d03pf:: Partial Differential Equations (NAG Toolbox)

nag_pde_1d_parab_convdiff (d03pf) integrates a system of linear or nonlinear convection-diffusion equations in one space dimension, with optional source terms. 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 PDEs to a system of ordinary differential equations (ODEs), and the resulting system is solved using a backward differentiation formula (BDF) method.

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 22:

lrsave and lisave were removed from the interface

nag_pde_1d_parab_convdiff (d03pf) 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},

(1)

or the hyperbolic convection-only system:

\frac{\partial U_{i}}{\partial t} + \frac{\partial F_{i}}{\partial x} = 0,

(2)

for

i = 1, 2, \dots, npde, a \leq x \leq b, t \geq t_{0}

, where the vector

U

is the set of solution values

U (x, t) = {[U_{1} (x, t), \dots, U_{npde} (x, t)]}^{T} .

The functions

P_{i, j}

,

F_{i}

,

C_{i}

and

S_{i}

depend on

x

,

t

and

U

; and

D_{i}

depends on

x

,

t

,

U

and

U_{x}

, where

U_{x}

is the spatial derivative of

U

. Note that

P_{i, j}

,

F_{i}

,

C_{i}

and

S_{i}

must not depend on any space derivatives; and none of the functions may depend on 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.

The integration in time is from

t_{0}

to

t_{out}

, over the space interval

a \leq x \leq b

, where

a = x_{1}

and

b = x_{npts}

are the leftmost and rightmost points of a user-defined mesh

x_{1}, x_{2}, \dots, x_{npts}

. The initial values of the functions

U (x, t)

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 nag_pde_1d_parab_fd (d03pc), nag_pde_1d_parab_dae_fd (d03ph) and nag_pde_1d_parab_remesh_fd (d03pp), 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_{j - 1 / 2} = (x_{j - 1} + x_{j}) / 2

, for

j = 2, 3, \dots, npts

. The left and right values are calculated by nag_pde_1d_parab_convdiff (d03pf) 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,

(3)

where

y = x - x_{j - 1 / 2}

, i.e.,

y = 0

corresponds to

x = x_{j - 1 / 2}

, with discontinuous initial values

U = U_{L}

for

y < 0

and

U = U_{R}

for

y > 0

, using an approximate Riemann solver. This applies for either of the systems (1) or (2); the numerical flux is independent of the functions

P_{i, j}

,

C_{i}

,

D_{i}

and

S_{i}

. A description of several approximate Riemann solvers can be found in LeVeque (1990) and Berzins et al. (1989). Roe's scheme (see Roe (1981)) is perhaps the easiest to understand and use, and a brief summary follows. Consider the system of PDEs

U_{t} + F_{x} = 0

or equivalently

U_{t} + A U_{x} = 0

. Provided the system is linear in

U

, i.e., the Jacobian matrix

A

does not depend on

U

, the numerical flux

\hat{F}

is given by

\hat{F} = \frac{1}{2} (F_{L} + F_{R}) - \frac{1}{2} \sum_{k = 1}^{npde} α_{k} |λ_{k}| e_{k},

(4)

where

F_{L}

(

F_{R}

) is the flux

F

calculated at the left (right) value of

U

, denoted by

U_{L}

(

U_{R}

); the

λ_{k}

are the eigenvalues of

A

; the

e_{k}

are the right eigenvectors of

A

; and the

α_{k}

are defined by

U_{R} - U_{L} = \sum_{k = 1}^{npde} α_{k} e_{k} .

(5)

An example is given in Example.

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 a pdedef. The numerical flux

{\hat{F}}_{i}

must be supplied in a separate numflx. For problems in the form (2), the actual argument nag_pde_1d_parab_convdiff_sample_pdedef (d03pfp) may be used for pdedef. nag_pde_1d_parab_convdiff_sample_pdedef (d03pfp) is included in the NAG Toolbox and sets the matrix with entries

P_{i, j}

to the identity matrix, and the functions

C_{i}

,

D_{i}

and

S_{i}

to zero.

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 conditions, i.e., a total of npde conditions at each boundary point.

The position of each boundary condition should be chosen with care. In simple terms, if information is flowing into the domain then a physical boundary condition is required at that boundary, and a numerical boundary condition is required at the other boundary. In many cases the boundary conditions are simple, e.g., for the linear advection equation. In general you should calculate the characteristics of the PDE system and specify a physical boundary condition for each of the characteristic variables associated with incoming characteristics, and a numerical boundary condition for each outgoing characteristic.

A common way of providing numerical boundary conditions is to extrapolate the characteristic variables from the inside of the domain. Note that only linear extrapolation is allowed in this function (for greater flexibility the function nag_pde_1d_parab_convdiff_dae (d03pl) should be used). 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. Examples can be found in Example.

