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Some notes on summation by parts time integration methods. (English) Zbl 1453.65161
Summary: Some properties of numerical time integration methods using summation by parts (SBP) operators and simultaneous approximation terms are studied. These schemes can be interpreted as implicit Runge-Kutta methods with desirable stability properties such as \(A\)-,\(B\)-,\(L\)-, and algebraic stability [J. Nordström and T. Lundquist, J. Comput. Phys. 251, 487–499 (2013; Zbl 1349.65399); T. Lundquist and J. Nordström, J. Comput. Phys. 270, 86–104 (2014; Zbl 1349.65550); P. D. Boom and D. W. Zingg, SIAM J. Sci. Comput. 37, No. 6, A2682–A2709 (2015; Zbl 1359.65127); A. A. Ruggiu and J. Nordström, J. Comput. Phys. 360, 192–201 (2018; Zbl 1395.65107)]. Here, insights into the necessity of certain assumptions, relations to known Runge-Kutta methods, and stability properties are provided by new proofs and counterexamples. In particular, it is proved that a) a technical assumption is necessary since it is not fulfilled by every SBP scheme, b) not every Runge-Kutta scheme having the stability properties of SBP schemes is given in this way, c) the classical collocation methods on Radau and Lobatto nodes are SBP schemes, and d) nearly no SBP scheme is strong stability preserving.

MSC:
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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