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Von Neumann Stability Analysis of DG-like and PNPM-like Schemes for PDEs that have Globally Curl-Preserving Evolution of Vector Fields

by D. S. Balsara and R. Käppeli

(Report number 2021-44)

Abstract
This paper examines a class of 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, have to be collocated at the edges of the mesh. They are updated by using potentials that are collocated at the vertices of the mesh. The resulting schemes : 1) do not blow up even after very long integration times, 2) do not need any special cleaning treatment, 3) can operate with large explicit timesteps, 4) do not require the solution of an elliptic system and 5) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, so crucial to the design of a DG method, is now carried out in the edges of the mesh and yields a weak form update that uses potentials that are obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is carried out and the limiting CFL numbers of this entire family of methods is catalogued in this work. The stability analysis confirms that with 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 the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods have also been shown to be curl-preserving over long integration times.

Keywords: Discontinuous Galerkin, Higher order schemes, Curl-preserving, von Neumann stability

@Techreport{BK21_986,
  author = {D. S. Balsara and R. K\"appeli},
  title = {Von Neumann Stability Analysis of DG-like and PNPM-like Schemes for PDEs that have Globally Curl-Preserving Evolution of Vector Fields},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-44.pdf },
  year = {2021}
}

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