NAG Library Function Document
nag_pde_parab_1d_keller_ode (d03pkc)
1
Purpose
nag_pde_parab_1d_keller_ode (d03pkc) 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). 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 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).
2
Specification
#include <nag.h> |
#include <nagd03.h> |
void |
nag_pde_parab_1d_keller_ode (Integer npde,
double *ts,
double tout,
void |
(*pdedef)(Integer npde,
double t,
double x,
const double u[],
const double ut[],
const double ux[],
Integer nv,
const double v[],
const double vdot[],
double res[],
Integer *ires,
Nag_Comm *comm),
|
|
void |
(*bndary)(Integer npde,
double t,
Integer ibnd,
Integer nobc,
const double u[],
const double ut[],
Integer nv,
const double v[],
const double vdot[],
double res[],
Integer *ires,
Nag_Comm *comm),
|
|
double u[],
Integer npts,
const double x[],
Integer nleft,
Integer nv,
void |
(*odedef)(Integer npde,
double t,
Integer nv,
const double v[],
const double vdot[],
Integer nxi,
const double xi[],
const double ucp[],
const double ucpx[],
const double ucpt[],
double r[],
Integer *ires,
Nag_Comm *comm),
|
|
Integer nxi,
const double xi[],
Integer neqn,
const double rtol[],
const double atol[],
Integer itol,
Nag_NormType norm,
Nag_LinAlgOption laopt,
const double algopt[],
double rsave[],
Integer lrsave,
Integer isave[],
Integer lisave,
Integer itask,
Integer itrace,
const char *outfile,
Integer *ind,
Nag_Comm *comm, Nag_D03_Save *saved,
NagError *fail) |
|
3
Description
nag_pde_parab_1d_keller_ode (d03pkc) integrates the system of first-order PDEs and coupled ODEs
In the PDE part of the problem given by
(1), the functions
must have the general form
where
,
and
depend on
and
.
The vector
is the set of PDE solution values
and the vector
is the partial derivative with respect to
. The vector
is the set of ODE solution values
and
denotes its derivative with respect to time.
In the ODE part given by
(2),
represents a vector of
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.
,
and
are the functions
,
and
evaluated at these coupling points. Each
may only depend linearly on time derivatives. Hence equation
(2) may be written more precisely as
where
,
is a vector of length
nv,
is an
nv by
nv matrix,
is an
nv by
matrix. The entries in
,
and
may depend on
,
,
,
and
. In practice you only need to supply a vector of information to define the ODEs and not the matrices
and
. (See
Section 5 for the specification of
odedef.)
The integration in time is from to , over the space interval , where and are the leftmost and rightmost points of a user-defined mesh .
The PDE system which is defined by the functions
must be specified in
pdedef.
The initial values of the functions and must be given at .
For a first-order system of PDEs, only one boundary condition is required for each PDE component
. The
npde boundary conditions are separated into
at the left-hand boundary
, and
at the right-hand boundary
, such that
. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of
at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for
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:
at the left-hand boundary, and
at the right-hand boundary.
Note that the functions
and
must not depend on
, since spatial derivatives are not determined explicitly in the Keller box scheme. 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
and
must be linear with respect to time derivatives, so that the boundary conditions have the general form:
at the left-hand boundary, and
at the right-hand boundary, where
,
,
,
,
and
depend on
and
only.
The boundary conditions must be specified in
bndary.
The problem is subject to the following restrictions:
(i) |
, and must not depend on any time derivatives; |
(ii) |
, so that integration is in the forward direction; |
(iii) |
The evaluation of the function is done approximately at the mid-points of the mesh
, for , 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 ; |
(iv) |
At least one of the functions 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
must be specified in
odedef. You must also specify the coupling points
in the array
xi.
The parabolic equations are approximated by a system of ODEs in time for the values of
at mesh points. 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
at each mesh point. In total there are
ODEs in time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.
4
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
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
5
Arguments
- 1:
– IntegerInput
-
On entry: the number of PDEs to be solved.
Constraint:
.
- 2:
– double *Input/Output
-
On entry: the initial value of the independent variable .
Constraint:
.
On exit: the value of
corresponding to the solution in
u. Normally
.
- 3:
– doubleInput
-
On entry: the final value of to which the integration is to be carried out.
- 4:
– function, supplied by the userExternal Function
-
pdedef must evaluate the functions
which define the system of PDEs.
pdedef is called approximately midway between each pair of mesh points in turn by
nag_pde_parab_1d_keller_ode (d03pkc).
