Schemes with Well-Controlled Dissipation

Abstract

We design schemes for the approximation of entropy solutions to nonlinear hyperbolic conservation laws in the regime where small-scale effects drive the dynamics of nonclassical shock waves in these solutions. Typically, the small-scale effects are modeled by adding higher-order dissipation terms taking into account the viscosity, capillarity, the Hall coefficient, etc., of the material under consideration. We analyze a strategy for the design of numerical methods adapted to these problems, referred to as schemes with well-controlled dissipation (WCD). Nonclassical entropy solutions are approximated by (small-scale) consistent and converging schemes with high accuracy---the main challenge being to capture physically relevant shocks. Following earlier works by LeFloch and collaborators, we rely on the equivalent equation, which provides a suitable tool in order to guarantee that small-scale dependent shocks are computed in a consistent and accurate way. WCD schemes are also intended to capture (nonclassical) shocks with arbitrary large strength. Our strategy is exemplified with examples and numerical experiments encompassing the cubic conservation law, the nonlinear elasticity system, and a reduced magnetohydrodynamics model. We compute the kinetic functions associated with the schemes and, as the order of the WCD schemes is increased, we observe convergence toward the exact kinetic function---even for strong shocks.