Multigrid Optimization for Space-Time Discontinuous Galerkin Discretizations of Advection Dominated Flows

Abstract

The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two-and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit RungeKutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p-and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated.

hp-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows: Part I. Multilevel analysis

“…In particular, Runge-Kutta methods can be developed into efficient multigrid smoothers when they are used as pseudo-time integrators, which was originally proposed in [13], see also [20]. Since time-accuracy is not important in pseudo-time significant freedom is available to optimize Runge-Kutta smoothers for good multigrid performance [16,25,29].…”

“…For problems containing boundary layers line smoothers are generally necessary to deal with large aspect ratio meshes [8,26]. Explicit and (semi)-implicit time integration methods have also been used as smoothers [2,3,16,25]. In particular, Runge-Kutta methods can be developed into efficient multigrid smoothers when they are used as pseudo-time integrators, which was originally proposed in [13], see also [20].…”

“…Many of these studies considered the advection-diffusion equation or linearized versions of the compressible Euler equations. The main analysis tool to understand the performance of the multigrid algorithm has been single grid and two-level Fourier analysis [8,9,16,18,24,25,33]. For a more general discussion of these techniques we refer to [11,28,35,37].…”

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge–Kutta smoother

“…The remaining coefficients can be chosen such that the multigrid performance for a selected class of problems is optimal. In [7,12,15] we performed this optimization for explicit Runge-Kutta methods. In this section we will discuss the optimization of the semi-implicit Runge-Kutta smoother used in the hp-MGS multigrid algorithm.…”

Efficient p‐multigrid discontinuous Galerkin solver for complex viscous flows on stretched grids

“…In order to improve the computational efficiency of DG methods, a considerable research effort has been recently devoted to devise more efficient computational strategies for the construction of DG space discretization operators [6][7][8] and for the integration in time of the space discretized DG equations, by means of implicit time discretization schemes [9][10][11][12] and multigrid (MG) solution strategies [13][14][15][16][17][18][19][20][21][22][23][24][25], both in the h and p variants.…”