Optimal Runge–Kutta smoothers for the p-multigrid discontinuous Galerkin solution of the 1D Euler equations

• Published in: Journal of computational physics Vol. 230; no. 11; pp. 4153 - 4175
• Main Authors: Bassi, F., Ghidoni, A., Rebay, S.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 20-05-2011
• Subjects: Algorithms; Damping; Euler equations; Explicit Runge–Kutta; Galerkin methods; High-order accurate discontinuous Galerkin; Mathematical models; Optimization; p-Multigrid; Runge-Kutta method; Steady state; High-order accurate discontinuous Galerkin; p-Multigrid; Explicit Runge–Kutta; Euler equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.04.030

This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a
p-multigrid solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the
h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.