Is Discontinuous Reconstruction Really a Good Idea?

Abstract

It has been almost automatically assumed for a quarter century that the numerical solution of hyperbolic conservation laws is best accomplished by making a reconstruction of the initial data that is only piecewise continuous. The effect of the discontinuities is taken into account by means of Riemann solvers. This strategy has enjoyed great practical success but introduces only one-dimensional physics as a guide to the discretization of multidimensional problems. This article points out some of the resulting defects and proposes an alternative viewpoint. The chief novelty of the new "Active Flux" method, apart from the elimination of discontinuities, is the division into advective and acoustic disturbances, with acoustics being handled by exploiting classical solutions to the scalar wave equation. Results for a standard test problem for the compressible Euler equations indicate that an order of magnitude increase in cost effectiveness may be possible by following this approach, compared to the traditional one.

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

“…[11,12] for parabolic equations, or [4]. This is what is done here, when Kirchhoff's exact solution yields a time-marching scheme for (1.1), in the form (1.2), see [27].…”

The Active Flux Scheme on Cartesian Grids and Its Low Mach Number Limit

“…t+=∆t cell boundaries due to (3.5), but has discontinuous derivative there. [VL79] discusses the application to linear advection; Burgers' equation has been considered in [ER11a,Roe17], nonlinear hyperbolic systems in [ER11b]. An extension to two-dimensional triangular grids has been introduced in [ER13,Eym13].…”

Some remarks concerning stabilization techniques for convection-diffusion problems

“…We already mentioned several times that no numerical method, even for linear scalar convection equation, can be both monotone and better than first-order accurate. Instead of too complicated high-order methods based on the multi-dimensional Riemann solvers, a number of alternative approaches have been developed based on the continuous approximation of the solution (see [18]). These approaches are Riemann solver-free.…”