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SIAM J. Numer. Anal. 50, pp. 79-104 (26 pages)
Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations
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Add to FacebookIn this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semidiscrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multidimensional Schrödinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization by using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.
© 2012 Society for Industrial and Applied Mathematics
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Received January 28, 2011
Accepted October 27, 2011
Published online January 19, 2012
Accepted October 27, 2011
Published online January 19, 2012
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