Numerical Methods For Hamilton-jacobi Equations

Numerical methods for Hamilton-Jacobi equations are studied.In general, the solutions of Hamilton-Jacobi (HJ) equations develop singularitiesat a finite time even with smooth data and numerically, one looks for a consistent monotonescheme to approximate viscosity solutions of HJ equations.In chapter2monotone schemes for Cauchy-Dirichlet problems of time-dependentHamilton-Jacobi equations are constructed. The problem of constructing numericalschemes turn out to have some theoretical difficulties because how to interpolatebetween the boundary and interior grid points is not obvious. For that, a new class ofabstract monotone schemes for HJ equation with weak Dirichlet boundary conditionis provided with a convergence rate of1/2. Based on abstract convergence results,one can construct numerically useful convergent schemes for Cauchy-Dirichletproblems of HJ equations. Firstly, a convergent finite volume scheme for Cauchy problemis constructed, which does not propose any requirements on triangulation except the usualregularity. Then, through newly constructing finite volume approximations with theright properties in the vicinity of the boundary, the convergent finite volume schemesare finally obtained for Cauchy-Dirichlet problems of HJ equations.The WENO scheme is one of the successful high order numerical methods forapproximating conservation laws and HJ equations. Based on the idea of adaptivestencils in the reconstruction procedure, WENO schemes achieve high order accuracyin smooth regions of solution and an essentially nonoscillatory and sharp singularityresolution. On the other hand, despite its good numerical properties, the convergenceto the viscosity solution of HJ equation could not be expected for certain nonconvexproblems. In chapter3, a general method of constructing convergent high orderschemes is proposed for time dependent Hamilton-Jacobi equations and the question of its convergence is discussed. The scheme relies on the reasonable combination of ahigh order scheme and a first order monotone scheme, which is determined so as tomake the scheme converge while achieving high order accuracy. The adaptivealgorithms are also provided for problems with nonconvex Hamiltonians. A detailednumerical study is performed to demonstrate its convergence. Furthermore, similaradaptive strategies could be applied with any couple of a dissipative and a high-order(compressive) reconstruction, including the cases of unstructured meshes, and thiswill become the theme of future study.In chapter4numerical methods are studied for level set like equations. Level setlike equations arise in curve (surface) evolution and image processing problems,together with many other applications. Finite volume (element) schemes are proposedfor the mean curvature flow level set equation, based on semi-implicit discretizationin time and using of primal-dual co-volumes for space approximation. Also, for levelset like image smoothing model, finite volume-element type schemes are constructed,based on a sort of operator-splitting according to isotropic and anisotropic diffusion.Some properties including stability and consistency are proved for proposed schemes.On the other hand, corner-preserving becomes a critical property in some applicationssuch as image smoothing. An image smoothing model based on corner-preservingflow is proposed with the purpose of achieving a superior image restoration. Thestudy aims to emphasize a role of locally estimated mean curvature of level set incorner-preserving flows and image processing applications. Numerical experimentsrelated to curve evolution and image denoising are performed to demonstrate thevalidity of the proposed models.