A p-multigrid discontinuous Galerkin method for the Euler equations on unstructured grids

Abstract

A $\rho$-multigrid ($\rho$ = polynomial degree) discontinuous Galerkin method is presented for the solution of the compressible Euler equations on unstructured grids. The method operates on a sequence of solution approximations of different polynomial orders. A distinct feature of this $\rho$-multigrid method is to use different time integration schemes on different approximation levels, resulting in an accurate, fast, and low memory method that can be used to accelerate the convergence of the Euler equations to a steady state for discontinuous Galerkin methods. The developed method is used to compute the compressible flows for a variety of test problems on unstructured grids. The numerical results obtained strongly indicate the order independent property of this $\rho$-multigrid method. An overall speed-up factor more than one order of magnitude for both second- and third-order solutions of all test cases in comparison with the explicit method is demonstrated.

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