Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians

Hendrik Ranocha; Philipp \"Offner; Thomas Sonar

Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians

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arxiv 1511.08408 v2 pith:IQ7R5I25 submitted 2015-11-26 math.NA cs.NA

Hendrik Ranocha , Philipp Öffner , Thomas Sonar This is my paper · ORCID

classification math.NAcs.NA

keywords methodsoperatorsdenseframeworknormssummation-by-partsanalyticalboundary

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A generalised analytical notion of summation-by-parts (SBP) methods is proposed, extending the concept of SBP operators in the correction procedure via reconstruction (CPR), a framework of high-order methods for conservation laws. For the first time, SBP operators with dense norms and not including boundary points are used to get an entropy stable split-form of Burgers' equation. Moreover, overcoming limitations of the finite difference framework, stability for curvilinear grids and dense norms is obtained for SBP CPR methods by using a suitable way to compute the Jacobian.

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Discrete Vector Calculus and Helmholtz Hodge Decomposition for Classical Finite Difference Summation by Parts Operators
math.NA 2019-08 accept novelty 7.0 of 10

For collocated SBP finite difference operators, discrete Helmholtz Hodge decompositions have an unavoidable remainder caused by grid oscillations, and iterative least-squares projections overcome this in practice.

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