High-Order Schemes for Multi-Dimensional Hamilton-Jacobi Equations

We are interested in high-order numerical approximations for the solution of multi-dimensional Hamilton-Jacobi (HJ) equations of the form $$ {{\phi }_{t}} + H(\nabla \phi ) = 0,\quad x = ({{x}_{1}}, \ldots {{x}_{d}}) \in {{\mathbb{R}}^{d}}. $$ Here, H is the Hamiltonian, which we assume depends on ∇ϕ and possibly on x and t. The main difficulty in approximating the solutions of HJ equations is the discontinuous derivatives that develop in the solution in finite time even when the initial data is smooth. From an analytical point of view, solutions of HJ equations past the discontinuity are treated via the machinery of viscosity solutions (see [3, 4, 11] and the references therein). As far as the numerics is concerned, the first significant result were the converging first-order approximations by Souganidis [18]. High-order upwind methods, that are based on the ENO reconstruction of Harten et al [6], were introduced by Osher, Sethian and Shu [16, 17]. Jiang and Peng combined the Weighted ENO (WENO) interpolant [8, 14] with a monotone fiux obtaining the upwind-WENO schemes for HJ equations in [7].