Computational and Applied Mathematics Seminar

Mar 23, 2017

3:30 pm

Allen Hall Room 14

Professor Manuel Sanchez Uribe from the Department of Mathematics at the University of Minnesota will lecture on "Hamiltonian HDG methods for wave propagation phenomena." 

 

Abstract: 

We devise the first symplectic Hamiltonian Hybridizable Discontinuous Galerkin (HDG) methods for the acoustic wave equation. We discretize in space by using a Hamiltonian HDG scheme and in time by using symplectic, diagonally implicit and explicit partitioned Runge-Kutta methods. The fundamental characteristic of the semi-discrete scheme is it preserves the Hamiltonian structure of the wave equation, which combined with symplectic time integrators guarantees the conservation of the energy. We obtain optimal approximations of order $k+1$ in the $L^{2}$-norm when polynomials of degree $k\geq0$ and Runge-Kutta formulae of order $k+1$ are used. In addition, by means of post-processing techniques and augmenting the order of the Runge-Kutta method to $k+2$, we obtain superconvergent approximations of order $k+2$ in the $L^2$ norm for the displacement and velocity. We provide numerical examples corroborating these convergence properties as well as depicting the conservative features of the methods.

 


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