naginterfaces.library.pde.dim1_​parab_​keller

naginterfaces.library.pde.dim1_parab_keller(ts, tout, pdedef, bndary, u, x, nleft, acc, comm, itask, itrace, ind, data=None, io_manager=None)

dim1_parab_keller integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable. The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a Backward Differentiation Formula (BDF) method.

For full information please refer to the NAG Library document for d03pe

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/d03/d03pef.html

Parameters

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, ires, data=None)

$pdedef$ must compute 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_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 $\text{i}=1,2,\dots ,\text{npde}$.

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

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

iresint

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

$ires=-1$

Equation (8) must be used.

$ires=1$

Equation (9) 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 $\text{i}=1,2,\dots ,\text{npde}$, where $G$ is defined as

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

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

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

i.e., all terms in equation (2).

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:

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

$bndary$ must compute the functions $G_i^L$ and $G_i^R$ which define the boundary conditions as in equations (4) and (5).

Parameters

tfloat

The current value of the independent variable $t$.

ibndint

Determines the position of the boundary conditions.

$ibnd=0$

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

$ibnd\ne 0$

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

nobcint

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

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

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

iresint

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

$ires=-1$

Equation (0) must be used.

$ires=1$

Equation (1) 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

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

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

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

i.e., all terms in equation (6), 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_keller returns to the calling function with the error indicator set to $errno$ = 4.

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

The initial values of $U(x,t)$ at $t=ts$ and the mesh points $x[\text{j}-1]$, for $\text{j}=1,2,\dots ,\text{npts}$.

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

The mesh points in the spatial 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 $x\left[0\right]$.

accfloat

A positive quantity for controlling the local error estimate in the time integration. If $E(i,j)$ is the estimated error for $U_i$ at the $j$th mesh point, the error test is:

$$\left|E(i,j)\right|=acc\times (1.0+\left|u\right[i-1,j-1\left]\right|)\text{.}$$

commdict, communication object, modified in place

Communication structure.

On initial entry: need not be set.

itaskint

Specifies the task to be performed by the ODE integrator.

$itask=1$

Normal computation of output values $u$ at $t=tout$.

$itask=2$

Take one step and return.

$itask=3$

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

itraceint

The level of trace information required from dim1_parab_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.

$itrace\ge 3$

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 $itrace=0$.

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_keller.

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

tsfloat

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

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

$u[\text{i}-1,\text{j}-1]$ will contain the computed solution at $t=ts$.

indint

$ind=1$.

Raises

NagValueError

(errno $1$)

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

Constraint: on initial entry $ind=0$.

(errno $1$)

On entry, $ind=⟨value⟩$.

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

(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, $acc=⟨value⟩$.

Constraint: $acc>0.0$.

(errno $1$)

On entry, $nleft=⟨value⟩$, $\text{npde}=⟨value⟩$.

Constraint: $nleft\le \text{npde}$.

(errno $1$)

On entry, $nleft=⟨value⟩$.

Constraint: $nleft\ge 0$.

(errno $1$)

On entry, $\text{npde}=⟨value⟩$.

Constraint: $\text{npde}\ge 1$.

(errno $1$)

On entry, $\text{npts}=⟨value⟩$.

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

(errno $1$)

On entry, $itask=⟨value⟩$.

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

(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 $7$)

$acc$ was too small to start integration: $acc=⟨value⟩$.

(errno $8$)

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

(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.

Warns

NagAlgorithmicWarning

(errno $2$)

Underlying ODE solver cannot make further progress from the point $ts$ with the supplied value of $acc$. $ts=⟨value⟩$, $acc=⟨value⟩$.

(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 $pdedef$ or $bndary$. Integration is successful as far as $ts$: $ts=⟨value⟩$.

(errno $10$)

Integration completed, but a small change in $acc$ is unlikely to result in a changed solution. $acc=⟨value⟩$.

Notes

dim1_parab_keller integrates the system of first-order PDEs

$$G_i(x,t,U,U_x,U_t)=0\text{,}i=1,2,\dots ,\text{npde}\text{.}$$

In particular the functions $G_i$ must have the general form

$$G_i=\sum _{j=1}^{\text{npde}}{P}_{i,j}\frac{\partial U_j}{\partial t}+Q_i\text{,}i=1,2,\dots ,\text{npde}\text{,}a\le x\le b,t\ge t_0\text{,}$$

where $P_{i,j}$ and $Q_i$ depend on $x$, $t$, $U$, $U_x$ and the vector $U$ is the set of solution values

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

and the vector $U_x$ is its partial derivative with respect to $x$. Note that $P_{i,j}$ and $Q_i$ must not depend on $\frac{\partial U}{\partial t}$.

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 mesh should be chosen in accordance with the expected behaviour of the solution.

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

The initial values of the functions $U(x,t)$ must be given at $t=t_0$. For a first-order system of PDEs, only one boundary condition is required for each PDE component $U_i$. The $\text{npde}$ 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 $n_a+n_b=\text{npde}$. 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 form:

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

at the left-hand boundary, and

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

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 (see Keller (1970)). 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

$$\sum _{j=1}^{\text{npde}}{E}_{i,j}^L\frac{\partial U_j}{\partial t}+S_i^L=0\text{,}i=1,2,\dots ,n_a$$

at the left-hand boundary, and

$$\sum _{j=1}^{\text{npde}}{E}_{i,j}^R\frac{\partial U_j}{\partial t}+S_i^R=0\text{,}i=1,2,\dots ,n_b$$

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

The boundary conditions must be specified in $bndary$.

The problem is subject to the following restrictions:

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

2. $P_{i,j}$ and $Q_i$ must not depend on any time derivatives;

3. The evaluation of the function $G_i$ is done at the mid-points of the mesh intervals by calling the $pdedef$ for each mid-point in turn. Any discontinuities in the function must, therefore, be at one or more of the mesh points $x_1,x_2,\dots ,x_{\text{npts}}$;

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

In this method of lines approach the Keller box scheme (see Keller (1970)) 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 $\text{npde}\times \text{npts}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF method.

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

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