A comparison of flux limited difference methods and characteristic galerkin methods for shock modelling

A comparison is made between flux-limited finite difference methods and characteristic Galerkin methods for approximating hyperbolic conservation laws. At the first-order level, the characteristic Galerkin scheme using piecewise constants is closely related to the difference schemes of Engquist, fisher, and Roe. Adaptive recovery techniques used to improve accuracy then have much in common with the flux limiters used with difference methods. These relationships are explored and comparisons made using the linear advection, inviscid Burgers and Euler equations. A new, simple formulation is given of the characteristic Galerkin method using piecewise constant elements with piecewise linear recovery: it reduces to the Engquist-Osher algorithm but with a modified flux function when the CFL number is no greater than one half.