\right]$, say, should be specified even if $itask$ subsequently specifies that $t_{crit}$ will not be used.

$algopt\left[11\right]$

Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, $algopt\left[11\right]$ should be set to $0.0$.

$algopt\left[12\right]$

Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, $algopt\left[12\right]$ should be set to $0.0$.

$algopt\left[13\right]$

Specifies the initial step size to be attempted by the integrator. If $algopt\left[13\right]=0.0$, the initial step size is calculated internally.

$algopt\left[14\right]$

Specifies the maximum number of steps to be attempted by the integrator in any one call. If $algopt\left[14\right]=0.0$, no limit is imposed.

$algopt\left[22\right]$

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 $\dot{V}$. If $algopt\left[22\right]=1.0$, a modified Newton iteration is used and if $algopt\left[22\right]=2.0$, functional iteration is used. The default value is $algopt\left[22\right]=1.0$.

$algopt\left[28\right]$ and $algopt\left[29\right]$ are used only for the sparse matrix algebra option, i.e., $laopt=\text{'S'}$.

$algopt\left[28\right]$

Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<algopt\left[28\right]<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $algopt\left[28\right]$ lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing $algopt\left[28\right]$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $algopt\left[28\right]=0.1$.

$algopt\left[29\right]$

Used as a relative pivot threshold during subsequent Jacobian decompositions (see $algopt\left[28\right]$) below which an internal error is invoked. $algopt\left[29\right]$ must be greater than zero, otherwise the default value is used. If $algopt\left[29\right]$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see $algopt\left[28\right]$). The default value is $algopt\left[29\right]=0.0001$.

remeshbool

Indicates whether or not spatial remeshing should be performed.

$remesh=True$

Indicates that spatial remeshing should be performed as specified.

$remesh=False$

Indicates that spatial remeshing should be suppressed.

Note: $remesh$ should not be changed between consecutive calls to dim1_parab_remesh_keller. Remeshing can be switched off or on at specified times by using appropriate values for the arguments $nrmesh$ and $trmesh$ at each call.

xfixfloat, array-like, shape $\left(\text{nxfix}\right)$

$xfix[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{nxfix}$, must contain the value of the $x$ coordinate at the $\text{i}$th fixed mesh point.

nrmeshint

Indicates the form of meshing to be performed.

$nrmesh<0$

Indicates that a new mesh is adopted according to the argument $dxmesh$. The mesh is tested every $\left|nrmesh\right|$ timesteps.

$nrmesh=0$

Indicates that remeshing should take place just once at the end of the first time step reached when $t>trmesh$.

$nrmesh>0$

Indicates that remeshing will take place every $nrmesh$ time steps, with no testing using $dxmesh$.

Note: $nrmesh$ may be changed between consecutive calls to dim1_parab_remesh_keller to give greater flexibility over the times of remeshing.

dxmeshfloat

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 $\left|nrmesh\right|$ time steps, but is adopted only if

$$x_i^{new}>x_i^{old}+dxmesh\times (x_{i+1}^{old}-x_i^{old})\text{,}$$

or

$$x_i^{new}<x_i^{old}-dxmesh\times (x_i^{old}-x_{i-1}^{old})\text{.}$$

$dxmesh$ thus imposes a lower limit on the difference between one mesh and the next.

trmeshfloat

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 dim1_parab_remesh_keller to force remeshing at several specified times.

ipminfint

The level of trace information regarding the adaptive remeshing. Details are directed to the file object associated with the advisory I/O unit (see FileObjManager).

$ipminf=0$

No trace information.

$ipminf=1$

Brief summary of mesh characteristics.

$ipminf=2$

More detailed information, including old and new mesh points, mesh sizes and monitor function values.

commdict, communication object, modified in place

Note: this argument will be (re-)initialized when it is an empty dict or under the following condition: $ind=0$.

Communication structure.

On initial entry: need not be set.

itaskint

The task to be performed by the ODE integrator.

$itask=1$

Normal computation of output values $u$ at $t=tout$ (by overshooting and interpolating).

$itask=2$

Take one step in the time direction and return.

$itask=3$

Stop at first internal integration point at or beyond $t=tout$.

$itask=4$

Normal computation of output values $u$ at $t=tout$ but without overshooting $t=t_{crit}$ where $t_{crit}$ is described under the argument $algopt$.

$itask=5$

Take one step in the time direction and return, without passing $t_{crit}$, where $t_{crit}$ is described under the argument $algopt$.

itraceint

The level of trace information required from dim1_parab_remesh_keller and the underlying ODE solver as follows:

$itrace\le -1$

No output is generated.

