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Solving one dimensional scalar conservation laws by particle management. (English) Zbl 1158.65339

Griebel, Michael (ed.) et al., Meshfree methods for partial differential equations IV. Papers based on the presentations at the 4th international workshop, Bonn, Germany, September 17–20, 2007. Berlin: Springer (ISBN 978-3-540-79993-1/pbk). Lecture Notes in Computational Science and Engineering 65, 95-109 (2008).
Summary: We present a meshfree numerical solver for scalar conservation laws in one space dimension. Points representing the solution are moved according to their characteristic velocities. Particle interaction is resolved by purely local particle management. Since no global remeshing is required, shocks stay sharp and propagate at the correct speed, while rarefaction waves are created where appropriate.
The method is total variation diminishing entropy decreasing, exactly conservative, and has no numerical dissipation. Difficulties involving transonic points do not occur, however inflection points of the flux function pose a slight challenge, which can be overcome by a special treatment. Away from shocks the method is second order accurate, while shocks are resolved with first order accuracy. A postprocessing step can recover the second order accuracy. The method is compared to CLAWPACK in test cases and is found to yield an increase in accuracy for comparable resolutions.
For the entire collection see [Zbl 1149.65002].

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations

Software:

CLAWPACK
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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