High Order Two Dimensional Nonoscillatory Methods for Solving Hamilton-Jacobi Scalar Equations

For the computation of nonlinear solutions of Hamilton–Jacobi scalar equations in two space dimensions, we develop high order accurate numerical schemes that can be applied to complicated geometries. Previously, the recently developed essentially nonoscillatory (ENO) technology has been applied in simple domains like squares or rectangles using dimension-by-dimension algorithms. On arbitrary two dimensional closed or multiply connected domains, first order monotone methods were used. In this paper, we propose two different techniques to construct high order accurate methods using the ENO philosophy. Namely, any arbitrary domain is triangulated by finite elements into which two dimensional ENO polynomials are constructed. These polynomials are then differentiated to compute a high order accurate numerical solution. These new techniques are shown to be very useful in the computation of numerical solutions of various applications without significantly increasing CPU running times as compared to dimension-by-dimension algorithms. Furthermore, these methods are stable and no spurious oscillations are detected near singular points or curves.