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." 



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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