The boundary conditions must be specified in bndary in the form

G_{i}^{L} (x, t, U) = 0 at ​ x = a, i = 1, 2, \dots, npde,

(6)

at the left-hand boundary, and

G_{i}^{R} (x, t, U) = 0 at ​ x = b, i = 1, 2, \dots, npde,

(7)

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 problem is subject to the following restrictions:

(i)	$P_{i, j}$ , $F_{i}$ , $C_{i}$ and $S_{i}$ must not depend on any space derivatives;
(ii)	$P_{i, j}$ , $F_{i}$ , $C_{i}$ , $D_{i}$ and $S_{i}$ must not depend on any time derivatives;
(iii)	$t_{0} < t_{out}$ , so that integration is in the forward direction;
(iv)	The evaluation of the terms $P_{i, j}$ , $C_{i}$ , $D_{i}$ and $S_{i}$ is done by calling the pdedef at a point approximately midway between each pair of mesh points in turn. Any discontinuities in these functions must therefore be at one or more of the mesh points $x_{1}, x_{2}, \dots, x_{npts}$ ;
(v)	At least one of the functions $P_{i, j}$ must be nonzero so that there is a time derivative present in the PDE problem.

In total there are

npde \times npts

ODEs in the time direction. This system is then integrated forwards in time using a BDF method.

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

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

Errors or warnings detected by the function:

nag_pde_1d_parab_convdiff (d03pf) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the components of the accuracy argument, acc.

nag_pde_1d_parab_convdiff (d03pf) is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional second-order diffusion terms. The use of the function to solve systems which are not naturally in this form is discouraged, and you are advised to use one of the central-difference schemes for such problems.

You should be aware of the stability limitations for hyperbolic PDEs. For most problems with small error tolerances the ODE integrator does not attempt unstable time steps, but in some cases a maximum time step should be imposed using tsmax. It is worth experimenting with this argument, particularly if the integration appears to progress unrealistically fast (with large time steps). Setting the maximum time step to the minimum mesh size is a safe measure, although in some cases this may be too restrictive.

Problems with source terms should be treated with caution, as it is known that for large source terms stable and reasonable looking solutions can be obtained which are in fact incorrect, exhibiting non-physical speeds of propagation of discontinuities (typically one spatial mesh point per time step). It is essential to employ a very fine mesh for problems with source terms and discontinuities, and to check for non-physical propagation speeds by comparing results for different mesh sizes. Further details and an example can be found in Pennington and Berzins (1994).

The time taken depends on the complexity of the system and on the accuracy requested.

For this function two examples are presented. There is a single example program for nag_pde_1d_parab_convdiff (d03pf), with a main program and the code to solve the two example problems given in Example 1 (EX1) and Example 2 (EX2).

Example 1 (EX1)

This example is a simple first-order system which illustrates the calculation of the numerical flux using Roe's approximate Riemann solver, and the specification of numerical boundary conditions using extrapolated characteristic variables. The PDEs are

\begin{array}{lcl} \frac{\partial U_{1}}{\partial t} + \frac{\partial U_{1}}{\partial x} + \frac{\partial U_{2}}{\partial x} & = & 0, \\ \frac{\partial U_{2}}{\partial t} + 4 \frac{\partial U_{1}}{\partial x} + \frac{\partial U_{2}}{\partial x} & = & 0, \end{array}

for

x \in [0, 1]

and

t \geq 0

. The PDEs have an exact solution given by

\begin{array}{lcl} U_{1} (x, t) & = & \frac{1}{2} \{\exp (x + t) + \exp (x - 3 t)\} + \frac{1}{4} \{\sin (2 π {(x - 3 t)}^{2}) - \sin (2 π {(x + t)}^{2})\} + 2 t^{2} - 2 x t, \\ U_{2} (x, t) & = & \exp (x - 3 t) - \exp (x + t) + \frac{1}{2} \{\sin (2 π {(x - 3 t)}^{2}) + \sin (2 π {(x - 3 t)}^{2})\} + x^{2} + 5 t^{2} - 2 x t . \end{array}

The initial conditions are given by the exact solution. The characteristic variables are

2 U_{1} + U_{2}

and

2 U_{1} - U_{2}

corresponding to the characteristics given by

d x / d t = 3

and

d x / d t = - 1

respectively. Hence a physical boundary condition is required for

2 U_{1} + U_{2}

at the left-hand boundary, and for

2 U_{1} - U_{2}

at the right-hand boundary (corresponding to the incoming characteristics); and a numerical boundary condition is required for

