Central DG methods for Hamilton-Jacobi equations and ideal MHD equations

Monday, November 1, 2010 - 3:00pm - 3:45pm
Keller 3-180
Fengyan Li (Rensselaer Polytechnic Institute)
In this talk, I will present our recent work in developing high order
central discontinuous Galerkin (DG) methods for Hamilton-Jacobi (H-J)
equations and ideal MHD equations. Originally introduced for hyperbolic
conservation laws, central DG methods combine ideas in central schemes
and DG methods. They avoid the use of exact or approximate Riemann
solvers, while evolving two copies of approximating solutions on
overlapping meshes.

To devise Galerkin type methods for H-J equations, the main difficulty is that these equations in general are not in the divergence form. By
recognizing a weighted-residual or stabilization-based formulation of
central DG methods when applied to hyperbolic conservation laws, we
propose a central DG method for H-J equations. Though the stability and
the error estimate are established only for linear cases, the accuracy
and reliability of the method in approximating the viscosity solutions
are demonstrated through general numerical examples. This work is jointly done with Sergey Yakovlev.

In the second part of the talk, we introduce a family of central DG
methods for ideal MHD equations which provide the exactly divergence-free magnetic field. Ideal MHD system consists of a set of nonlinear
hyperbolic equations, with a divergence-free constraint on the magnetic
field. Though such constraint holds for the exact solution as long as it
does initially, neglecting this condition
numerically may lead to nonphysical features of approximating solutions
or numerical instability. This work is jointly done with Liwei Xu and
Sergey Yakovlev.
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