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Schemes with Well controlled Dissipation (WCD) I: Non-classical shock waves

by J. Ernest and P. LeFloch and S. Mishra

(Report number 2013-41)

Abstract
We consider the approximation of entropy solutions to nonlinear hyperbolic conservation laws, in the regime that small--scale effects drive the dynamics of shock waves in these solutions. We introduce and analyze a new class of numerical methods, referred to as the schemes with well-controled dissipation (WCD), which approximate entropy solutions with high--accuracy and can capture small scale dependent shock waves of arbitrary strength. Following earlier work by LeFloch and collaborators, we rely on the equivalent equation associated with a finite difference scheme which provides us with the proper tool in order to ensure that small-scale dependent shock waves are computed accurately. Examples involving nonclassical shocks for cubic conservation laws, the nonlinear elasticity system, and a reduced model of magnetohydrodynamics are investigated with our approach.

Keywords: Small scale dependent shock waves, finite differences, Equivalent equation, high-order schemes.

@Techreport{ELM13_539,
  author = {J. Ernest and P. LeFloch and S. Mishra},
  title = {Schemes with Well controlled Dissipation (WCD) I: Non-classical shock waves},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-41},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-41.pdf },
  year = {2013}
}

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