On moving mesh WENO schemes with characteristic boundary conditions for Hamilton-Jacobi equations

Abstract In this paper, we are concerned with the study of efficient and high order accurate numerical methods for solving Hamilton-Jacobi (HJ) equations with initial conditions defined in the whole domain. One of the commonly used strategy is to solve the problem only in a finite domain, but the determination of boundary conditions at the artificial boundary of the finite computational domain is a problem. If the initial condition decays fast in space, one could use zero boundary condition at the artificial boundary if the domain is large enough, but this may not be very efficient since the computational domain may need to be very large to justify this choice. In this paper we use the high order moving mesh arbitrary Lagrangian Eulerian (ALE) weighted essentially non-oscillatory (WENO) finite difference scheme, recently developed in [13], in a finite and moving computational domain, with numerical boundary conditions obtained by solving the characteristic ordinary differential equations (ODEs) along the artificial boundary of the moving computational domain. The usage of this moving characteristic boundary conditions allows us to solve the HJ equations in any initial finite domain that we are interested in, regardless of the magnitude of the initial condition at the artificial domain boundary. This method works well when singularities do not appear at the artificial boundary. Extensive numerical tests in one and two dimensions are given to demonstrate the flexibility and efficiency of our method in solving both smooth problems and problems with corner singularities.