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 Award | May 7 2018 |
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| Original language | English (US) |
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| Awarding Institution | - Computer, Electrical and Mathematical Sciences and Engineering
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| Supervisor | Diogo Gomes (Supervisor) |
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- Hamilton-Jacobi equations
- difference schemes
- Viscosity solutions
- numerical methods