Damped, direction-dependent multigrid for hypersonic flow computations

B. Koren, P.W. Hemker

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Abstract

A nonlinear multigrid technique with improved robustness is developed for the solution of the steady Euler equations. The system of nonlinear equations is discretized by an upwind finite volume method. Collective symmetric point Gauss-Seidel relaxation is applied as the standard smoothing technique. In case of failure of the point relaxation, a switch is made to a local evolution technique. The novel robustness improvements to the nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a global upwind restriction of the defect. The defect damping operator is derived from a two-grid convergence analysis. The upwind prolongation operator is made such that it is consistent with the upwind finite volume discretization. It makes efficient use of the P-variant of Osher's approximate Riemann solver. The upwind restriction operator is an approximate adjoint of the upwind prolongation operator. Satisfactory convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For all test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping.
Original languageEnglish
Pages (from-to)309-328
JournalApplied Numerical Mathematics
Volume7
Issue number4
DOIs
Publication statusPublished - 1991

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Hypersonic Flow
Hypersonic flow
Damped
Defects
Dependent
Damping
Prolongation
Multigrid Method
Reentry
Euler equations
Hypersonic aerodynamics
Finite volume method
Nonlinear equations
Operator
Restriction
Switches
Robustness
Riemann Solver
Smoothing Techniques
Gauss-Seidel

Cite this

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title = "Damped, direction-dependent multigrid for hypersonic flow computations",
abstract = "A nonlinear multigrid technique with improved robustness is developed for the solution of the steady Euler equations. The system of nonlinear equations is discretized by an upwind finite volume method. Collective symmetric point Gauss-Seidel relaxation is applied as the standard smoothing technique. In case of failure of the point relaxation, a switch is made to a local evolution technique. The novel robustness improvements to the nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a global upwind restriction of the defect. The defect damping operator is derived from a two-grid convergence analysis. The upwind prolongation operator is made such that it is consistent with the upwind finite volume discretization. It makes efficient use of the P-variant of Osher's approximate Riemann solver. The upwind restriction operator is an approximate adjoint of the upwind prolongation operator. Satisfactory convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For all test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping.",
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Damped, direction-dependent multigrid for hypersonic flow computations. / Koren, B.; Hemker, P.W.

In: Applied Numerical Mathematics, Vol. 7, No. 4, 1991, p. 309-328.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Hemker, P.W.

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AB - A nonlinear multigrid technique with improved robustness is developed for the solution of the steady Euler equations. The system of nonlinear equations is discretized by an upwind finite volume method. Collective symmetric point Gauss-Seidel relaxation is applied as the standard smoothing technique. In case of failure of the point relaxation, a switch is made to a local evolution technique. The novel robustness improvements to the nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a global upwind restriction of the defect. The defect damping operator is derived from a two-grid convergence analysis. The upwind prolongation operator is made such that it is consistent with the upwind finite volume discretization. It makes efficient use of the P-variant of Osher's approximate Riemann solver. The upwind restriction operator is an approximate adjoint of the upwind prolongation operator. Satisfactory convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For all test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping.

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