High-order weighted compact nonlinear scheme for one- and two-dimensional Hamilton-Jacobi equations

This paper designs a fifth-order weighted compact nonlinear scheme (WCNS) on a five-point stencil to solve one- and two-dimensional Hamilton-Jacobi equations. The five-point WCNS is used to compute the left and right limits of first-order spatial derivatives of the HJ equations in the Lax-Friedrichs monotone numerical Hamiltonian. The WENO-Z type interpolation for cell-edge values of the solutions is used to suppress numerical oscillations which may appear near discontinuities. Five- and seven-point WENO-Z schemes for Hamilton-Jacobi equations are also designed for comparisons. The performance of the WCNS and the two WENO-Z schemes is demonstrated by several numerical examples in one-dimensional and two-dimensional cases.