A space–time smooth artificial viscosity method for nonlinear conservation laws

We introduce a new methodology for adding localized, space–time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the C-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a 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, and strength of the artificial viscosity. Near shock discontinuities, C ( x , t ) is large and localized, and transitions smoothly in space–time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. onstrate the effectiveness of the C-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher–Shu shock-tube, the Woodward–Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our C-method framework, WENO-C. Third, we use WENO with the Lax–Friedrichs flux together with the C-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.