# Multigrid Solution of the Steady Euler Equations

• P. W. Hemker
• S. P. Spekreijse
Chapter
Part of the Notes on Numerical Fluid Mechanics book series (NNFM, volume 11)

## Summary

A multigrid (MG) method for the approximation of steady solutions to the full 2-D Euler equations is described. The space discretization is obtained by the finite volume technique and Osher’s approximate Riemann-solver. Symmetric Gauss-Seidel relaxation is applied to solve the nonlinear discrete system of equations. A multigrid method, the full approximation scheme, accelerates this iterative process.

In a few two-dimensional testproblems, (subsonic, transsonic and supersonic) the multigrid iteration is applied to an initial estimate that was obtained by means of the FMG-technique (nested iteration). For the discretization on the different levels, a fully consistent sequence of nested discretizations is used. The prolongations and restrictions selected are in agreement with this consistency.

It turns out that the total amount of work required to obtain a solution, that is accurate upto truncation error, corresponds to a small number of nonlinear Gauss-Seidel iterations. In the case of transsonic flow the rate of convergence of the MG-iteration appears independent of N, i.e. the number of cells in the discretization.

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