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Mathematics of Computation

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A kernel-based discretisation method for first order partial differential equations


Authors: Tobias Ramming and Holger Wendland
Journal: Math. Comp. 87 (2018), 1757-1781
MSC (2010): Primary 65M12, 65M15, 35F50
DOI: https://doi.org/10.1090/mcom/3265
Published electronically: October 26, 2017
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Abstract: We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique and is a typical kernel-based method. It differs, however, significantly from the SPH method since it employs an Eulerian and not a Lagrangian approach. We prove stability and convergence for the resulting semi-discrete scheme under certain smoothness assumptions on the defining function of the PDE. The approximation order depends on the underlying kernel and the smoothness of the solution. Hence, we also review an easy way of constructing smooth kernels yielding arbitrary convergence orders. Finally, we give a numerical example by testing our method in the case of a one-dimensional Burgers equation.


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Additional Information

Tobias Ramming
Affiliation: Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany
Email: tobias.ramming@uni-bayreuth.de

Holger Wendland
Affiliation: Department of Mathematics, University of Bayreuth, D-95440 Bayreuth, Germany
Email: holger.wendland@uni-bayreuth.de

DOI: https://doi.org/10.1090/mcom/3265
Received by editor(s): November 28, 2015
Received by editor(s) in revised form: December 22, 2016, and February 8, 2017
Published electronically: October 26, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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