HIGH ORDER WEIGHT COMPACR NONLINEAR SCHEME FOR TIME-DEPENDENT HAMILTON-JACOBI EQUATIONS

Hamilton-Jacobi (HJ) equations are an important class of nonlinear partial differential equations. They are often used in various applications, such as physics, fluid mechanics, image processing, differential geometry, financial mathematics, optimal control theory, and so on. Because the weak solutions of the HJ equations exist but are not unique, and the spatial derivatives of the solutions may be discontinuous, numerical difficulties arise in numerical solutions of these equations. This paper presents a seventh-order weighted compact nonlinear scheme (WCNS) for the time-dependent HJ equations. This scheme is composed of the monotone Lax-Friedrichs flux splitting method for the Hamilton functions and the high-order hybrid cell-node and cell-edge central differencing for the left and right limits of first-order spatial derivatives in the numerical Hamilton functions. A high-order linear approximation scheme and four low-order linear approximation schemes for the unknowns at half nodes are derived based on a seven-point global stencil and four four-point sub-stencils, respectively. The smoothness indicators of the global stencil and four sub-stencils are also derived. In order to avoid non-physical oscillations of numerical solutions near the discontinuities and improve the numerical stability of the designed scheme, the WENO-type nonlinear interpolation technique is adopted to compute the unknowns at half nodes. The third-order TVD Runge-Kutta method is used for time discretization. The presented WCNS scheme is verified to have the optimal seventh order of accuracy for smooth solutions by theoretical analysis. For the sake of comparison, the classical seventh-order WENO scheme for solving hyperbolic conservation laws is also extended to solve the HJ equations. Numerical results show that the presented WCNS scheme can well simulate the exact solutions and can achieve seventh-order accuracy in smooth regions. Compared with the classical WENO scheme of the same order, the presented WCNS scheme has better accuracy, convergence and resolution, and its computational efficiency is slightly higher.