A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular 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 (p = polynomial degree) approach. A p-multigrid method is investigated in this paper for solving SD formulations of the scalar wave and Euler equations on unstructured grids. A fast preconditioned lower–upper symmetric Gauss–Seidel (LU-SGS) relaxation method is implemented as an iterative smoother. Meanwhile, a Runge–Kutta explicit method is employed for comparison. The multigrid method considered here is nonlinear and utilizes full approximation storage (FAS) [Ta’asan S. Multigrid one-shot methods and design strategy, Von Karman Institute Lecture Note, 1997 [28]] scheme. For some p-multigrid calculations, blending implicit and explicit smoothers for different p-levels is also studied. The p-multigrid method is firstly validated by solving both linear and nonlinear 2D wave equations. Then the same idea is extended to 2D nonlinear Euler equations. Generally speaking, we are able to achieve speedups of up to two orders using the p-multigrid method with the implicit smoother.