Two classes of third-order weighted compact nonlinear schemes for Hamilton-Jacobi equations

The solutions of Hamilton -Jacobi (HJ) equations may contain discontinuous derivatives, which brings numerical difficulties in capturing sharply these derivatives. This paper presents a thirdorder weighted compact nonlinear scheme (WCNS) and a third -order perturbed WCNS (WCNS-P) for solving HJ equations. A linear combination of hybrid cell -edge and cell -node values of the function is applied to approximate the spatial derivatives of the function. For the WCNS, the upwind -biased nonlinear weighted interpolation is used for the unknown cell -edge values, while for the WCNS-P perturbations with a free parameter is introduced to the upwind -biased linear interpolation. The free parameter can be optimized to reduce numerical errors in smooth regions. To enhance the numerical stability, a monotone polynomial interpolation method is designed to switch the WCNS-P to the WCNS. Several numerical experiments are performed to test the order of accuracy and the discontinuity -capturing ability of the two schemes.