The specification of
pdedef is:
void |
pdedef (Integer npde,
double t,
double x,
const double u[],
const double ut[],
const double ux[],
Integer nv,
const double v[],
const double vdot[],
double res[],
Integer *ires,
Nag_Comm *comm)
|
|
- 1:
– IntegerInput
-
On entry: the number of PDEs in the system.
- 2:
– doubleInput
-
On entry: the current value of the independent variable .
- 3:
– doubleInput
-
On entry: the current value of the space variable .
- 4:
– const doubleInput
-
On entry: contains the value of the component , for .
- 5:
– const doubleInput
-
On entry: contains the value of the component , for .
- 6:
– const doubleInput
-
On entry: contains the value of the component , for .
- 7:
– IntegerInput
-
On entry: the number of coupled ODEs in the system.
- 8:
– const doubleInput
-
On entry: if , contains the value of the component , for .
- 9:
– const doubleInput
-
On entry: if , contains the value of component , for .
- 10:
– doubleOutput
-
On exit:
must contain the
th component of
, for
, where
is defined as
i.e., only terms depending explicitly on time derivatives, or
i.e., all terms in equation
(3).
The definition of
is determined by the input value of
ires.
- 11:
– Integer *Input/Output
-
On entry: the form of
that must be returned in the array
res.
- Equation (9) must be used.
- Equation (10) must be used.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions, as described below:
- Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
- Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set when a physically meaningless input or output value has been generated. If you consecutively set , nag_pde_parab_1d_keller_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
- 12:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
pdedef.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_pde_parab_1d_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by
pdedef when called from
nag_pde_parab_1d_keller_ode (d03pkc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_pde_parab_1d_keller_ode (d03pkc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_keller_ode (d03pkc) is likely to produce unexpected results.
- 5:
– function, supplied by the userExternal Function
-
bndary must evaluate the functions
and
which describe the boundary conditions, as given in
(5) and
(6).
The specification of
bndary is:
void |
bndary (Integer npde,
double t,
Integer ibnd,
Integer nobc,
const double u[],
const double ut[],
Integer nv,
const double v[],
const double vdot[],
double res[],
Integer *ires,
Nag_Comm *comm)
|
|
- 1:
– IntegerInput
-
On entry: the number of PDEs in the system.
- 2:
– doubleInput
-
On entry: the current value of the independent variable .
- 3:
– IntegerInput
-
On entry: specifies which boundary conditions are to be evaluated.
- bndary must compute the left-hand boundary condition at .
- bndary must compute the right-hand boundary condition at .
- 4:
– IntegerInput
-
On entry: specifies the number of boundary conditions at the boundary specified by
ibnd.
- 5:
– const doubleInput
-
On entry:
contains the value of the component
at the boundary specified by
ibnd, for
.
- 6:
– const doubleInput
-
On entry:
contains the value of the component
at the boundary specified by
ibnd, for
.
- 7:
– IntegerInput
-
On entry: the number of coupled ODEs in the system.
- 8:
– const doubleInput
-
On entry: if , contains the value of the component , for .
- 9:
– const doubleInput
-
On entry: if
,
contains the value of component
, for
.
Note:
, for
, may only appear linearly as in
(7) and
(8).
- 10:
– doubleOutput
-
On exit:
must contain the
th component of
or
, depending on the value of
ibnd, for
, where
is defined as
i.e., only terms depending explicitly on time derivatives, or
i.e., all terms in equation
(7), and similarly for
.
The definitions of
and
are determined by the input value of
ires.
- 11:
– Integer *Input/Output
-
On entry: the form of
(or
) that must be returned in the array
res.
- Equation (11) must be used.
- Equation (12) must be used.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
- Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
- Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set when a physically meaningless input or output value has been generated. If you consecutively set , nag_pde_parab_1d_keller_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
- 12:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
bndary.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_pde_parab_1d_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by
bndary when called from
nag_pde_parab_1d_keller_ode (d03pkc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_pde_parab_1d_keller_ode (d03pkc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_keller_ode (d03pkc) is likely to produce unexpected results.
- 6:
– doubleInput/Output
-
On entry: the initial values of the dependent variables defined as follows:
-
contain , for and , and
-
contain , for .
On exit: the computed solution , for and , and
, for , evaluated at .
- 7:
– IntegerInput
-
On entry: the number of mesh points in the interval .
