The representation of shock-like solutions in an Eulerian mesh

Methods are presented for integrating differential equations in conservation form in an Eulerian mesh in the presence of discontinuities and shocks. We make use of the monotonicity property, namely, that a solution is assumed to be monotone between mesh points. Monotonicity is preserved by convective equations and can be rigorously implemented in a code. In more general systems of conservative equations, monotonicity is enforced when interpolating a curve, given on an integer mesh, onto a half-integer mesh. This can be interpreted as introducing a local diffusive term whenever a discontinuity or shock occurs. The integration of a Riemann problem shows that with this method shocks are represented within two zones and contact discontinuities exhibit little diffusion.