Graduate Seminar "Aktuelle Themen der Numerik"

Thursday, Jul 4, 2013, 02:00 pm

Adaptive Multiresolution Discontinuous-Galerkin schemes for
compressible flows

Nils Gerhard, M.Sc. (RWTH Aachen)

Abstract:
Abstract. In the present work, we extend the concept of adaptive multiresolution-based DG (MR-DG) schemes originally developed for one-dimensional conservation laws in [4] to multidi-mensional systems. Motivated by data compression, the key idea is to perform a multiresolution analysis of the data using Alpert’s multiwavelets and performing hard thresholding. From a reduced set of significant multi-scale coefficients we then design a locally refined grid, on which we apply a DG discretization.
Our goal is to reduce the number of degrees of freedom by adaptivity, but at the same time we want to preserve the accuracy of the reference scheme. For this purpose we balance the perturbation error, i.e. the dfference between the adaptive scheme and the non-adaptive scheme on the reference mesh, with the discretization error of the reference scheme. Thereby we can determine an optimal threshold parameter by balancing these errors. In the one-dimensional case we perform parameter studies for the Euler equations to numerically estimate an optimal
threshold parameter. Furthermore we apply our scheme to the two-dimensional shallow water, Euler and compressible Navier-Stokes equations.

References
[1]  B. Alpert. A class of bases in L² for the sparse representation of integral operators. SIAM J. Math. Anal.,
24:246–262, 1993.
[2]  B. Alpert, G. Beylkin, D. Gines and L. Vozovoi. Adaptive solution of partial dfferential equation in multi-
wavelet bases. J. Comp. Phys., 182:149–190, 2002.
[3]  B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element
method for conservation laws II: General framework. Mathematics of Computation, 52(186):411–435, 1989.
[4]  N. Hovhannisyan, S. Müller and R. Schäfer. Adaptive multiresolution Discontinuous Galerkin schemes for
conservation laws. Mathematics of Computation, accepted, 2012.

Time: 02:00 pm

Location: Room 149, Hauptgebäude, RWTH Aachen, Templergraben 55, 52056 Aachen