Implicit-Explicit Schemes for Compressible Cahn–Hilliard–Navier–Stokes Equations

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The isentropic compressible Cahn–Hilliard–Navier–Stokes equations are a system of fourth-order partial differential equations that model the evolution of some binary fluids under convection. The purpose of this paper is the design of efficient numerical schemes to approximate the solution of initial-boundary value problems with these equations. The efficiency stems from the implicit treatment of the high-order terms in the equations. Our proposal is a second-order linearly implicit-explicit time stepping scheme applied in a method of lines approach, in which the convective terms are treated explicitly and only linear systems have to be solved. Some experiments are performed to assess the validity and efficiency of this proposal.