$itrace=0$

Only warning messages from the PDE solver are printed.

$itrace=1$

Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

$itrace=2$

Output from the underlying ODE solver is similar to that produced when $itrace=1$, except that the advisory messages are given in greater detail.

$itrace\ge 3$

The output from the underlying ODE solver is similar to that produced when $itrace=2$, except that the advisory messages are given in greater detail.

You are advised to set $itrace=0$, unless you are experienced with submodule ode.

indint

Indicates whether this is a continuation call or a new integration.

$ind=0$

Starts or restarts the integration in time.

$ind=1$

Continues the integration after an earlier exit from the function. In this case, only the argument $tout$ and the remeshing arguments $nrmesh$, $dxmesh$, $trmesh$, $xratio$ and $con$ may be reset between calls to dim1_parab_remesh_keller.

odedefNone or callable (r, ires) = odedef(t, v, vdot, xi, ucp, ucpx, ucpt, ires, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$odedef$ 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., $nv=0$), $odedef$ must be None.

Parameters

tfloat

The current value of the independent variable $t$.

vfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $v[\text{i}-1]$ contains the value of the component $V_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

vdotfloat, ndarray, shape $\left(\text{nv}\right)$

If $nv>0$, $vdot[\text{i}-1]$ contains the value of component ${\dot{V}}_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

xifloat, ndarray, shape $\left(\text{nxi}\right)$

If $\text{nxi}>0$, $xi[\text{i}-1]$ contains the ODE/PDE coupling point, ${\xi }_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{nxi}$.

ucpfloat, ndarray, shape $(\text{npde},\text{nxi})$

If $\text{nxi}>0$, $ucp[\text{i}-1,\text{j}-1]$ contains the value of $U_{\text{i}}(x,t)$ at the coupling point $x={\xi }_{\text{j}}$, for $\text{j}=1,2,\dots ,\text{nxi}$, for $\text{i}=1,2,\dots ,npde$.

ucpxfloat, ndarray, shape $(\text{npde},\text{nxi})$

If $\text{nxi}>0$, $ucpx[\text{i}-1,\text{j}-1]$ contains the value of $\frac{\partial U_{\text{i}}(x,t)}{\partial x}$ at the coupling point $x={\xi }_{\text{j}}$, for $\text{j}=1,2,\dots ,\text{nxi}$, for $\text{i}=1,2,\dots ,npde$.

ucptfloat, ndarray, shape $(\text{npde},\text{nxi})$

If $\text{nxi}>0$, $ucpt[\text{i}-1,\text{j}-1]$ contains the value of $\frac{\partial U_{\text{i}}}{\partial t}$ at the coupling point $x={\xi }_{\text{j}}$, for $\text{j}=1,2,\dots ,\text{nxi}$, for $\text{i}=1,2,\dots ,npde$.

iresint

The form of $R$ that must be returned in the array $r$.

$ires=-1$

Equation (3) must be used.

$ires=1$

Equation (4) must be used.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

rfloat, array-like, shape $\left(\text{nv}\right)$

If $nv>0$, $r[\text{i}-1]$ must contain the $\text{i}$th component of $R$, for $\text{i}=1,2,\dots ,nv$, where $R$ is defined as

$$R=-B\dot{V}-C{U}_t^{\ast }\text{,}$$

i.e., only terms depending explicitly on time derivatives, or

$$R=A-B\dot{V}-C{U}_t^{\ast }\text{,}$$

i.e., all terms in equation (4). The definition of $R$ is determined by the input value of $ires$.

iresint

Should usually remain unchanged. However, you may reset $ires$ to force the integration function to take certain actions, as described below:

$ires=2$

Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $errno$ = 6.

$ires=3$

Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires=3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires=3$, dim1_parab_remesh_keller returns to the calling function with the error indicator set to $errno$ = 4.

xratiofloat, optional

An input bound on the adjacent mesh ratio (greater than $1.0$ and typically in the range $1.5$ to $3.0$). The remeshing functions will attempt to ensure that

$$(x_i-x_{i-1})/xratio<x_{i+1}-x_i<xratio\times (x_i-x_{i-1})\text{.}$$

conNone or float, optional

Note: if this argument is None then a default value will be used, determined as follows: $2.0/(\text{npts}-1)$.

