# Exponential Scaling and the Time Growth of the Error of DG for Advection-Reaction Problems

## 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.

## Notes

### Acknowledgements

The work of V. Kučera was supported by the J. William Fulbright Commission in the Czech Republic and research project No. 17-01747S of the Czech Science Foundation. The work of C.-W. Shu was supported by DOE grant DE-FG02-08ER25863 and NSF grants DMS-1418750 and DMS-1719410.

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