In this paper, we address the full discretization of Friedrichs' systems with a two-field structure, such as Maxwell's equations or the acoustic wave equation in div-grad form, cf. [14]. We focus on a discontinuous Galerkin space discretization applied to a locally refined mesh or a small region with high wave speed. This results in a stiff system of ordinary differential equations, where the stiffness is mainly caused by a small region of the spatial mesh. When using explicit time-integration schemes, the time step size is severely restricted by a few spatial elements, leading to a loss of efficiency. As a remedy, we propose and analyze a general leapfrog-based scheme which is motivated by [5]. The new, fully explicit, local time-integration method filters the stiff part of the system in such a way that its CFL condition is significantly weaker than that of the leapfrog scheme while its computational cost is only slightly larger. For this scheme, the filter function is a suitably scaled and shifted Chebyshev polynomial. While our main interest is in explicit local-time stepping schemes, the filter functions can be much more general, for instance, a certain rational function leads to the locally implicit method, proposed and analyzed in [24]. Our analysis provides sufficient conditions on the filter function to ensure full order of convergence in space and second order in time for the whole class of local time-integration schemes.
Friedrichs' systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions
Explicit Time Integration of Transient Eddy Current Problems
University of Wuppertal（伍珀塔尔大学）；Technische Universität Darmstadt（达姆施塔特工业大学）；Hochschule für angewandte Wissenschaften Würzburg-Schweinfurt（维尔茨堡-施韦因富特应用科学大学）
In a recent paper (see [7]), a quasi-nonlocal coupling method was introduced to seamlessly bridge a nonlocal diffusion model with the classical local diffusion counterpart in a one-dimensional space. The proposed coupling framework removes inconsistency on the interface, preserves the balance of fluxes, and satisfies the maximum principle of diffusion problem. However, the numerical scheme proposed in that paper does not maintain all of these properties on a discrete level. In this paper we resolve these issues by proposing a new finite difference scheme that ensures the balance of fluxes and …
A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations
Universität zu Köln（科隆大学）；Imperial College London（帝国理工学院）；Instituto Tecnológico de Aeronáutica (ITA)（巴西航空技术学院）；California Institute of Technology（加州理工学院）
A local approach to the time integration of PDEs by exponential methods is proposed, motivated by theoretical estimates by A.Iserles on the decay of off-diagonal terms in the exponentials of sparse matrices. An overlapping domain decomposition technique is outlined, that allows to replace the computation of a global exponential matrix by a number of independent and easily parallelizable local problems. Advantages and potential problems of the proposed technique are discussed. Numerical experiments on simple, yet relevant model problems show that the resulting method allows to increase computat…
