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A positivity-preserving fractional algorithm is presented for solving thefour-equation homogeneous relaxation model (HRM) with an arbitrary number ofideal gases and a liquid governed by the stiffened gas equation of state. Thefractional algorithm consists of a time step of the hyperbolic five-equationmodel by Allaire et al. and an algebraic numerical thermal relaxation step atan infinite relaxation rate. Interpolation and flux limiters are proposed forthe use of high-order Cartesian finite difference or finite volume schemes in ageneral form such that the positivity of the partial densities and squaredsound speed, as well as the boundedness of the volume fractions and massfractions, are preserved with the algorithm. A conservative solution update forthe four-equation HRM is also guaranteed by the algorithm which is advantageousfor certain applications such as those with phase transition. The accuracy androbustness of the algorithm with a high-order explicit finite differenceweighted compact nonlinear scheme (WCNS) using the incremental-stencil weightedessentially non-oscillatory (WENO) interpolation, are demonstrated with variousnumerical tests.

A positivity-preserving Eulerian two-phase approach with thermal relaxation for compressible flows with a liquid and gases