Constraint:
.
- 8:
– const doubleInput
-
On entry: the mesh points in the space direction. must specify the left-hand boundary, , and must specify the right-hand boundary, .
Constraint:
.
- 9:
– IntegerInput
-
On entry: the number of boundary conditions at the left-hand mesh point .
Constraint:
.
- 10:
– IntegerInput
-
On entry: the number of coupled ODE components.
Constraint:
.
- 11:
– function, supplied by the userExternal Function
-
odedef must evaluate the functions
, which define the system of ODEs, as given in
(4).
If
,
odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to
nag_pde_parab_1d_keller_ode (d03pkc).
The specification of
odedef is:
void |
odedef (Integer npde,
double t,
Integer nv,
const double v[],
const double vdot[],
Integer nxi,
const double xi[],
const double ucp[],
const double ucpx[],
const double ucpt[],
double r[],
Integer *ires,
Nag_Comm *comm)
|
|
- 1:
– IntegerInput
-
On entry: the number of PDEs in the system.
- 2:
– doubleInput
-
On entry: the current value of the independent variable .
- 3:
– IntegerInput
-
On entry: the number of coupled ODEs in the system.
- 4:
– const doubleInput
-
On entry: if , contains the value of the component , for .
- 5:
– const doubleInput
-
On entry: if , contains the value of component , for .
- 6:
– IntegerInput
-
On entry: the number of ODE/PDE coupling points.
- 7:
– const doubleInput
-
On entry: if , contains the ODE/PDE coupling points, , for .
- 8:
– const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: if , contains the value of at the coupling point , for and .
- 9:
– const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: if , contains the value of at the coupling point , for and .
- 10:
– const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: if , contains the value of at the coupling point , for and .
- 11:
– doubleOutput
-
On exit: if
,
must contain the
th component of
, for
, where
is defined as
i.e., only terms depending explicitly on time derivatives, or
i.e., all terms in equation
(4). The definition of
is determined by the input value of
ires.
- 12:
– Integer *Input/Output
-
On entry: the form of
that must be returned in the array
r.
- Equation (13) must be used.
- Equation (14) must be used.
On exit: should usually remain unchanged. However, you may reset
ires to force the integration function to take certain actions, as described below:
- Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to NE_USER_STOP.
- Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set when a physically meaningless input or output value has been generated. If you consecutively set , nag_pde_parab_1d_keller_ode (d03pkc) returns to the calling function with the error indicator set to NE_FAILED_DERIV.
- 13:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
odedef.
- user – double *
- iuser – Integer *
- p – Pointer
The type Pointer will be
void *. Before calling
nag_pde_parab_1d_keller_ode (d03pkc) you may allocate memory and initialize these pointers with various quantities for use by
odedef when called from
nag_pde_parab_1d_keller_ode (d03pkc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by
nag_pde_parab_1d_keller_ode (d03pkc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_keller_ode (d03pkc) is likely to produce unexpected results.
- 12:
– IntegerInput
-
On entry: the number of ODE/PDE coupling points.
Constraints:
- if , ;
- if , .
- 13:
– const doubleInput
-
On entry: , for , must be set to the ODE/PDE coupling points, .
Constraint:
.
- 14:
– IntegerInput
-
On entry: the number of ODEs in the time direction.
Constraint:
.
- 15:
– const doubleInput
-
Note: the dimension,
dim, of the array
rtol
must be at least
- when or ;
- when or .
On entry: the relative local error tolerance.
Constraint:
for all relevant .
- 16:
– const doubleInput
-
Note: the dimension,
dim, of the array
atol
must be at least
- when or ;
- when or .
On entry: the absolute local error tolerance.
Constraint:
for all relevant
.
Note: corresponding elements of
rtol and
atol cannot both be
.
- 17:
– IntegerInput
-
On entry: a value to indicate the form of the local error test.
itol indicates to
nag_pde_parab_1d_keller_ode (d03pkc) whether to interpret either or both of
rtol or
atol as a vector or scalar. The error test to be satisfied is
, where
is defined as follows:
itol | rtol | atol | |
1 | scalar | scalar | |
2 | scalar | vector | |
3 | vector | scalar | |
4 | vector | vector | |
In the above, denotes the estimated local error for the th component of the coupled PDE/ODE system in time, , for .
The choice of norm used is defined by the argument
norm.
Constraint:
.
