Convergent Difference Schemes for Hamilton-Jacobi equations

  • Serikbolsyn Duisembay

Student thesis: Master's Thesis

Abstract

In this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.
Date of AwardMay 7 2018
Original languageEnglish
Awarding Institution
  • Computer, Electrical and Mathematical Science and Engineering
SupervisorDiogo Gomes (Supervisor)

Keywords

  • Hamilton-Jacobi equations
  • difference schemes
  • Viscosity solutions
  • numerical methods

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