2020/11/30 by Shinhoo Kang, Tan Bui–Thanh, Tan Bui-Thanh
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Burgers' equation #Computational Fluid Dynamics and Aerodynamics #Computer science #Conservation law #Differential algebraic equation #Differential equation #Discontinuous Galerkin method #Discretization #Euler's formula #Exponential function #Exponential integrator #Finite element method #Integrator #Krylov subspace #Linear system #Mathematical analysis #Mathematics #Nonlinear system #Numerical methods for differential equations #Ordinary differential equation #Partial differential equation #Preconditioner #cs.NA #math-ph #math.MP #math.NA #msc:65M60 #msc:65Y05 #msc:76M10 #physics.flu-dyn
paper · pdf · doi:10.1016/j.cma.2021.114031
39 pages, 14 figures and 14 tables
arxiv created 2021/04/17 · openalex publication_date 2021/07/29 · arxiv updated 2021/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr > 1); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present detailed stability and convergence analyses for the exponential Euler DG scheme.