- 18:
– Nag_NormTypeInput
-
On entry: the type of norm to be used.
- Maximum norm.
- Averaged norm.
If
denotes the norm of the vector
u of length
neqn, then for the averaged
norm
while for the maximum norm
See the description of
itol for the formulation of the weight vector
.
Constraint:
or .
- 19:
– Nag_LinAlgOptionInput
-
On entry: the type of matrix algebra required.
- Full matrix methods to be used.
- Banded matrix methods to be used.
- Sparse matrix methods to be used.
Constraint:
,
or
.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., ).
- 20:
– const doubleInput
-
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options,
should be set to
. 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:
- Selects the ODE integration method to be used. If , a BDF method is used and if , a Theta method is used. The default value is .
If , then
, for , are not used.
- Specifies the maximum order of the BDF integration formula to be used. may be , , , or . The default value is .
- Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If a modified Newton iteration is used and if 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 .
- 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
, for , for some or when there is no dependence in the coupled ODE system. If , the Petzold test is used. If , the Petzold test is not used. The default value is .
If ,
, for , are not used.
- Specifies the value of Theta to be used in the Theta integration method. . The default value is .
- Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If , a modified Newton iteration is used and if , a functional iteration method is used. The default value is .
- 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 , switching is allowed and if , switching is not allowed. The default value is .
- Specifies a point in the time direction, , beyond which integration must not be attempted. The use of is described under the argument itask. If , a value of , for , say, should be specified even if itask subsequently specifies that will not be used.
- Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, should be set to .
- Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, should be set to .
- Specifies the initial step size to be attempted by the integrator. If , the initial step size is calculated internally.
- Specifies the maximum number of steps to be attempted by the integrator in any one call. If , no limit is imposed.
- Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of , , and . If , a modified Newton iteration is used and if , functional iteration is used. The default value is .
and are used only for the sparse matrix algebra option, i.e., .
- Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range , with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing towards may help, but at the cost of increased fill-in. The default value is .
- Used as a relative pivot threshold during subsequent Jacobian decompositions (see ) below which an internal error is invoked. must be greater than zero, otherwise the default value is used. If is greater than 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 ). The default value is .
- 21:
– doubleCommunication Array
-
If
,
rsave need not be set on entry.
If
,
rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
- 22:
– IntegerInput
-
On entry: the dimension of the array
rsave.
Its size depends on the type of matrix algebra selected.
If , .
If , .
If , .
Where
| and are the lower and upper half bandwidths given by such that , for problems involving PDEs only; or , for coupled PDE/ODE problems. |
| |
| |
Note: when using the sparse option, the value of
lrsave may be too small when supplied to the integrator. An estimate of the minimum size of
lrsave is printed on the current error message unit if
and the function returns with
NE_INT_2.
- 23:
– IntegerCommunication Array
-
If
,
isave need not be set.
If
,
isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular the following components of the array
isave concern the efficiency of the integration:
- Contains the number of steps taken in time.
- Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves evaluating the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
- Contains the number of Jacobian evaluations performed by the time integrator.
- Contains the order of the ODE method last used in the time integration.
- Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the decomposition of the Jacobian matrix.
- 24:
– IntegerInput
-
On entry: the dimension of the array
isave. Its size depends on the type of matrix algebra selected:
- if , ;
- if , ;
- if , .
Note: when using the sparse option, the value of
lisave may be too small when supplied to the integrator. An estimate of the minimum size of
lisave is printed if
and the function returns with
NE_INT_2.
- 25:
– IntegerInput
-
On entry: the task to be performed by the ODE integrator.
- Normal computation of output values u at (by overshooting and interpolating).
- Take one step in the time direction and return.
- Stop at first internal integration point at or beyond .
- Normal computation of output values u at but without overshooting where is described under the argument algopt.
- Take one step in the time direction and return, without passing , where is described under the argument algopt.
Constraint:
, , , or .
- 26:
– IntegerInput
-
On entry: the level of trace information required from
nag_pde_parab_1d_keller_ode (d03pkc) and the underlying ODE solver as follows:
- No output is generated.
- Only warning messages from the PDE solver are printed.
- 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.
- Output from the underlying ODE solver is similar to that produced when , except that the advisory messages are given in greater detail.
- Output from the underlying ODE solver is similar to that produced when , except that the advisory messages are given in greater detail.
- 27:
– const char *Input
-
On entry: the name of a file to which diagnostic output will be directed. If
outfile is
NULL the diagnostic output will be directed to standard output.
