d03pjc integrates a system of linear or nonlinear parabolic partial differential equations (PDEs), in one space variable with scope for coupled ordinary differential equations (ODEs). The spatial discretization is performed using a Chebyshev collocation method, 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).
The function may be called by the names: d03pjc, nag_pde_dim1_parab_dae_coll or nag_pde_parab_1d_coll_ode.
3Description
d03pjc integrates the system of parabolic-elliptic equations and coupled ODEs
(1)
(2)
where (1) defines the PDE part and (2) generalizes the coupled ODE part of the problem.
In (1), and depend on , , , , and ; depends on , , , , and linearly on . The vector is the set of PDE solution values
and the vector is the partial derivative with respect to . Note that , and must not depend on . The vector is the set of ODE solution values
and denotes its derivative with respect to time.
In (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 the equation (2) may be written more precisely as
(3)
where , is a vector of length nv, is an nv by nv matrix, is an nv by matrix and the entries in , and may depend on , , , and . In practice you need only 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 of a user-defined set of break-points . The coordinate system in space is defined by the value of ; for Cartesian coordinates, for cylindrical polar coordinates and for spherical polar coordinates.
The PDE system which is defined by the functions , and must be specified in pdedef.
The initial values of the functions and must be given at . These values are calculated in uvinit.
The functions which may be thought of as fluxes, are also used in the definition of the boundary conditions. The boundary conditions must have the form
(4)
where or . The functions may only depend linearly on .
The boundary conditions must be specified in bndary.
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. Thus, the problem is subject to the following restrictions:
(i)in (1), , for , may only appear linearly in the functions
, for , with a similar restriction for ;
(ii) and the flux must not depend on any time derivatives;
(iii), so that integration is in the forward direction;
(iv)the evaluation of the functions , and is done at both the break-points and internally selected points for each element in turn, that is , and are evaluated twice at each break-point. Any discontinuities in these functions must, therefore, be at one or more of the mesh points;
(v)at least one of the functions must be nonzero so that there is a time derivative present in the PDE problem;
(vi)if and , which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done either by specifying the solution at or by specifying a zero flux there, that is and .
The parabolic equations are approximated by a system of ODEs in time for the values of at the mesh points. This ODE system is obtained by approximating the PDE solution between each pair of break-points by a Chebyshev polynomial of degree npoly. The interval between each pair of break-points is treated by d03pjc as an element, and on this element, a polynomial and its space and time derivatives are made to satisfy the system of PDEs at spatial points, which are chosen internally by the code and the break-points. The user-defined break-points and the internally selected points together define the mesh. The smallest value that npoly can take is one, in which case, the solution is approximated by piecewise linear polynomials between consecutive break-points and the method is similar to an ordinary finite element method.
In total there are mesh points in the spatial direction, and ODEs in the time direction; one ODE at each break-point for each PDE component, ODEs for each PDE component between each pair of break-points, and nv coupled ODEs. The system is then integrated forwards in time using a Backward Differentiation Formula (BDF) method or a Theta method.
4References
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 and Dew P M (1991) Algorithm 690: Chebyshev polynomial software for elliptic-parabolic systems of PDEs ACM Trans. Math. Software17 178–206
Berzins M, Dew P M and Furzeland R M (1988) Software tools for time-dependent equations in simulation and optimization of large systems Proc. IMA Conf. Simulation and Optimization (ed A J Osiadcz) 35–50 Clarendon Press, Oxford
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
Zaturska N B, Drazin P G and Banks W H H (1988) On the flow of a viscous fluid driven along a channel by a suction at porous walls Fluid Dynamics Research4
5Arguments
1: – IntegerInput
On entry: the number of PDEs to be solved.
Constraint:
.
2: – IntegerInput
On entry: the coordinate system used:
Indicates Cartesian coordinates.
Indicates cylindrical polar coordinates.
Indicates spherical polar coordinates.
