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A p-Multigrid Spectral Difference method with explicit and implicit smoothers on unstructured grids

The convergence of high-order methods, such as recently developed Spectral Difference (SD) method, can be accelerated using both implicit temporal advancement and a p-Multigrid approach. A good combination of these two can significantly improve the efficiency of the SD method for steady flow problems. In this paper, we demonstrate a p-Multigrid approach developed for a 2D Euler solver using the SD method. A fast preconditioned Lower-Upper Symmetric Gauss Seidel (LU-SGS) implicit method is developed and tested for both linear scalar, and nonlinear Euler equations. Meanwhile, a five-stage Runge-Kutta explicit method is employed for comparison. We are able to achieve significant speedup (up to two orders) using both p-Multigrid method and the implicit smoother while maintaining very stable convergence property and nearly ideal order of accuracy. The numerical results are very promising, and indicate that the approach has great potential for 3D flow problems.