High order methods play important roles in the modelling of compressible multi-component flows. However, they may generate negative sound speed, which leads to an instability of the numerical schemes. In this paper, we propose bound- and positivity-preserving limiters for high order finite difference schemes, based on which the equilibriums of the velocity and pressure are preserved throughout the whole computation of contact moving interface problems with the ideal and stiffened equations of state. For illustration purpose, high order alternative WENO scheme is taken for example. Numerical examples verify the theory and demonstrate the robustness of the proposed bound- and positivity-preserving limiters.

A Robust High Order Alternative WENO Scheme for the Five-Equation Model

A bound-preserving and positivity-preserving finite volume WENO scheme for solving five-equation model of two-medium flows

This work focuses on the analysis and development of a high-order physical-constraint-preserving discontinuous Galerkin (DG) method for compressible two-fluid flows by solving Kapila's five-equation model with the ideal gas equation of state. The proposed method is proved to possess the following two attractive advantages: first, the high-order DG discretization intrinsically satisfies the mechanical equilibrium criterion around an isolated material interface, and the numerical oscillations due to the use of high-order approximations are well suppressed by using a cell-based characteristic-wise weighted essentially non-oscillatory (WENO) limiter without destroying the mechanical equilibrium criterion; second, it maintains the physical-constraint-preserving property, including the bound-preserving property of volume fractions and positivity-preserving property of partial densities and internal energy. A thorough analysis on the sufficient condition is given to achieve the physical-constraint-preserving property. Several benchmark test cases are examined to validate the accuracy and robustness of the proposed method. Further discussion on the application of the physical-constraint-preserving DG method for solving Kapila's five-equation model with the stiffened gas equation of state is also presented. 

Analysis on physical-constraint-preserving high-order discontinuous Galerkin method for solving Kapila's five-equation model

We propose and analyze a seamless extended Discontinuous Galerkin (DG) discretization of advection–diffusion equations on semi-infinite domains. The semi-infinite half line is split into a finite subdomain where the model uses a standard polynomial basis, and a semi-unbounded subdomain where scaled Laguerre functions are employed as basis and test functions. Numerical fluxes enable the coupling at the interface between the two subdomains in the same way as standard single domain DG interelement fluxes. A novel linear analysis on the extended DG model yields unconditional stability with respect to the Peclet number. Errors due to the use of different sets of basis functions on different portions of the domain are negligible, as highlighted in numerical experiments with the linear advection–diffusion and viscous Burgers’ equations. With an added damping term on the semi-infinite subdomain, the extended framework is able to efficiently simulate absorbing boundary conditions without additional conditions at the interface. A few modes in the semi-infinite subdomain are found to suffice to deal with outgoing single wave and wave train signals more accurately than standard approaches at a given computational cost, thus providing an appealing model for fluid flow simulations in unbounded regions.

A Seamless, Extended DG Approach for Advection–Diffusion Problems on Unbounded Domains

/pdf/diffuse-interface-relaxation-model-for-two-phase-7xa0y5nu.pdf

Diffuse interface relaxation model for two-phase compressible flows with diffusion processes

A fifth order finite difference alternative weighted essentially non-oscillatory scheme is studied for a five-equation model, which plays an important role in the modelling of compressible multi-component flows. In our algorithm, the primitive variables are used in the weighted essentially non-oscillatory interpolation, from which it can be proved that the equilibriums of the velocity and pressure are preserved respectively throughout the whole computation of contact moving interface problems, under the ideal and stiffened gases equations of state. In addition, it is found that with the interpolation being performed on the characteristic variables which are the projections of the primitive variables, non-physical oscillations appearing around the contact waves can be significantly restrained. Numerical examples verify the theory and illustrate the robustness of the proposed scheme.

High Order Finite Difference Alternative WENO Scheme for Multi-component Flows

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A Robust High Order Alternative WENO Scheme for the Five-Equation Model

High Order Finite Difference Alternative WENO Scheme for Multi-component Flows