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Journal of Scientific Computing

Erschienen in:

16.10.2019

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

verfasst von: Dylan Abrahamsen, Bengt Fornberg

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2019

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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.

Vorheriger Artikel Modified Ghost Fluid Method with Acceleration Correction (MGFM/AC)

Nächster Artikel High Order Anchoring and Reinitialization of Level Set Function for Simulating Interface Motion

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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”.

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

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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Titel: Explicit Time Stepping of PDEs with Local Refinement in Space-Time
verfasst von: Dylan Abrahamsen
Bengt Fornberg
Publikationsdatum: 16.10.2019
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-01065-3

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