Construction of Higher Order Residual Distribution Schemes

Summary

We present the construction of Residual Distribution (RD) schemes of order higher than 2. The RD schemes have several advantages: robustness, a compact stencil and the residual property. Going to higher order can be easily done by increasing the polynomial approximation in the cell. We focus here on a systematic procedure to build higher order schemes from first order monotone ones. Assuming we are given a low order monotone finite volume (rewritten in the RD framework), or RD discretization, we show how to automatically generate a high order monotone scheme. The formal order of accuracy is tuned by the degree of the local polynomial interpolation of the numerical solution. This construction is done in three steps: given a polynomial interpolation, compute the total residual and the first order RD scheme, limit and stabilize (if needed). This is illustrated by an extension of the Lax-Friedrichs scheme with P ^k approximation of the solution on triangles.