Constraint:
, or .
3: – double *Input/Output
On entry: the initial value of the independent variable .
On exit: the value of corresponding to the solution values in u. Normally .
Constraint:
.
4: – doubleInput
On entry: the final value of to which the integration is to be carried out.
5: – function, supplied by the userExternal Function
pdedef must compute the functions , and which define the system of PDEs. The functions may depend on , , , and ; may depend linearly on . The functions must be evaluated at a set of points.
On entry: the current value of the independent variable .
3: – const doubleInput
On entry: contains a set of mesh points at which , and are to be evaluated. and contain successive user-supplied break-points and the elements of the array will satisfy .
4: – IntegerInput
On entry: the number of points at which evaluations are required (the value of ).
5: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: contains the value of the component where , for and .
6: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: contains the value of the component where , for and .
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 .
where appears in this document, it refers to the array element .
On exit: must be set to the value of where , for , and .
11: – doubleOutput
Note: the th element of the matrix is stored in .
On exit: must be set to the value of where , for and .
12: – doubleOutput
Note: the th element of the matrix is stored in .
On exit: must be set to the value of where , for and .
13: – Integer *Input/Output
On entry: set to or .
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 , d03pjc returns to the calling function with the error indicator set to NE_FAILED_DERIV.
14: – 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 d03pjc you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03pjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjc. If your code inadvertently does return any NaNs or infinities, d03pjc is likely to produce unexpected results.
6: – function, supplied by the userExternal Function
bndary must compute the functions and which define the boundary conditions as in equation (4).
On entry: the current value of the independent variable .
3: – const doubleInput
On entry: contains the value of the component at the boundary specified by ibnd, for .
4: – const doubleInput
On entry: contains the value of the component at the boundary specified by ibnd, for .
5: – IntegerInput
On entry: the number of coupled ODEs in the system.
6: – const doubleInput
On entry: if , contains the value of the component , for .
7: – const doubleInput
On entry: if , contains the value of component , for .
Note:
, for , may only appear linearly in
, for .
8: – IntegerInput
On entry: specifies which boundary conditions are to be evaluated.
bndary must set up the coefficients of the left-hand boundary, .
bndary must set up the coefficients of the right-hand boundary, .
9: – doubleOutput
On exit: must be set to the value of at the boundary specified by ibnd, for .
10: – doubleOutput
On exit: must be set to the value of at the boundary specified by ibnd, for .
11: – Integer *Input/Output
On entry: set to or .
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 , d03pjc 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 d03pjc you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03pjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjc. If your code inadvertently does return any NaNs or infinities, d03pjc is likely to produce unexpected results.
7: – doubleInput/Output
On entry: if the value of u must be unchanged from the previous call.
On exit: the computed solution
, for and , and
, for , evaluated at , as follows:
contain , for and , and
contain , for .
8: – IntegerInput
On entry: the number of break-points in the interval .
Constraint:
.
9: – const doubleInput
On entry: the values of the break-points in the space direction. must specify the left-hand boundary, , and must specify the right-hand boundary, .
Constraint:
.
10: – IntegerInput
On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break-points.
Constraint:
.
11: – IntegerInput
On entry: the number of mesh points in the interval .
Constraint:
.
12: – doubleOutput
On exit: the mesh points chosen by d03pjc in the spatial direction. The values of x will satisfy .
13: – IntegerInput
On entry: the number of coupled ODE components.
Constraint:
.
14: – function, supplied by the userExternal Function
odedef must evaluate the functions , which define the system of ODEs, as given in (3).
odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to d03pjc.
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 , d03pjc returns to the calling function with the error indicator set to NE_FAILED_DERIV.
15: – 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 d03pjc you may allocate memory and initialize these pointers with various quantities for use by odedef when called from d03pjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjc. If your code inadvertently does return any NaNs or infinities, d03pjc is likely to produce unexpected results.