An input bound on the sub-integral of the monitor function $F^{mon}\left(x\right)$ over each space step. The remeshing functions will attempt to ensure that

$${\int }_{x_1}^{x_{i+1}}{F}^{mon}\left(x\right)dx\le con{\int }_{x_1}^{x_{\text{npts}}}{F}^{mon}\left(x\right)dx\text{,}$$

(see Furzeland (1984)). $con$ gives you more control over the mesh distribution e.g., decreasing $con$ allows more clustering. A typical value is $2/(\text{npts}-1)$, 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.

monitfNone or callable fmon = monitf(t, x, u, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$monitf$ must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.

If you specify $remesh=False$, i.e., no remeshing, $monitf$ will not be called and may be None.

Parameters

tfloat

The current value of the independent variable $t$.

xfloat, ndarray, shape $\left(\text{npts}\right)$

The current mesh. $x[\text{i}-1]$ contains the value of $x_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{npts}$.

ufloat, ndarray, shape $(\text{npde},\text{npts})$

$u[\text{i}-1,\text{j}-1]$ contains the value of $U_{\text{i}}(x,t)$ at $x=x[\text{j}-1]$ and time $t$, for $\text{j}=1,2,\dots ,\text{npts}$, for $\text{i}=1,2,\dots ,npde$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

fmonfloat, array-like, shape $\left(\text{npts}\right)$

$fmon[i-1]$ must contain the value of the monitor function $F^{mon}\left(x\right)$ at mesh point $x=x[i-1]$.

lrsave_estimint, optional

When performing a new integration, the size to use for the communication array $comm$[‘rsave’].

Otherwise, the value has no effect.

An initial estimate for an adequate $lrsave\_estim$ is computed by the function.

If your supplied $lrsave\_estim$ is too small, the estimated value will be used instead.

In some cases the estimated value will be sufficient for continuation calls to the function.

When $laopt=\text{'S'}$, even the function’s initial estimated value of $lrsave\_estim$ may be too small.

If so, the function returns with $errno$ = 15.

You are advised to call the function again with $ind=0$ and $lrsave\_estim$ set to at least the lower-bound value returned in $lrsave\_min$, then make the desired subsequent calls with $ind=1$, then repeat the process if necessary.

lisave_estimint, optional

When performing a new integration, the size to use for the communication array $comm$[‘isave’].

Otherwise, the value has no effect.

An initial estimate for an adequate $lisave\_estim$ is computed by the function.

If your supplied $lisave\_estim$ is too small, the estimated value will be used instead.

In some cases the estimated value will be sufficient for continuation calls to the function.

When $laopt=\text{'S'}$, even the function’s initial estimated value of $lisave\_estim$ may be too small.

If so, the function returns with $errno$ = 15.

You are advised to call the function again with $ind=0$ and $lisave\_estim$ set to at least the lower-bound value returned in $lisave\_min$, then make the desired subsequent calls with $ind=1$, then repeat the process if necessary.

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

spiked_sorderstr, optional

If $u$ in $uvinit$ is spiked (i.e., has unit extent in all but one dimension, or has size $1$), $spiked\_sorder$ selects the storage order to associate with it in the NAG Engine:

spiked_sorder = $\text{'C'}$

row-major storage will be used;

spiked_sorder = $\text{'F'}$

column-major storage will be used.

Returns

tsfloat

The value of $t$ corresponding to the solution values in $u$. Normally $ts=tout$.

ufloat, ndarray, shape $\left(\text{neqn}\right)$

The computed solution $U_{\text{i}}(x_{\text{j}},t)$, for $\text{j}=1,2,\dots ,\text{npts}$, for $\text{i}=1,2,\dots ,npde$, and $V_{\text{k}}\left(t\right)$, for $\text{k}=1,2,\dots ,nv$, evaluated at $t=ts$, as follows:

$u[npde\times (\text{j}-1)+\text{i}-1]$ contain $U_{\text{i}}(x_{\text{j}},t)$, for $\text{j}=1,2,\dots ,\text{npts}$, for $\text{i}=1,2,\dots ,npde$, and

$u[\text{npts}\times npde+\text{i}-1]$ contain $V_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$.

xfloat, ndarray, shape $\left(\text{npts}\right)$

The final values of the mesh points.

indint

$ind=1$.

lrsave_minint

A lower bound on the sufficient size for $comm$[‘rsave’].

lisave_minint

A lower bound on the sufficient size for $comm$[‘isave’].

Raises

NagValueError

(errno $1$)

On entry, on initial entry $ind=1$.

