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The subcharacteristic condition for hyperbolic relaxation systems states that wave velocities of an equilibrium system cannot exceed the corresponding wave velocities of its relaxation system. This condition is central to the stability of hyperbolic relaxation systems, and is expected to hold for most such models describing natural phenomena. In this paper, we study a hierarchy of two-phase flow models. We consider relaxation with respect to volume transfer, heat transfer and mass transfer. We formally verify that our relaxation processes are consistent with the first and second laws of thermodynamics, and present analytical expressions for the wave velocities for each model in the hierarchy. Through an appropriate choice of variables, we prove directly by sums-of-squares that for all relaxation processes considered, the subcharacteristic condition holds for any thermodynamically stable equation of state.

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.

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Abstract. Large time step explicit schemes in the form originally proposed by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463-477] have seen a significant revival in recent years. In this paper we consider a general framework of local 2k + 1 point schemes containing LeVeque's scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient.We identify the least and most diffusive TVD schemes in this framework. The most diffusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The least diffusive scheme is the Large Time Step scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes.We discuss the nature of entropy violations associated with the LTS-Roe scheme, in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions, and discuss numerical entropy fixes.Finally, we propose a one-parameter family of Large TimeStep TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The 1D Burgers equation and the Euler system are used as numerical illustrations.