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Fast methods for the Eikonal and related Hamilton- Jacobi equations on unstructured meshes
1Department of Mathematics, University of California, Berkeley, CA 94720, USA.
Summary
The Fast Marching Method efficiently solves the Eikonal equation using Dijkstra-like algorithms. This study explores extensions for higher-order accuracy on complex meshes and manifolds.
Area of Science:
- Numerical analysis
- Computational mathematics
- Partial differential equations
Background:
- The Eikonal equation is fundamental in various scientific fields.
- The Fast Marching Method (FMM) provides an efficient numerical solution.
- Existing methods face challenges with complex geometries and higher-order accuracy.
Purpose of the Study:
- To extend the Fast Marching Method for broader applicability.
- To investigate higher-order accurate FMM versions.
- To explore FMM on unstructured meshes and manifolds.
Main Methods:
- Utilized upwind finite difference approximations for the gradient.
- Employed a Dijkstra-like programming approach based on causality.
- Developed higher-order schemes for unstructured grids in Rn and on manifolds.
Main Results:
- Achieved O(M log M) complexity for FMM on rectangular meshes.
- Demonstrated extensions to handle unstructured meshes and manifolds.
- Established connections to generalized static Hamilton-Jacobi equations.
Conclusions:
- The Fast Marching Method is a powerful tool for solving the Eikonal equation.
- Extensions enhance FMM's applicability to complex problems and geometries.
- Further research can explore connections to broader classes of Hamilton-Jacobi equations.