- 28:
– Integer *Input/Output
-
On entry: indicates whether this is a continuation call or a new integration.
- Starts or restarts the integration in time.
- Continues the integration after an earlier exit from the function. In this case, only the arguments tout and fail should be reset between calls to nag_pde_parab_1d_keller_ode (d03pkc).
Constraint:
or .
On exit: .
- 29:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 30:
– Nag_D03_Save *Communication Structure
-
saved must remain unchanged following a previous call to a
Chapter d03 function and prior to any subsequent call to a
Chapter d03 function.
- 31:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ACC_IN_DOUBT
-
Integration completed, but small changes in
atol or
rtol are unlikely to result in a changed solution.
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_FAILED_DERIV
-
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting
in
pdedef or
bndary.
- NE_FAILED_START
-
atol and
rtol were too small to start integration.
- NE_FAILED_STEP
-
Error during Jacobian formulation for ODE system. Increase
itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as
ts:
.
Underlying ODE solver cannot make further progress from the point
ts with the supplied values of
atol and
rtol.
.
- NE_INT
-
ires set to an invalid value in call to
pdedef,
bndary, or
odedef.
On entry, .
Constraint: or .
On entry, .
Constraint: , , , or .
On entry, .
Constraint: , , or .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, on initial entry .
Constraint: on initial entry .
- NE_INT_2
-
On entry, corresponding elements and are both zero: and .
On entry,
lisave is too small:
. Minimum possible dimension:
.
On entry,
lrsave is too small:
. Minimum possible dimension:
.
On entry, ,
.
Constraint: .
On entry, and .
Constraint: when .
On entry, and .
Constraint: when .
When using the sparse option
lisave or
lrsave is too small:
,
.
- NE_INT_4
-
On entry, , , and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Serious error in internal call to an auxiliary. Increase
itrace for further details.
- NE_ITER_FAIL
-
In solving ODE system, the maximum number of steps has been exceeded. .
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_NOT_CLOSE_FILE
-
Cannot close file .
- NE_NOT_STRICTLY_INCREASING
-
On entry, , and .
Constraint: .
On entry, mesh points
x badly ordered:
,
,
and
.
- NE_NOT_WRITE_FILE
-
Cannot open file for writing.
- NE_REAL_2
-
On entry, at least one point in
xi lies outside
:
and
.
On entry, and .
Constraint: .
On entry, is too small:
and .
- NE_REAL_ARRAY
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_SING_JAC
-
Singular Jacobian of ODE system. Check problem formulation.
- NE_USER_STOP
-
In evaluating residual of ODE system,
has been set in
pdedef,
bndary, or
odedef. Integration is successful as far as
ts:
.
- NE_ZERO_WTS
-
Zero error weights encountered during time integration.
7
Accuracy
nag_pde_parab_1d_keller_ode (d03pkc) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy arguments,
atol and
rtol.
8
Parallelism and Performance
nag_pde_parab_1d_keller_ode (d03pkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_pde_parab_1d_keller_ode (d03pkc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The Keller box scheme can be used to solve higher-order problems which have been reduced to first-order by the introduction of new variables (see the example in
Section 10). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (see
nag_pde_parab_1d_fd (d03pcc) or
nag_pde_parab_1d_fd_ode (d03phc) for example), but at the expense of increased CPU time due to the larger number of function evaluations required.
It should be noted that the Keller box scheme, in common with other central-difference schemes, may be unsuitable for some hyperbolic first-order problems such as the apparently simple linear advection equation
, where
is a constant, resulting in spurious oscillations due to the lack of dissipation. This type of problem requires a discretization scheme with upwind weighting (
nag_pde_parab_1d_cd_ode (d03plc) for example), or the addition of a second-order artificial dissipation term.
The time taken depends on the complexity of the system and on the accuracy requested. For a given system and a fixed accuracy it is approximately proportional to
neqn.
10
Example
This example provides a simple coupled system of two PDEs and one ODE.
for
, for
. The left boundary condition at
is
and the right boundary condition at
is
The initial conditions at
are defined by the exact solution:
and the coupling point is at
.
This problem is exactly the same as the
nag_pde_parab_1d_fd_ode (d03phc) example problem, but reduced to first-order by the introduction of a second PDE variable (as mentioned in
Section 9).
10.1
Program Text
Program Text (d03pkce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (d03pkce.r)