Hamilton-jacobi Equation, Numerical Methods

The Hamilton-Jacobi equations have been widely applied to various fields, Such as, optimal control theory, computational fluid dynamics, computing vision, differential geometry, crystal growth, mesh generation. In the past few years, the research on the l-Iamilton-Jacobi equations has attracted much attention. It is well known that the solutions of the Hamilton-Jacobi equations are continuous but with discontinuous derivative, even the initial values and the Hamiltonian are smooth. And such solutions are in general not unique. This thesis is concerned with the numerical solutions of the Hamilton-Jacobi equations. We focus on the numerical schemes, their stability, resolution for singular points and convergence of the numerical solutions. The thesis consists of five chapters. In chapter 1, we briefly review the developments on theoretical analysis and numerical methods for the Hamilton-Jacobi equations, and introduce some important application fields of the Hamilton-Jacobi equations. In chapter 2, we discuss the finite element methods with high order for solving one-dimensional Hamilton-Jacobi equations. Since the three types of basis functions are adopted, we can obtain the three schemes. All schemes belong to the TVD type. We also prove that the numerical solutions of the semi-discrete finite element schemes with the continuous basis functions converge to the viscosity solutions of the Hamilton-Jacobi equations under some conditions. The numerical tests show the accuracy, stability and resolution of three schemes. In chapter 3, we study the finite element methods for the Hamilton-Jacobi equations on the unstructured meshes. A numerical scheme is obtained by applying the finite element method to the viscosity equations of the Hamilton-Jacobi equations on unstructured meshes. By improving the finite element scheme, we can construct another numerical scheme. Under some limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations. The latter need weaker limitations than that of the former. Numerical examples test the stability, the convergence and the sensitivity to meshes. In chapter 4, the schemes with high order accuracy are presented for solving the Hamilton-Jacobi equations on the unstructured meshes. Based on the numerical schemes constructed in chapter 3, we present a building block on the triangular meshes. By means of the building block, an ENO scheme and two WENO schemes are constructed. Since the reasonable weight is adopted for each substencil, a third-order WENO scheme is presented on the second-order ENO stencil. Numerical examples are shown to demonstrate the accuracy and resolution of the ENO and WENO schemes. In the final chapter, the local adaptive refinement method on two-dimensional structured meshes is studied for solving the Hamilton-Jacobi equations. By using the ENO philosophy on unstructured meshes in chapter 4, we construct the non-oscillatory numerical schemes with high order accuracy on structured meshes. The local adaptive refinement method is developed for local regions where the solutions of the Hamilton-Jacobi equations vary sharply. And it is consistent with the non-oscillatory numerical schemes. The advantages of the refinement method are that the local refinement meshes trace the discontinuities and when less computation is increased, the accuracy and resolution can be improved. Numerical examples illustrate these advantages.