Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields  

作　　者：Dinshaw S.Balsara Roger Käppeli 

机构地区：[1]Physics and ACMS Departments,University of Notre Dame,Notre Dame,IN,USA [2]Seminar for Applied Mathematics(SAM),Department of Mathematics,ETH Zürich,CH-8092 Zurich,Switzerland

出　　处：《Communications on Applied Mathematics and Computation》2022年第3期945-985,共41页应用数学与计算数学学报（英文）

基　　金：Open Access funding provided by ETH Zurich.The funding has been acknowledged.DSB acknowledges support via NSF grants NSF-19-04774,NSF-AST-2009776 and NASA-2020-1241.

摘　　要：This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of

关 键 词：PDES Numerical schemes MIMETIC Discontinuous Galerkin 

分 类 号：O17[理学—数学]