NAG Library Function Document

nag_ode_bvp_ps_lin_solve (d02uec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_ode_bvp_ps_lin_solve (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>

#include <nagd02.h>

void	nag_ode_bvp_ps_lin_solve (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)

3 Description

nag_ode_bvp_ps_lin_solve (d02uec) solves the constant linear coefficient ordinary differential problem

\sum_{j = 0}^{m} f_{j + 1} \frac{d^{j} u}{d x^{j}} = f (x), x \in [a, b]

subject to a set of

m

linear constraints at points

y_{i} \in [a, b]

, for

i = 1, 2, \dots, m

\sum_{j = 0}^{m} B_{i, j + 1} {(\frac{d^{j} u}{{d x}^{j}})}_{(x = y_{i})} = β_{i},

where

1 \leq m \leq 4

B

is an

m \times (m + 1)

matrix of constant coefficients and

β_{i}

are constants. The points

y_{i}

are usually either

a

b

The function

f (x)

is supplied as an array of Chebyshev coefficients

c_{j}

j = 0, 1, \dots, n

for the function discretized on

n + 1

Chebyshev Gauss–Lobatto points (as returned by nag_ode_bvp_ps_lin_cgl_grid (d02ucc)); the coefficients are normally obtained by a previous call to nag_ode_bvp_ps_lin_coeffs (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 nag_ode_bvp_ps_lin_cgl_vals (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 nag_ode_bvp_ps_lin_grid_vals (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 – IntegerInput

On entry:

n

, where the number of grid points is

n + 1

Constraint:

n \geq 8

and n is even.

2: a – doubleInput

On entry:

a

, the lower bound of domain

[a, b]

Constraint:

a < b

3: b – doubleInput

On entry:

b

, the upper bound of domain

[a, b]

Constraint:

b > a

4: m – IntegerInput

On entry: the order,

m

, of the boundary value problem to be solved.

Constraint:

1 \leq m \leq 4

5: c[ $n + 1$ ] – const doubleInput

On entry: the Chebyshev coefficients

c_{j}

j = 0, 1, \dots, n

, for the right hand side of the boundary value problem. Usually these are obtained by a previous call of nag_ode_bvp_ps_lin_coeffs (d02uac).

6: bmat[ $m \times (m + 1)$ ] – doubleInput/Output

Note: the

(i, j)

th element of the matrix is stored in

bmat [(j - 1) \times m + i - 1]

On entry:

bmat [j \times m + i - 1]

must contain the coefficients

B_{i, j + 1}

, for

i = 1, 2, \dots, m

and

j = 0, 1, \dots, 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 doubleInput

On entry: the points,

y_{i}

, for

i = 1, 2, \dots, m

, where the boundary conditions are discretized.

8: bvec[m] – const doubleInput

On entry: the values,

β_{i}

, for

i = 1, 2, \dots, m

, in the formulation of the boundary conditions given in Section 3.

9: f[ $m + 1$ ] – doubleInput/Output

On entry: the coefficients,

f_{j}

, for

j = 1, 2, \dots, 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[ $(n + 1) \times (m + 1)$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

uc [(j - 1) \times (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} (|\sum_{j = 0}^{m} B_{i, j + 1} {(\frac{d^{j} u}{d x^{j}})}_{(x = y_{i})} - β_{i}|), \max_{i = 1, n + 1} (|\sum_{j = 0}^{m} f_{j + 1} {(\frac{d^{j} u}{{d x}^{j}})}_{(x = x_{i})} - f (x)|)\} .

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: During iterative refinement, convergence was achieved, but the residual is more than $100 \times machine precision$ . $Residual achieved on convergence = ⟨value⟩$ .
NE_INT: On entry, $m = ⟨value⟩$ .
Constraint: $1 \leq m \leq 4$ .
On entry, $n = ⟨value⟩$ .
Constraint: n is even.
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 8$ .
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.
Internal error while unpacking matrix during iterative refinement.
Please contact NAG.
NE_REAL_2: On entry, $a = ⟨value⟩$ and $b = ⟨value⟩$ .
Constraint: $a < b$ .
NE_REAL_ARRAY: On entry, $f [m] = 0.0$ .
NE_SINGULAR_MATRIX: Singular matrix encountered during iterative refinement.
Please check that your system is well posed.
NE_TOO_MANY_ITER: 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

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

nag_ode_bvp_ps_lin_solve (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 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 nag_ode_bvp_ps_lin_solve (d02uec) should be faster for larger discretizations.

10 Example

This example solves the third-order problem

4 U_{x x x} + 3 U_{x x} + 2 U_{x} + U = 2 \sin x - 2 \cos x

[- π / 2, π / 2]

subject to the boundary conditions

U [- π / 2] = 0

3 U_{x x} [- π / 2] + 2 U_{x} [- π / 2] + U [- π / 2] = 2

, and

3 U_{x x} [π / 2] + 2 U_{x} [π / 2] + U [π / 2] = - 2

using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid of order

16

NAG Library Function Documentnag_ode_bvp_ps_lin_solve (d02uec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_ode_bvp_ps_lin_solve (d02uec)