Abstract

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 operate 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 special 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 vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability 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 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 are also shown to be curl-preserving over long integration times.

Highlights

  • Novel PDEs, important to science and engineering problems are routinely discovered

  • By taking a close look at Eq (3) we realize that the von Neumann stability analysis depends on the angle that the velocity vector vx, vy makes with respect to the x-axis of the mesh

  • Eqs. (18) and (19) show us that the von Neumann stability analysis depends on the angle that the wave vector kx, ky makes with respect to the velocity vector vx, vy

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Summary

Introduction

Novel PDEs, important to science and engineering problems are routinely discovered. Many of those PDE systems have involution constraints whose deeper study leads to mimetic numerical solution strategies that retain greater fidelity with the physics of those systems. The structure of the evolutionary equation for the vector field of interest is such that it either keeps the curl of the vector field zero, or evolves it in proportion to certain source terms. The former PDEs are referred to as curl-free whereas the latter class of PDEs are referred to as curl-preserving. It has recently become possible to recast Schrödinger’s equation in first-order hyperbolic form, and the time-evolution of this very important equation has curl-preserving constraints (Dhaouadi et al [25], Busto et al [19])

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