A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

摘要

We introduce a new methodology for adding localized, space-time smooth,artificial viscosity to nonlinear systems of conservation laws which propagateshock waves, rarefactions, and contact discontinuities, which we call the$C$-method. We shall focus our attention on the compressible Euler equations inone space dimension. The novel feature of our approach involves the coupling ofa linear scalar reaction-diffusion equation to our system of conservation laws,whose solution $C(x,t)$ is the coefficient to an additional (and artificial)term added to the flux, which determines the location, localization, andstrength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ islarge and localized, and transitions smoothly in space-time to zero away fromdiscontinuities. Our approach is a provably convergent, spacetime-regularizedvariant of the original idea of Richtmeyer and Von Neumann, and is provided atthe level of the PDE, thus allowing a host of numerical discretization schemesto be employed. We demonstrate the effectiveness of the $C$-method with threedifferent numerical implementations and apply these to a collection ofclassical problems: the Sod shock-tube, the Osher-Shu shock-tube, theWoodward-Colella blast wave and the Leblanc shock-tube. First, we use aclassical continuous finite-element implementation using second-orderdiscretization in both space and time, FEM-C. Second, we use a simplified WENOscheme within our $C$-method framework, WENO-C. Third, we use WENO with theLax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C.All three schemes yield higher-order discretization strategies, which providesharp shock resolution with minimal overshoot and noise, and compare well withhigher-order WENO schemes that employ approximate Riemann solvers,outperforming them for the difficult Leblanc shock tube experiment.