An improved iterative HDG approach for partial differential equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:367

Article

Muralikrishnan, Sriramkrishnan¹  Minh-Binh Tran³  Tan Bui-Thanh^1,2 

[1] Univ Texas Austin, Dept Aerosp Engn & Engn Mech, Austin, TX 78712 USA

[2] Univ Texas Austin, Inst Computat Engn & Sci, Austin, TX 78712 USA

[3] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

关键词: Iterative solvers;    Schwarz methods;    Hybridized Discontinuous Galerkin methods;    Transport equation;    Shallow water equation;    Convection-diffusion equation;   

DOI  :  10.1016/j.jcp.2018.04.033

来源: Elsevier

【 摘 要 】

We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work [45] in several directions: 1) the improved iHDG approach converges in a finite number of iterations for the scalar transport equation; 2) it is unconditionally convergent for both the linearized shallow water system and the convection-diffusion equation; 3) it has improved stability and convergence rates; 4) we uncover a relationship between the number of iterations and time stepsize, solution order, meshsize and the equation parameters. This allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize; and 5) we provide both strong and weak scalings of the improved iHDG approach up to 16,384 cores. A connection between iHDG and time integration methods such as parareal and implicit/explicit methods are discussed. Extensive numerical results for linear (and nonlinear) PDEs are presented to verify the theoretical findings. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free