Gauss-Konrod Wave Form Relaxation

Allan Struthers

Outline

The purpose of this project is to implement and test "waveform" relaxation with relatively high-order Gauss-Kronrod Integration.  The advantage of Gauss-Kronrod over Gauss integration is in the shared points for refinement and I will test this refinement as an error estimation tool. The algorithm will be tested in local extrapolation as well as a correction mode.

Details

Explain and compute Gauss-Kronrod evaluation points and weights

Implement algorithm(s):  Low Accuracy Advance, High Accuracy Advance, and Incorporating the correction. If possible I will do this in Jacobi, Block Jacobi, and Gauss-Seidel like algorithms.

Points and weights from Mathematica

Explain interpolation versus no-interpolation

Examine accuracy/stability of algorithm(s):  Examine the accuracy of the fully converged iteration on a variety of problems, with a variety of computational intervals and a variety of discretization numbers.

Make sure that the error growth is appropriate for different accuracies.

Define and explore stability issues

Examine behavior for stiff problems.

Examine convergence of algorithm(s):  Examine the convergence behavior of the iteration on a variety of problems, with a variety of computational intervals and a variety of discretization numbers.

Examine the changes of convergence along the computational interval .

Examine behavior with different initial conditions.

Examine accelerating the convergence.

Examine behavior for stiff problems:  does it need good initial conditions.  does it converge much more slowly.

Further work could include modifying the computation to use Clenshaw-Curtis integration, lower precision arithmetic for the preliminary steps, composite panel versions of the algorithm etc..

References

Wolfram Mathematica: tutorial/NIntegrateIntegrationRules #617627612

Numerical recipes in C