naginterfaces.library.pde.dim1_​parab_​convdiff_​dae

naginterfaces.library.pde.dim1_parab_convdiff_dae(npde, ts, tout, numflx, bndary, u, x, nv, xi, rtol, atol, itol, norm, laopt, algopt, comm, itask, itrace, ind, pdedef=None, odedef=None, lrsave_estim=0, lisave_estim=0, data=None, io_manager=None, spiked_sorder='C')

dim1_parab_convdiff_dae 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. 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.

For full information please refer to the NAG Library document for d03pl

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d03/d03plf.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.

numflxcallable (flux, ires) = numflx(t, x, v, uleft, uright, ires, data=None)

$numflx$ must supply the numerical flux for each PDE given the left and right values of the solution vector $u$. $numflx$ is called approximately midway between each pair of mesh points in turn by dim1_parab_convdiff_dae.

Parameters

tfloat

The current value of the independent variable $t$.

xfloat

The current value of the space variable $x$.

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 $\text{i}=1,2,\dots ,nv$.

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

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

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

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

iresint

Set to $-1$ or $1$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

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

$flux[\text{i}-1]$ must be set to the numerical flux ${\hat{F}}_{\text{i}}$, for $\text{i}=1,2,\dots ,npde$.

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_convdiff_dae returns to the calling function with the error indicator set to $errno$ = 4.

bndarycallable (g, ires) = bndary(t, x, u, v, vdot, ibnd, ires, data=None)

$bndary$ must evaluate the functions $G_i^L$ and $G_i^R$ which describe the physical and numerical boundary conditions, as given by (9) and (0).

Parameters

tfloat

The current value of the independent variable $t$.

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

The mesh points in the spatial direction. $x\left[0\right]$ corresponds to the left-hand boundary, $a$, and $x[\text{npts}-1]$ corresponds to the right-hand boundary, $b$.

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

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

Note: if banded matrix algebra is to be used then the functions $G_{\text{i}}^L$ and $G_{\text{i}}^R$ may depend on the value of $U_{\text{i}}(x,t)$ at the boundary point and the two adjacent points only.

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: ${\dot{V}}_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$, may only appear linearly in $G_{\text{j}}^L$ and $G_{\text{j}}^R$, for $\text{j}=1,2,\dots ,npde$.

ibndint

Specifies which boundary conditions are to be evaluated.

$ibnd=0$

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

$ibnd\ne 0$

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

iresint

Set to $-1$ or $1$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

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

$g[\text{i}-1]$ must contain the $\text{i}$th component of either $G_{\text{i}}^L$ or $G_{\text{i}}^R$ in (9) and (0), depending on the value of $ibnd$, for $\text{i}=1,2,\dots ,npde$.

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_convdiff_dae returns to the calling function with the error indicator set to $errno$ = 4.

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

The initial values of the dependent variables defined as follows:

$u[npde\times (\text{j}-1)+\text{i}-1]$ contain $U_{\text{i}}(x_{\text{j}},t_0)$, 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 $V_{\text{i}}\left(t_0\right)$, for $\text{i}=1,2,\dots ,nv$.

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

The 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, $b$.

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.

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. If $e_{\text{i}}$ is the estimated local error for $u[\text{i}-1]$, for $\text{i}=1,2,\dots ,\text{neqn}$, and $\parallel \parallel$ denotes the norm, the error test to be satisfied is $\parallel e_{\text{i}}\parallel <1.0$. $itol$ indicates to dim1_parab_convdiff_dae whether to interpret either or both of $rtol$ and $atol$ as a vector or scalar in the formation of the weights $w_{\text{i}}$ used in the calculation of the norm (see the description of $norm$):

$itol$	$rtol$	$atol$	$w_{\text{i}}$
1	scalar	scalar	$rtol\left[0\right]\times \left|u\right[\text{i}-1\left]\right|+atol\left[0\right]$
2	scalar	vector	$rtol\left[0\right]\times \left|u\right[\text{i}-1\left]\right|+atol[\text{i}-1]$
3	vector	scalar	$rtol[\text{i}-1]\times \left|u\right[\text{i}-1\left]\right|+atol\left[0\right]$
4	vector	vector	$rtol[\text{i}-1]\times \left|u\right[\text{i}-1\left]\right|+atol[\text{i}-1]$

normstr, length 1

The type of norm to be used.

