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A MUSTA scheme for a nonconservative two-fluid model. (English) Zbl 1387.76106

Summary: We present a multistage centered scheme, of the kind proposed by E. F. Toro [Appl. Numer. Math. 56, No. 10–11, 1464–1479 (2006; Zbl 1101.65088)], for numerically resolving the simultaneous flow of two fluids through a transport pipeline. This model contains nonconservative terms in both the temporal and spatial derivatives, and an extension of the standard numerical framework for conservation laws is needed. In this paper, we rewrite the model in an equivalent mathematical form, eliminating the nonconservative time derivatives. This allows us to use the framework described by C. Parés [SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089)]. We develop FORCE and MUSTA-type schemes which are consistent with Parés’ formalism. Numerical simulations demonstrate a high degree of stability of our proposed schemes. Comparisons with the Roe and Rusanov schemes indicate that convergence to near-identical solutions is obtained when the nonconservative terms are discretized with respect to the same evaluation of the path-dependent integrals. However, if the schemes are not mutually formally path-consistent in the sense of Parés, different converged solutions are obtained.

MSC:

76T10 Liquid-gas two-phase flows, bubbly flows
35L60 First-order nonlinear hyperbolic equations
35Q35 PDEs in connection with fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Software:

CATHARE; MUSTA; AUSM
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