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Convergence of finite-difference schemes behind a shock front. (English. Russian original) Zbl 1122.76355
Comput. Math. Math. Phys. 37, No. 10, 1161-1172 (1997); translation from Zh. Vychisl. Mat. Mat. Fiz. 37, No. 10, 1201-1212 (1997).
In this paper for Harten’s TVD scheme, it is shown that finite-difference schemes that provide high-order approximation of smooth solutions are generally first-order accurate on average in the domain of smooth behavior of the generalized solution behind a shock front propagating with varying velocity. A comparative analysis of the accuracies of the TVD scheme and a cross-type first-order scheme with specially adjusted linear artificial viscosity is conducted. The analysis shows that the guaranteed order of accuracy of the TVD scheme proves to be substantially lower, as compared to that of the cross-type scheme, in a fairly wide region behind an unsteady shock front.

76M20 Finite difference methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76L05 Shock waves and blast waves in fluid mechanics