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Published in:
2019 | OriginalPaper | Chapter
Exponential Scaling and the Time Growth of the Error of DG for Advection-Reaction Problems
Authors : Václav Kučera, Chi-Wang Shu
Published in: Numerical Mathematics and Advanced Applications ENUMATH 2017
Publisher: Springer International Publishing
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Abstract
We present an overview of the results of the authors’ paper (Kučera and Shu, IMA J Numer Anal, to appear) on the time growth of the error of the discontinuous Galerkin (DG) method and set them in appropriate context. The application of Gronwall’s lemma gives estimates which grow exponentially in time even for problems where such behavior does not occur. In the case of a nonstationary advection-diffusion equation we can circumvent this problem by considering a general space-time exponential scaling argument. Thus we obtain error estimates for DG which grow exponentially not in time, but in the time particles carried by the flow field spend in the spatial domain. If this is uniformly bounded, one obtains an error estimate of the form C(h
p+1∕2), where C is independent of time. We discuss the time growth of the exact solution and the exponential scaling argument and give an overview of results from Kučera and Shu (IMA J Numer Anal, to appear) and the tools necessary for the analysis.