$norm=\text{'1'}$

Averaged $L_1$ norm.

$norm=\text{'2'}$

Averaged $L_2$ norm.

If $U_{norm}$ denotes the norm of the vector $u$ of length $\text{neqn}$, then for the averaged $L_1$ norm

$$U_{\text{norm}}=\frac{1}{\text{neqn}}\sum _{i=1}^{\text{neqn}}u[i-1]/w_i\text{,}$$

and for the averaged $L_2$ norm

$$U_{\text{norm}}=\sqrt{\frac{1}{\text{neqn}}\sum _{i=1}^{\text{neqn}}{\left(u\right[i-1\left]/w_i\right)}^2}\text{.}$$

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.

Note: you are recommended to use the banded option when no coupled ODEs are present ($nv=0$). 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.

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 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 the range then the default value is used. If the functions regard the Jacobian matrix as numerically singular, 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 the 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 matrix is found to be numerically singular (see $algopt\left[28\right]$). The default value is $algopt\left[29\right]=0.0001$.

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_convdiff_dae and the underlying ODE solver. $itrace$ may take the value $-1$, $0$, $1$, $2$ or $3$.

$itrace=-1$

No output is generated.

$itrace=0$

Only warning messages from the PDE solver are printed.

$itrace>0$

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.

If $itrace<-1$, $-1$ is assumed and similarly if $itrace>3$, $3$ is assumed.

The advisory messages are given in greater detail as $itrace$ increases. 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 $tout$ should be reset between calls to dim1_parab_convdiff_dae.

pdedefNone or callable (p, c, d, s, ires) = pdedef(t, x, u, ux, v, vdot, ires, data=None), optional

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

$pdedef$ 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 $\dot{V}$. $pdedef$ is called approximately midway between each pair of mesh points in turn by dim1_parab_convdiff_dae. None may be used for $pdedef$ for problems in the form (2).

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 $\text{i}=1,2,\dots ,npde$.

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 $\text{i}=1,2,\dots ,nv$.

Note: ${\dot{V}}_{\text{i}}\left(t\right)$, for $\text{i}=1,2,\dots ,nv$, may only appear linearly in $S_{\text{j}}$, for $\text{j}=1,2,\dots ,npde$.

iresint

Set to $-1$ or $1$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

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

$p[\text{i}-1,\text{j}-1]$ must be set to the value of $P_{\text{i},\text{j}}(x,t,U,V)$, for $\text{j}=1,2,\dots ,npde$, for $\text{i}=1,2,\dots ,npde$.

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

$c[\text{i}-1]$ must be set to the value of $C_{\text{i}}(x,t,U,V)$, for $\text{i}=1,2,\dots ,npde$.

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

$d[\text{i}-1]$ must be set to the value of $D_{\text{i}}(x,t,U,U_x,V)$, for $\text{i}=1,2,\dots ,npde$.

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

$s[\text{i}-1]$ must be set to the value of $S_{\text{i}}(x,t,U,V,\dot{V})$, for $\text{i}=1,2,\dots ,npde$.

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_convdiff_dae returns to the calling function with the error indicator set to $errno$ = 4.

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 (1) must be used.

$ires=-1$

Equation (2) must be used.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

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

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

$$R=L-M\dot{V}-N{U}_t^{\ast }\text{,}$$

or

$$R=-M\dot{V}-N{U}_t^{\ast }\text{.}$$

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_convdiff_dae returns to the calling function with the error indicator set to $errno$ = 4.

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 $p$ in $pdedef$ 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 $\text{i}=1,2,\dots ,nv$.

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, 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, $nv=⟨value⟩$ 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, $npde=⟨value⟩$.

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: $norm=\text{'1'}$ or $\text{'2'}$.

(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 $pdedef$, $numflx$, $bndary$ or $odedef$.

(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 a call to functions $pdedef$, $numflx$, $bndary$ 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 $14$)

The functions $P$, $D$, or $C$ appear to depend on time derivatives.