Constraint: on initial entry $ind=0$.

(errno $1$)

On entry, the point $xfix[\text{i}-1]$ does not coincide with any $x[\text{j}-1]$: $\text{i}=⟨value⟩$ and $xfix[\text{i}-1]=⟨value⟩$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$, $xfix\left[\text{i}\right]=⟨value⟩$ and $xfix[\text{i}-1]=⟨value⟩$.

Constraint: $xfix\left[\text{i}\right]>xfix[\text{i}-1]$.

(errno $1$)

On entry, at least one point in $xi$ lies outside $[x\left[0\right],x[\text{npts}-1]]$: $x\left[0\right]=⟨value⟩$ and $x[\text{npts}-1]=⟨value⟩$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$, $xi\left[\text{i}\right]=⟨value⟩$ and $xi[\text{i}-1]=⟨value⟩$.

Constraint: $xi\left[\text{i}\right]>xi[\text{i}-1]$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$ and $\text{j}=⟨value⟩$.

Constraint: corresponding elements $atol[\text{i}-1]$ and $rtol[\text{j}-1]$ cannot both be $0.0$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$ and $rtol[\text{i}-1]=⟨value⟩$.

Constraint: $rtol[\text{i}-1]\ge 0.0$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$ and $atol[\text{i}-1]=⟨value⟩$.

Constraint: $atol[\text{i}-1]\ge 0.0$.

(errno $1$)

On entry, $itol=⟨value⟩$.

Constraint: $itol=1$, $2$, $3$ or $4$.

(errno $1$)

On entry, $\text{i}=⟨value⟩$, $x[\text{i}-1]=⟨value⟩$, $\text{j}=⟨value⟩$ and $x[\text{j}-1]=⟨value⟩$.

Constraint: $x\left[0\right]<x\left[1\right]<\cdots <x[\text{npts}-1]$.

(errno $1$)

On entry, $\text{neqn}=⟨value⟩$, $npde=⟨value⟩$, $\text{npts}=⟨value⟩$ and $nv=⟨value⟩$.

Constraint: $\text{neqn}=npde\times \text{npts}+nv$.

(errno $1$)

On entry, $con=⟨value⟩$, $\text{npts}=⟨value⟩$.

Constraint: $con\le 10.0/(\text{npts}-1)$.

(errno $1$)

On entry, $con=⟨value⟩$, $\text{npts}=⟨value⟩$.

Constraint: $con\ge 0.1/(\text{npts}-1)$.

(errno $1$)

On entry, $xratio=⟨value⟩$.

Constraint: $xratio>1.0$.

(errno $1$)

On entry, $ipminf=⟨value⟩$.

Constraint: $ipminf=0$, $1$ or $2$.

(errno $1$)

On entry, $dxmesh=⟨value⟩$.

Constraint: $dxmesh\ge 0.0$.

(errno $1$)

On entry, $\text{nxfix}=⟨value⟩$, $\text{npts}=⟨value⟩$.

Constraint: $\text{nxfix}\le \text{npts}-2$.

(errno $1$)

On entry, $\text{nxfix}=⟨value⟩$.

Constraint: $\text{nxfix}\ge 0$.

(errno $1$)

On entry, $nv=⟨value⟩$ and $\text{nxi}=⟨value⟩$.

Constraint: $\text{nxi}=0$ when $nv=0$.

(errno $1$)

On entry, $nv=⟨value⟩$ and $\text{nxi}=⟨value⟩$.

Constraint: $\text{nxi}\ge 0$ when $nv>0$.

(errno $1$)

On entry, $nleft=⟨value⟩$, $npde=⟨value⟩$.

Constraint: $nleft\le npde$.

(errno $1$)

On entry, $nleft=⟨value⟩$.

Constraint: $nleft\ge 0$.

(errno $1$)

On entry, $npde=⟨value⟩$.

Constraint: $npde\ge 1$.

(errno $1$)

On entry, $\text{npts}=⟨value⟩$.

Constraint: $\text{npts}\ge 3$.

(errno $1$)

On entry, $laopt=⟨value⟩$.

Constraint: $laopt=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.

(errno $1$)

On entry, $norm=⟨value⟩$.

Constraint: $norm=\text{'A'}$ or $\text{'M'}$.

(errno $1$)

On entry, $ind=⟨value⟩$.

Constraint: $ind=0$ or $1$.

(errno $1$)

On entry, $itask=⟨value⟩$.

