A local piecewise parabolic method for Hamilton–Jacobi equations

Abstract

In this paper we present, analyze and implement a new local piecewise parabolic method for nonlinear Hamilton–Jacobi equations. The scheme is third-order accurate in smooth regions, uses a concept of local smoothing to prevent the excessive increase of the total variation at discontinuity, and has a local stencil in the sense that it does not extrapolate from data of the smoothest neighboring cells. One and two-dimensional numerical experiments, accuracy tests, and the behavior of the total variation of the approximate solution are presented to prove the accuracy and good resolution of our method.