2 U_{1} - U_{2}

at the left-hand boundary, and for

2 U_{1} + U_{2}

at the right-hand boundary (outgoing characteristics). The physical boundary conditions are obtained from the exact solution, and the numerical boundary conditions are calculated by linear extrapolation of the appropriate characteristic variable. The numerical flux is calculated using Roe's approximate Riemann solver: Using the notation in Description, the flux vector

F

and the Jacobian matrix

A

are

F = [\begin{matrix} U_{1} + U_{2} \\ 4 U_{1} + U_{2} \end{matrix}] and A = [\begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}],

and the eigenvalues of

A

are

3

and

- 1

with right eigenvectors

{[\begin{matrix} 1 & 2 \end{matrix}]}^{T}

and

{[\begin{matrix} - 1 & 2 \end{matrix}]}^{T}

respectively. Using equation (4) the

α_{k}

are given by

[\begin{matrix} U_{1 R} - U_{1 L} \\ U_{2 R} - U_{2 L} \end{matrix}] = α_{1} [\begin{matrix} 1 \\ 2 \end{matrix}] + α_{2} [\begin{array}{r} - 1 \\ 2 \end{array}],

that is

α_{1} = \frac{1}{4} (2 U_{1 R} - 2 U_{1 L} + U_{2 R} - U_{2 L}) and α_{2} = \frac{1}{4} (- 2 U_{1 R} + 2 U_{1 L} + U_{2 R} - U_{2 L}) .

F_{L}

is given by

F_{L} = [\begin{array}{r} U_{1 L} + U_{2 L} \\ 4 U_{1 L} + U_{2 L} \end{array}],

and similarly for

F_{R}

. From equation (4), the numerical flux vector is

\hat{F} = \frac{1}{2} [\begin{array}{r} U_{1 L} + U_{2 L} + U_{1 R} + U_{2 R} \\ 4 U_{1 L} + U_{2 L} + 4 U_{1 R} + U_{2 R} \end{array}] - \frac{1}{2} α_{1} |3| [\begin{matrix} 1 \\ 2 \end{matrix}] - \frac{1}{2} α_{2} |- 1| [\begin{array}{r} - 1 \\ 2 \end{array}],

that is

\hat{F} = \frac{1}{2} [\begin{array}{r} 3 U_{1 L} - U_{1 R} + \frac{3}{2} U_{2 L} + \frac{1}{2} U_{2 R} \\ 6 U_{1 L} + 2 U_{1 R} + 3 U_{2 L} - U_{2 R} \end{array}] .

Example 2 (EX2)

This example is an advection-diffusion equation in which the flux term depends explicitly on

x

:

\frac{\partial U}{\partial t} + x \frac{\partial U}{\partial x} = ε \frac{\partial^{2} U}{\partial x^{2}},

for

x \in [- 1, 1]

and

0 \leq t \leq 10

. The argument

ε

is taken to be

0.01

. The two physical boundary conditions are

U (- 1, t) = 3.0

and

U (1, t) = 5.0

and the initial condition is

U (x, 0) = x + 4

. The integration is run to steady state at which the solution is known to be

U = 4

across the domain with a narrow boundary layer at both boundaries. In order to write the PDE in conservative form, a source term must be introduced, i.e.,

\frac{\partial U}{\partial t} + \frac{\partial (x U)}{\partial x} = ε \frac{\partial^{2} U}{\partial x^{2}} + U .

As in Example 1, the numerical flux is calculated using the Roe approximate Riemann solver. The Riemann problem to solve locally is

\frac{\partial U}{\partial t} + \frac{\partial (x U)}{\partial x} = 0 .

The

x

in the flux term is assumed to be constant at a local level, and so using the notation in Description,

F = x U

and

A = x

. The eigenvalue is

x

and the eigenvector (a scalar in this case) is

1

. The numerical flux is therefore

\hat{F} = \{\begin{cases} x U_{L} & if ​ x \geq 0, \\ x U_{R} & if ​ x < 0 . \end{cases}

On entry,	$ts \geq tout$ ,
or	$tout - ts$ is too small,
or	$itask = 1$ , $2$ or $3$ ,
or	$npts < 3$ ,
or	$npde < 1$ ,
or	$ind \neq 0$ or $1$ ,
or	incorrect user-defined mesh, i.e., $x (i) \geq x (i + 1)$ for some $i = 1, 2, \dots, npts - 1$ ,
or	lrsave or lisave are too small,
or	$ind = 1$ on initial entry to nag_pde_1d_parab_convdiff (d03pf),
or	$acc (1)$ or $acc (2) < 0.0$ ,
or	$acc (1)$ or $acc (2)$ are both zero,
or	$tsmax < 0.0$ .

NAG Toolbox: nag_pde_1d_parab_convdiff (d03pf)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example