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On the construction of combined finite-difference schemes of high accuracy. (English. Russian original) Zbl 1393.65014
Dokl. Math. 97, No. 1, 77-81 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 478, No. 5, 517-522 (2018).
Summary: A method is proposed for constructing combined shock-capturing finite-difference schemes that localize shock fronts with high accuracy and preserve the high order of convergence in all domains where the computed weak solution is smooth. A particular combined scheme is considered in which a nonmonotone compact scheme with a third-order weak approximation is used as a basis one, while the internal scheme is the second-order accurate (for smooth solutions) monotone CABARET. The advantages of the new scheme are demonstrated using test computations.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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