NAG Library Routine Document

D02UEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D02UEF 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

SUBROUTINE D02UEF (

N, A, B, M, C, BMAT, Y, BVEC, F, UC, RESID, IFAIL)

INTEGER	N, M, IFAIL
REAL (KIND=nag_wp)	A, B, C(N+1), BMAT(M,M+1), Y(M), BVEC(M), F(M+1), UC(N+1,M+1), RESID

3 Description

D02UEF 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 D02UCF); the coefficients are normally obtained by a previous call to D02UAF. 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 D02UBF 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 routine D02UWF.

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 Parameters

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 – REAL (KIND=nag_wp)Input

On entry:

a

, the lower bound of domain

[a, b]

Constraint:

A < B

3: B – REAL (KIND=nag_wp)Input

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( $N + 1$ ) – REAL (KIND=nag_wp) arrayInput

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 D02UAF.

6: BMAT(M, $M + 1$ ) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

BMAT (i, j + 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) – REAL (KIND=nag_wp) arrayInput

On entry: the points,

y_{i}

, for

i = 1, 2, \dots, m

, where the boundary conditions are discretized.

8: BVEC(M) – REAL (KIND=nag_wp) arrayInput

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( $M + 1$ ) – REAL (KIND=nag_wp) arrayInput/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 + 1)

, 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( $N + 1$ , $M + 1$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The

n + 1

elements

UC (1 : N + 1, 1)

contain the coefficients representing the solution

U (x_{i})

, for

i = 0, 1, \dots, n

UC (1 : N + 1, j + 1)

contains the coefficients representing the

j

th derivative of

U

, for

j = 1, 2, \dots, m

11: RESID – REAL (KIND=nag_wp)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: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $N < 8$ or N is odd.

$IFAIL = 2$

On entry,

A \geq B

$IFAIL = 3$

On entry,

F (M + 1) = 0

$IFAIL = 6$

On entry,

M \neq 1

2

3

4

$IFAIL = 7$: Internal error unpacking a matrix, seek expert advice.

$IFAIL = 8$: Singular system when factoring a matrix, check your system is well posed, contact NAG.

$IFAIL = 9$: Warning, iterative refinement failed to converge within the maximum number of iterations (50).

$IFAIL = 10$: Warning, iterative refinement converged but did not achieve a residual close to machine precision.

$IFAIL = - 999$: Internal memory allocation failed.

7 Accuracy

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

8 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 D02UEF should be faster for larger discretizations.

9 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 Routine DocumentD02UEF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D02UEF