A MUSTA Scheme for a Nonconservative Two-Fluid Model

Abstract. We present a multi-stage centred scheme, of the kind proposed by Toro [Appl. Numer. Math. 56 (2006) 1464], for numerically resolving the simultaneous flow of two fluids through a transport pipeline. This model contains non-conservative 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 Parés [SIAM J. Numer. Anal. 44 (2006) 300]. 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 are obtained when the non-conservative 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.