SUMMARYThe Navier-Stokes-Korteweg (NSK) system is a classical diffuse-interface model for compressible twophase flow. However, the direct numerical simulation based on the NSK system is quite expensive and in some cases even not possible. We propose a lower-order relaxation of the NSK system with hyperbolic firstorder part. This allows applying numerical methods for hyperbolic conservation laws and removing some of the difficulties of the original NSK system. To illustrate the new ansatz, we first present a local discontinuous Galerkin method in one and two spatial dimensions. It is shown that we can compute initial boundary value problems with realistic density ratios and perform stable computations for small interfacial widths. Second, we show that it is possible to construct a semi-discrete finite-volume scheme that satisfies a discrete entropy inequality.

The Navier-Stokes-Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flow. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicitexplicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.

Abstract. We present a numerical scheme for immiscible two-phase flows with one compressible and one incompressible phase. Special emphasis lies in the discussion of the coupling strategy for compressible and incompressible Euler equations to simulate inviscid liquid-vapour flows. To reduce the computational effort further, we also introduce two approximate coupling strategies. The resulting schemes are compared numerically to a fully compressible scheme and show good agreement with these standard algorithm at lower numerical costs.

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