Hamilton-Jacobi equations have repeatedly emerged in many fields of physics, most notably, optimal control, differential games, geometric optics, and image processing. This thesis presents a new numerical method to solve a new class of Hamilton-Jacobi equation that has recently appeared in the context of nonlinear electroelastostatics.In a pioneering contribution, Crandall and Lions (1983) proved that a certain type of first-order finite difference method converges to the viscosity solution of a special class of Hamilton-Jacobi equations. From then on several successful methods of high-order approximation have been proposed in the literature, including the so-called WENO finite difference schemes. These schemes, however, were developed and tested for special classes of Hamilton-Jacobi equations, which do not include the general type of Hamilton-Jacobi equation of interest in this work. The objective of this thesis is to extend high-order WENO finite difference schemes to the most general type of Hamilton-Jacobi equations involving non-periodic boundary conditions in the "space" variables.Following its derivation, the proposed WENO scheme is tested for several cases involving one and two "space" variables for which there are analytical solutions available for arbitrarily large values of the "time" variable. These numerical experiments provide insight into the stability and rate of convergence of the method as "time" increases. They also provide insight into how errors propagate into the domain of computation due to non-periodic boundary conditions.This thesis concludes with the application of the method to compute the effective stored-energy function of an elastomer containing an isotropic distribution of vacuous pores under arbitrary 3D deformations.
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A WENO finite difference scheme for a new class of Hamilton-Jacobi equations in electroelastostatics
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