NAG CL Interface
d02uec (bvp_​ps_​lin_​solve)

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1 Purpose

d02uec finds the solution of a linear constant coefficient boundary value problem by using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid.

2 Specification

#include <nag.h>
void  d02uec (Integer n, double a, double b, Integer m, const double c[], double bmat[], const double y[], const double bvec[], double f[], double uc[], double *resid, NagError *fail)
The function may be called by the names: d02uec or nag_ode_bvp_ps_lin_solve.

3 Description

d02uec solves the constant linear coefficient ordinary differential problem
j=0 m fj+1 dju dxj = f(x) ,  x [a,b]  
subject to a set of m linear constraints at points yi[a,b] , for i=1,2,,m:
j=0 m B i,j+1 ( dju dxj ) (x=yi) = βi ,  
where 1m4, B is an m×(m+1) matrix of constant coefficients and βi are constants. The points yi are usually either a or b.
The function f(x) is supplied as an array of Chebyshev coefficients cj, j=0,1,,n for the function discretized on n+1 Chebyshev Gauss–Lobatto points (as returned by d02ucc); the coefficients are normally obtained by a previous call to d02uac. The solution and its derivatives (up to order m) are returned, in the form of their Chebyshev series representation, as arrays of Chebyshev coefficients; subsequent calls to d02ubc will return the corresponding function and derivative values at the Chebyshev Gauss–Lobatto discretization points on [a,b]. Function and derivative values can be obtained on any uniform grid over the same range [a,b] by calling the interpolation function d02uwc.

4 References

Clenshaw C W (1957) The numerical solution of linear differential equations in Chebyshev series Proc. Camb. Phil. Soc. 53 134–149
Coutsias E A, Hagstrom T and Torres D (1996) An efficient spectral method for ordinary differential equations with rational function coefficients Mathematics of Computation 65(214) 611–635
Greengard L (1991) Spectral integration and two-point boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80
Lundbladh A, Hennigson D S and Johannson A V (1992) An efficient spectral integration method for the solution of the Navier–Stokes equations Technical report FFA–TN 1992–28 Aeronautical Research Institute of Sweden
Muite B K (2010) A numerical comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems Journal of Computational and Applied Mathematics 234(2) 317–342

5 Arguments

1: n Integer Input
On entry: n, where the number of grid points is n+1.
Constraint: n8 and n is even.
2: a double Input
On entry: a, the lower bound of domain [a,b].
Constraint: a<b.
3: b double Input
On entry: b, the upper bound of domain [a,b].
Constraint: b>a.
4: m Integer Input
On entry: the order, m, of the boundary value problem to be solved.
Constraint: 1m4.
5: c[n+1] const double Input
On entry: the Chebyshev coefficients cj, j=0,1,,n, for the right-hand side of the boundary value problem. Usually these are obtained by a previous call of d02uac.
6: bmat[m×(m+1)] double Input/Output
Note: the (i,j)th element of the matrix is stored in bmat[(j-1)×m+i-1].
On entry: bmat[j×m+i-1] must contain the coefficients Bi,j+1, for i=1,2,,m and j=0,1,,m, in the problem formulation of Section 3.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain [−1,1].
7: y[m] const double Input
On entry: the points, yi, for i=1,2,,m, where the boundary conditions are discretized.
8: bvec[m] const double Input
On entry: the values, βi, for i=1,2,,m, in the formulation of the boundary conditions given in Section 3.
9: f[m+1] double Input/Output
On entry: the coefficients, fj, for j=1,2,,m+1, in the formulation of the linear boundary value problem given in Section 3. The highest order term, f[m], needs to be nonzero to have a well posed problem.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain [−1,1].
10: uc[(n+1)×(m+1)] double Output
Note: the (i,j)th element of the matrix is stored in uc[(j-1)×(n+1)+i-1].
On exit: the Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The coefficients of U are stored as the first n+1 elements of uc, the first derivative coefficients are stored as the next n+1 elements of uc, and so on.
11: resid double * Output
On exit: the maximum residual resulting from substituting the solution vectors returned in uc into both linear equations of Section 3 representing the linear boundary value problem and associated boundary conditions. That is
max { max i=1,m (| j=0 m B i,j+1 ( dju dxj ) (x=yi) - βi |) , max i=1, n+1 (| j=0 m f j+1 ( dju dxj ) (x=xi) - f(x) |) } .  
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
During iterative refinement, convergence was achieved, but the residual is more than 100×machine precision. Residual achieved on convergence=value.
On entry, m=value.
Constraint: 1m4.
On entry, n=value.
Constraint: n is even.
On entry, n=value.
Constraint: n8.
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.
Internal error while unpacking matrix during iterative refinement.
Please contact NAG.
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.
On entry, a=value and b=value.
Constraint: a<b.
On entry, f[m]=0.0.
Singular matrix encountered during iterative refinement.
Please check that your system is well posed.
During iterative refinement, the maximum number of iterations was reached.
Number of iterations=value and residual achieved=value.

7 Accuracy

The accuracy should be close to machine precision for well conditioned boundary value problems.

8 Parallelism and Performance

d02uec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02uec 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.

9 Further Comments

The number of operations is of the order n log(n) and the memory requirements are O(n); thus the computation remains efficient and practical for very fine discretizations (very large values of n). Collocation methods will be faster for small problems, but the method of d02uec should be faster for larger discretizations.

10 Example

This example solves the third-order problem 4Uxxx + 3Uxx + 2Ux + U = 2sinx - 2cosx on [-π/2,π/2] subject to the boundary conditions U [-π/2] = 0 , 3Uxx [-π/2] + 2Ux [-π/2] + U [-π/2] =2, and 3Uxx [π/2] + 2Ux [π/2] + U [π/2] =−2 using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid of order 16.

10.1 Program Text

Program Text (d02uece.c)

10.2 Program Data

Program Data (d02uece.d)

10.3 Program Results

Program Results (d02uece.r)