Construction of Convergent High Order Schemes for Time Dependent Hamilton–Jacobi Equations

Abstract

We construct a class of convergent high order schemes for time dependent Hamilton---Jacobi equations. In general, high order schemes such as WENO scheme achieve high order accuracy in smooth regions of the solution and an essentially non-oscillatory resolution at singularities of the solution, but despite its good numerical properties, the convergence to the viscosity solution could not be expected for certain nonconvex problems. We propose a general method of constructing convergent high order schemes and discuss the question of its convergence. The scheme relies on the reasonable combination of a high order scheme and a first order monotone scheme, which is determined so as to make the scheme converge while achieving high order accuracy. We provide adaptive algorithms for problems with nonconvex Hamiltonians and perform a detailed numerical study to demonstrate its convergence.