A Second-Order IMEX Method for Multi-Phase Flow Problems

We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Ne...