Constraint: $itask=1$, $2$, $3$, $4$ or $5$.

(errno $1$)

On entry, $nv=⟨value⟩$.

Constraint: $nv\ge 0$.

(errno $1$)

On entry, $tout-ts$ is too small: $tout=⟨value⟩$ and $ts=⟨value⟩$.

(errno $1$)

On entry, $tout=⟨value⟩$ and $ts=⟨value⟩$.

Constraint: $tout>ts$.

(errno $4$)

In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting $ires=3$ in $pdedef$ or $bndary$.

(errno $5$)

Singular Jacobian of ODE system. Check problem formulation.

(errno $7$)

$atol$ and $rtol$ were too small to start integration.

(errno $8$)

$ires$ set to an invalid value in call to $pdedef$, $bndary$, or $odedef$.

(errno $9$)

Serious error in internal call to an auxiliary. Increase $itrace$ for further details.

(errno $11$)

Error during Jacobian formulation for ODE system. Increase $itrace$ for further details.

(errno $12$)

In solving ODE system, the maximum number of steps $algopt\left[14\right]$ has been exceeded. $algopt\left[14\right]=⟨value⟩$.

(errno $16$)

$remesh$ has been changed between calls to dim1_parab_remesh_keller.

Warns

NagAlgorithmicWarning

(errno $2$)

Underlying ODE solver cannot make further progress from the point $ts$ with the supplied values of $atol$ and $rtol$. $ts=⟨value⟩$.

(errno $3$)

Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as $ts$: $ts=⟨value⟩$.

(errno $6$)

In evaluating residual of ODE system, $ires=2$ has been set in $pdedef$, $bndary$, or $odedef$. Integration is successful as far as $ts$: $ts=⟨value⟩$.

(errno $10$)

Integration completed, but small changes in $atol$ or $rtol$ are unlikely to result in a changed solution.

(errno $13$)

Zero error weights encountered during time integration.

(errno $15$)

When using the sparse option, $max(lisave\_min,lisave\_estim)$ or $max(lrsave\_min,lrsave\_estim)$ is too small: $max(lisave\_min,lisave\_estim)=⟨value⟩$, $max(lrsave\_min,lrsave\_estim)=⟨value⟩$.

Notes

dim1_parab_remesh_keller integrates the system of first-order PDEs and coupled ODEs given by the master equations:

$$G_i(x,t,U,U_x,U_t,V,\dot{V})=0\text{,}i=1,2,\dots ,npde\text{,}a\le x\le b,t\ge t_0\text{,}$$

$$R_i(t,V,\dot{V},\xi ,U^{\ast },U_x^{\ast },U_t^{\ast })=0\text{,}i=1,2,\dots ,nv\text{.}$$

In the PDE part of the problem given by (1), the functions $G_i$ must have the general form

$$G_i=\sum _{j=1}^{npde}{P}_{i,j}\frac{\partial U_j}{\partial t}+\sum _{j=1}^{nv}{Q}_{i,j}{\dot{V}}_j+S_i=0\text{,}i=1,2,\dots ,npde\text{,}$$

where $P_{i,j}$, $Q_{i,j}$ and $S_i$ depend on $x$, $t$, $U$, $U_x$ and $V$.

The vector $U$ is the set of PDE solution values

$$U(x,t)={[U_1(x,t),\dots ,U_{npde}(x,t)]}^T\text{,}$$

and the vector $U_x$ is the partial derivative with respect to $x$. The vector $V$ is the set of ODE solution values

$$V\left(t\right)={[V_1\left(t\right),\dots ,V_{nv}\left(t\right)]}^T\text{,}$$

and $\dot{V}$ denotes its derivative with respect to time.

In the ODE part given by (2), $\xi$ 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 some of the PDE spatial mesh points. $U^{\ast }$, $U_x^{\ast }$ and $U_t^{\ast }$ are the functions $U$, $U_x$ and $U_t$ evaluated at these coupling points. Each $R_i$ may only depend linearly on time derivatives. Hence equation (2) may be written more precisely as

$$R=A-B\dot{V}-C{U}_t^{\ast }\text{,}$$

where $R={[R_1,\dots ,R_{nv}]}^T$, $A$ is a vector of length $nv$, $B$ is an $nv$ by $nv$ matrix, $C$ is an $nv$ by $(n_{\xi }\times npde)$ matrix and the entries in $A$, $B$ and $C$ may depend on $t$, $\xi$, $U^{\ast }$, $U_x^{\ast }$ and $V$. In practice you only need to supply a vector of information to define the ODEs and not the matrices $B$ and $C$. (See Parameters for the specification of $odedef$.)

