A rarefaction-tracking method for hyperbolic conservation laws

We present a numerical method for scalar conservation laws in one space dimension. The solution is approximated by local similarity solutions. While many commonly used approaches are based on shocks, the presented method uses rarefaction and compression waves. The solution is represented by particles that carry function values and move according to the method of characteristics. Between two neighboring particles, an interpolation is defined by an analytical similarity solution of the conservation law. An interaction of particles represents a collision of characteristics. The resulting shock is resolved by merging particles so that the total area under the function is conserved. The method is variation diminishing, nevertheless, it has no numerical dissipation away from shocks. Although shocks are not explicitly tracked, they can be located accurately. We present numerical examples, and outline specific applications and extensions of the approach.Comment: 21 pages, 7 figures. Similarity 2008 conference proceeding

Initial Boundary-Value Problems for a Pair of Conservation Laws

We describe a multiple-scale technique for solving the initial boundary-value problem over the positive x-axis for a one-dimensional pair of hyperbolic conservation laws. This technique involves decomposing the solution into waves and incorporating slow temporal and stretched spatial scales in different parts of the solution domain. We apply these ideas to a wavemaker problem for shallow water flow and show why the presence of source terms in the conservation laws makes the analytic solution more complicated