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 functions $pdedef$, $numflx$, $bndary$ 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_convdiff_dae integrates the system of convection-diffusion equations in conservative form:

$$\sum _{j=1}^{npde}{P}_{i,j}\frac{\partial U_j}{\partial t}+\frac{\partial F_i}{\partial x}=C_i\frac{\partial D_i}{\partial x}+S_i\text{,}$$

or the hyperbolic convection-only system:

$$\frac{\partial U_i}{\partial t}+\frac{\partial F_i}{\partial x}=0\text{,}$$

for $i=1,2,\dots ,npde\text{,}a\le x\le b\text{,}t\ge t_0$, where the vector $U$ is the set of PDE solution values

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

The optional coupled ODEs are of the general form

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

where the vector $V$ is the set of ODE solution values

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

$\dot{V}$ denotes its derivative with respect to time, and $U_x$ is the spatial derivative of $U$.

In (1), $P_{i,j}$, $F_i$ 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 $\dot{V}$. 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. $U^{\ast }$, $U_x^{\ast }$ and $U_t^{\ast }$ are the functions $U$, $U_x$ and $U_t$ evaluated at these coupling points. Each $R_i$ may depend only linearly on time derivatives. Hence (3) may be written more precisely as

$$R=L-M\dot{V}-N{U}_t^{\ast }\text{,}$$

where $R={[R_1,\dots ,R_{nv}]}^T$, $L$ is a vector of length $nv$, $M$ is an $nv$ by $nv$ matrix, $N$ is an $nv$ by $(n_{\xi }\times npde)$ matrix and the entries in $L$, $M$ and $N$ 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 $L$, $M$ and $N$. (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 user-defined mesh $x_1,x_2,\dots ,x_{\text{npts}}$. The initial values of the functions $U(x,t)$ and $V\left(t\right)$ must be given at $t=t_0$.

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 dim1_parab_fd(), dim1_parab_dae_fd() and dim1_parab_remesh_fd(), 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 vector, ${\hat{F}}_i$ 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 $x_{\text{j}-\frac{1}{2}}=(x_{\text{j}-1}+x_{\text{j}})/2$, for $\text{j}=2,3,\dots ,\text{npts}$. The left and right values are calculated by dim1_parab_convdiff_dae 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

$$\frac{\partial U_i}{\partial t}+\frac{\partial F_i}{\partial y}=0\text{,}$$

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) and (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

$$\hat{F}=\frac{1}{2}(F_L+F_R)-\frac{1}{2}\sum _{k=1}^{npde}{\alpha }_k\left|{\lambda }_k\right|e_k\text{,}$$

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

$$U_R-U_L=\sum _{k=1}^{npde}{\alpha }_k{e}_k\text{.}$$

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 a separate $numflx$. For problems in the form (2), the actual argument None may be used for $pdedef$. This sets the matrix with entries $P_{i,j}$ to the identity matrix, and the functions $C_i$, $D_i$ and $S_i$ to zero.

The boundary condition specification has sufficient flexibility to allow for different types of problems. For second-order problems, i.e., $D_i$ depending on $U_x$, a boundary condition is required for each PDE at both boundaries for the problem to be well-posed. If there are no second-order 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 condition, 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 dim1_parab_convdiff() documentation.

The boundary conditions must be specified in $bndary$ in the form

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

at the left-hand boundary, and

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

at the right-hand boundary.

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 $xi$.

The problem is subject to the following restrictions:

1. In (1), ${\dot{V}}_j\left(t\right)$, for $j=1,2,\dots ,nv$, may only appear linearly in the functions $S_i$, for $i=1,2,\dots ,npde$, with a similar restriction for $G_i^L$ and $G_i^R$;

2. $P_{i,j}$, $F_i$, $C_i$ 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;

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

4. 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 mesh points $x_1,x_2,\dots ,x_{\text{npts}}$;

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

In total there are $npde\times \text{npts}+nv$ 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)).

For further details of the scheme, see Pennington and Berzins (1994) and the references therein.

References

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

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

Sod, G A, 1978, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. (27), 1–31