A Compact Upwind Second Order Scheme for the Eikonal Equation (Journal of Computational Mathematics, 2018)
We present a compact upwind second order scheme for computing the viscosity solution of the Eikonal equation. This new scheme is based on: 1. the numerical observation that classical first order monotone upwind schemes for the Eikonal equation yield numerical upwind gradient which is also first order accurate up to singularities; 2. a remark that partial information on the second derivatives of the solution is known and given in the structure of the Eikonal equation and can be used to reduce the size of the stencil.  We implement the second order scheme as a correction to the well known sweeping method but it should be applicable to any first order monotone upwind scheme. Care is needed to choose the appropriate stencils to avoid instabilities. Numerical examples are presented.
Keywords: Eikonal equation; Upwind scheme; Hamilton-Jacobi; Viscosity Solution; Sweeping method.