Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes

摘要

We build a simple and general class of finite difference schemes for firstorder Hamilton-Jacobi (HJ) Partial Differential Equations. These filteredschemes are convergent to the unique viscosity solution of the equation. Theschemes are accurate: we implement second, third and fourth order accurateschemes in one dimension and second order accurate schemes in two dimensions,indicating how to build higher order ones. They are also explicit, which meansthey can be solved using the fast sweeping method or the fast marchingmethod.The accuracy of the method is validated with computational results forthe eikonal equation in one and two dimensions, using filtered schemes madefrom standard centered differences, higher order upwinding and ENOinterpolation.