Discontinuous Galerkin methods for Hamilton-Jacobi equations

Hamilton-Jacobi (H-J) equations provide important mathematical models for many applications. The solutions of such equations may develop discontinuous derivatives even for the smooth initial and boundary data. And the concept of viscosity solution was introduced in the early 1980s. In this talk, I will present our recent work in developing high order numerical methods for H-J equations.

One part of the talk concerns the design of discontinuous Galerkin (DG) methods for directly solving time-dependent H-J equations. The main difficulty comes from the fact that these equations in general are not in the divergence form. By recognizing and following a weighted-residual or stabilization-based formulation of central DG methods when applied to hyperbolic conservation laws, a central DG method is designed for H-J equations. Though the stability and the error estimate are established only for linear cases, the high order accuracy and reliability of the method in approximating the viscosity solutions are demonstrated through general numerical examples. This work is jointly done with S. Yakovlev (RPI).

The other part of the talk focuses on the development of efficient high order methods for static H-J equations. In particular, second order DG-based fast sweeping methods are proposed for one family of H-J equations - Eikonal equations. These iterative methods demonstrate the linear computational complexity, namely, the number of iterations for the convergence is independent of the number of total unknowns. Besides its accuracy, DG discretization is chosen more for its compactness, which is important for the overall efficiency of the algorithms. This work is collaborated with S. Chen (Indiana U., South Bend), C.-W. Shu (Brown U.), Y.-T. Zhang (Notre Dame) and H.-K. Zhao (UC Irvine).