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Keywords:

  • Godunov scheme;
  • conservation laws;
  • memory terms;
  • Volterra integral terms

Abstract

We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014