15: – IntegerInput
On entry: the number of ODE/PDE coupling points.
Constraints:
if , ;
if , .
16: – const doubleInput
On entry: , for , must be set to the ODE/PDE coupling points.
Constraint:
.
17: – IntegerInput
On entry: the number of ODEs in the time direction.
Constraint:
.
18: – function, supplied by the userExternal Function
uvinit must compute the initial values of the PDE and the ODE components
, for and , and
, for .
On entry: the number of mesh points in the interval .
3: – const doubleInput
On entry: , for , contains the current values of the space variable .
4: – doubleOutput
Note: the th element of the matrix is stored in .
On exit: contains the value of the component , for and .
5: – IntegerInput
On entry: the number of coupled ODEs in the system.
6: – doubleOutput
On exit: contains the value of component , for .
7: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to uvinit.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling d03pjc you may allocate memory and initialize these pointers with various quantities for use by uvinit when called from d03pjc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note:uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjc. If your code inadvertently does return any NaNs or infinities, d03pjc is likely to produce unexpected results.
19: – 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 .
20: – 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 .
21: – IntegerInput
On entry: a value to indicate the form of the local error test. itol indicates to d03pjc 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:
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:
.
22: – 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 .
23: – 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., ).
24: – 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 ,
, 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, .
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 .
Is used as a relative pivot threshold during subsequent Jacobian decompositions (see ) below which an internal error is invoked. 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 .
If , rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
26: – IntegerInput
On entry: the dimension of the array rsave.
Its size depends on the type of matrix algebra selected.
If , .
If , .
If , .
Where is the lower or upper half bandwidths such that
for PDE problems only (no coupled ODEs),
for coupled PDE/ODE problems,
Where is defined by
if ,
if ,
if ,
Where is defined by
if the BDF method is used,
if the Theta method is used,
Note: when , 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.
If , isave must be unchanged from the previous call to the function because it contains required information about the iteration required for subsequent calls. In particular:
Contains the number of steps taken in time.
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing 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.
28: – 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.
29: – IntegerInput
On entry: specifies the task to be performed by the ODE integrator.
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 .
30: – IntegerInput
On entry: the level of trace information required from d03pjc and the underlying ODE solver. itrace may take the value , , , or .
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.
If , is assumed and similarly if , is assumed.
The advisory messages are given in greater detail as itrace increases.
31: – 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.
32: – 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 argument tout should be reset between calls to d03pjc.
Constraint:
or .
On exit: .
33: – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
34: – 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.
35: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ACC_IN_DOUBT
Integration completed, but small changes in atol or rtol are unlikely to result in a changed solution.
The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when or .)
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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:
.
In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as . The problem may have a singularity, or the error requirement may be inappropriate.
Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol.
.
On entry, on initial entry .
Constraint: on initial entry .
NE_INT_2
On entry, and .
Constraint: corresponding elements and cannot both be .
On entry, .
Constraint: .
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_3
On entry, , and .
Constraint: .
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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_CLOSE_FILE
Cannot close file .
NE_NOT_STRICTLY_INCREASING
On entry, ,
, and .
Constraint: .
On entry, , and .
Constraint: .
NE_NOT_WRITE_FILE
Cannot open file for writing.
NE_REAL
On entry, .
Constraint: , or .
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_TIME_DERIV_DEP
Flux function appears to depend on time derivatives.
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.
Some error weights became zero during the time integration (see the description of itol). Pure relative error control () was requested on a variable (the th) which has become zero. The integration was successful as far as .
7Accuracy
d03pjc 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 argument atol and rtol.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d03pjc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pjc 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.
9Further Comments
The argument specification allows you to include equations with only first-order derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem.
The time taken depends on the complexity of the parabolic system and on the accuracy requested.
10Example
This example provides a simple coupled system of one PDE and one ODE.
for .
The left boundary condition at is
The right boundary condition at is
The initial conditions at are defined by the exact solution: