HIGH-ORDER 2-DIMENSIONAL NONOSCILLATORY METHODS FOR SOLVING HAMILTON-JACOBI SCALAR EQUATIONS

For the computation of nonlinear solutions of Hamilton-Jacobi scalar e quations in two space dimensions, we develop high order accurate numer ical schemes that can be applied to complicated geometries. Previously , the recently developed essentially nonoscillatory (ENO) technology h as been applied in simple domains like squares or rectangles using dim ension-by-dimension algorithms. On arbitrary two dimensional closed or multiply connected domains, first order monotone methods were used. I n this paper, we propose two different techniques to construct high or der accurate methods using the ENO philosophy. Namely, any arbitrary d omain is triangulated by finite elements into which two dimensional EN O polynomials are constructed. These polynomials are then differentiat ed to compute a high order accurate numerical solution. These new tech niques are shown to be very useful in the computation of numerical sol utions of various applications without significantly increasing CPU ru nning times as compared to dimension-by-dimension algorithms. Furtherm ore, these methods are stable and no spurious oscillations are detecte d near singular points or curves. (C) 1996 Academic Press, Inc.