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Explicit Time Stepping of PDEs with Local Refinement in Space-Time

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Abstract

Traditional numerical time stepping allows variable node densities in space, but not also in time. Having the ability to utilize nodes that are placed irregularly in the space-time domain leads to many advantages when solving time dependent problems. In this paper we introduce a new method utilizing the radial basis function generated finite difference approach in order to accomplish this goal. Benefits include improved stability conditions and the option to use small time steps only in select spatial regions.

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Notes

  1. Although similar in many respects, RBF-FD (in its form known as PHS + poly, see Sections 5.1.5 and 5.1.7 in [16]) and MLS have also significant differences, as analyzed in [3]. The last paragraph in the Conclusions of this paper notes: “Overall, PHS + poly has performed superior than MLS. It can not only achieve at least the same accuracy than MLS, but can also overcome the harmful Runge’s phenomenon for any polynomial degree. This result potentially opens new opportunities for PHS + poly in areas of application where MLS is the preferred choice”.

  2. A code for producing this figure can be downloaded from https://github.com/DylanAbrahamsen/RBF-TD.

  3. We focus here on AB-FD4 schemes (rather than on RK-FD4 schemes, with internal stages) since these can be displayed as space-time stencils.

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  1. Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA

    Dylan Abrahamsen & Bengt Fornberg

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  1. Dylan Abrahamsen
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  2. Bengt Fornberg
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Correspondence to Dylan Abrahamsen.

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Abrahamsen, D., Fornberg, B. Explicit Time Stepping of PDEs with Local Refinement in Space-Time. J Sci Comput 81, 1945–1962 (2019). https://doi.org/10.1007/s10915-019-01065-3

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