The integration in time is from $t_0$ to $t_{out}$, over the space interval $a\le x\le b$, where $a=x_1$ and $b=x_{\text{npts}}$ are the leftmost and rightmost points of a mesh $x_1,x_2,\dots ,x_{\text{npts}}$ defined initially by you and (possibly) adapted automatically during the integration according to user-specified criteria.

The PDE system which is defined by the functions $G_i$ must be specified in $pdedef$.

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 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.

For a first-order system of PDEs, only one boundary condition is required for each PDE component $U_i$. The $npde$ boundary conditions are separated into $n_a$ at the left-hand boundary $x=a$, and $n_b$ at the right-hand boundary $x=b$, such that $n_a+n_b=npde$. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of $U_i$ at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for $U_i$ should be specified at the left-hand boundary. Incorrect positioning of boundary conditions generally results in initialization or integration difficulties in the underlying time integration functions.

The boundary conditions have the master equation form:

$$G_i^L(x,t,U,U_t,V,\dot{V})=0\text{at}x=a\text{,}i=1,2,\dots ,n_a\text{,}$$

at the left-hand boundary, and

$$G_i^R(x,t,U,U_t,V,\dot{V})=0\text{at}x=b\text{,}i=1,2,\dots ,n_b\text{,}$$

at the right-hand boundary.

Note that the functions $G_i^L$ and $G_i^R$ must not depend on $U_x$, since spatial derivatives are not determined explicitly in the Keller box scheme functions. If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that $G_i^L$ and $G_i^R$ must be linear with respect to time derivatives, so that the boundary conditions have the general form:

$$\sum _{j=1}^{npde}{E}_{i,j}^L\frac{\partial U_j}{\partial t}+\sum _{j=1}^{nv}{H}_{i,j}^L{\dot{V}}_j+K_i^L=0\text{,}i=1,2,\dots ,n_a\text{,}$$

at the left-hand boundary, and

$$\sum _{j=1}^{npde}{E}_{i,j}^R\frac{\partial U_j}{\partial t}+\sum _{j=1}^{nv}{H}_{i,j}^R{\dot{V}}_j+K_i^R=0\text{,}i=1,2,\dots ,n_b\text{,}$$

at the right-hand boundary, where $E_{i,j}^L$, $E_{i,j}^R$, $H_{i,j}^L$, $H_{i,j}^R$, $K_i^L$ and $K_i^R$ depend on $x,t,U$ and $V$ only.

The boundary conditions must be specified in $bndary$.

The problem is subject to the following restrictions:

1. $P_{i,j}$, $Q_{i,j}$ and $S_i$ must not depend on any time derivatives;

2. $t_0<t_{out}$, so that integration is in the forward direction;

3. The evaluation of the function $G_{\text{i}}$ is done approximately at the mid-points of the mesh $x[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{npts}$, by calling $pdedef$ for each mid-point in turn. Any discontinuities in the function must, therefore, be at one or more of the fixed mesh points specified by $xfix$;

4. At least one of the functions $P_{i,j}$ must be nonzero so that there is a time derivative present in the PDE problem.

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 $\xi$ in the array $xi$.

The first-order equations are approximated by a system of ODEs in time for the values of $U_i$ at mesh points. In this method of lines approach the Keller box scheme is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of $U_i$ at each mesh point. In total there are $npde\times \text{npts}+nv$ ODEs in time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.

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 $monitf$ which specifies in an analytic or numeric form the particular aspect of the solution behaviour you wish to track. This so-called monitor function is used 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 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 along 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.

References

Berzins, M, 1990, Developments in the NAG Library software for parabolic equations, Scientific Software Systems, (eds J C Mason and M G Cox), 59–72, Chapman and Hall

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

Berzins, M and Furzeland, R M, 1992, An adaptive theta method for the solution of stiff and nonstiff differential equations, Appl. Numer. Math. (9), 1–19

Furzeland, R M, 1984, The construction of adaptive space meshes, TNER.85.022, Thornton Research Centre, Chester

Keller, H B, 1970, A new difference scheme for parabolic problems, Numerical Solutions of Partial Differential Equations, (ed J Bramble) (2), 327–350, Academic Press

Pennington, S V and Berzins, M, 1994, New NAG Library software for first-order partial differential equations, ACM Trans. Math. Softw. (20), 63–99