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