The Methods For The Hamil-Jacobi Equations On The Unstructured Meshes

The Hamilton-Jacobi equations arises in many applications such as optimal control Theory, differential games,image processing and computer vision,geometric optics,and mesh generation. It is well known that the solutions of the Hamilton-Jacobi equations are continuous but with discontinuous derivatives even when the initial condition is smooth. This thesis is concerned with the non-oscillatory numerical method for Hamilton-Jacobi equations on unstructured meshes. Finite volume schemes and the local adaptive refinement method are included in this thesis.The thesis consists of five chapers. In chapter 1, we briefly review the applications on theoretical analysis and numerical methods for the Hamilton-Jacobi equations.In chapter 2, we present the numerical schemes for resolving Hamilton-Jacobi equations on structured and unstructured meshes, such as ENO and WENO schemes.In chapter 3,we present the finite volume method for the Hamilton-Jacobi equations on unstructured meshes. We get the secondary interpolation using the least square method and ensure that the derivative of the solution will not produce the new extremum using the maximum principle. Finally we construct the numerical scheme. Less computatation is increased. The method improves resolving power of the discontinuous domain. Numerical examples are shown to demonstrate the accuracy and resolution of the method.In chapter 4, we present an adaptive method for Hamilton-Jacobi equations on unstructured meshes. We get the non-oscillatory scheme by using the ENO philosophy on unstructured meshes. We conform the subdivisible triangles with smooth indicator and use a non-oscillatory interpolation on the new points. The advantages of the refinement method are that the local refinement meshes trace the discontinueties and when less computation is increased, the accuracy and resolution can be improved. Numerical examples verify these advantages.In the final chapter, we summarize the advantages and disadvantages in chapter3,4, and show our task in the future.