Colloquium April 10
Date: Friday, April 10
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Fengyan Li of Rensselaer Polyech
Title: A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations
Abstract: Hamilton-Jacobi equations provide important mathematical models for many areas such as optimal control, computer vision, and geometric optics. The solutions of such equations may develop discontinuous derivatives in finite time even when the initial data is smooth. To ensure the uniqueness and existence of the physically relevant solutions, the concept of viscosity solution was established for these equations.
In this work, a central discontinuous Galerkin (DG) method is proposed and investigated for solving Hamilton-Jacobi equations. Central DG methods are recently introduced for hyperbolic conservation laws, and they combine the methodology of the central scheme and the DG framework. Such methods avoid the use of Riemann solvers which can be complicated and costly for complex systems, at the expense of using two copies of approximating solutions defined on overlapping meshes. The co-existence of two copies of numerical solutions also provides new opportunities. Besides, the methods carry many features of standard DG methods.
In this talk, the theoretical results on accuracy and stability will be presented for the proposed central DG method when Hamiltonian is linear. In addition, we will use extensive numerical examples to demonstrate the performance of the method when approximating the viscosity solutions of more general Hamilton-Jacobi equations, which may involve linear, nonlinear, smooth, non-smooth, convex, or non-convex Hamiltonians.
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