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Schemes with Well-Controlled Dissipation

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 (nonclassica...