Efficiency of diagonally implicit Runge-Kutta time integration schemes in incompressible two-phase flow simulations

Abstract

Following a rapidly growing popularity of the use of computational fluid dynamics (CFD) for free surface flow applications, especially in the naval field, cost efficient numerical schemes are highly desirable. In the present study, diagonally implicit Runge-Kutta (DIRK) time integration schemes up to third-order are implemented in OpenFOAM for incompressible two-phase flow simulations. The coupling of these time integration schemes with solving the incompressible Navier-Stokes equations and the Volume of Fluid (VOF) convection equation is described in detail. After detailed convergence study, the efficiency of these schemes is evaluated in a number of one-phase and two-phase benchmark cases in terms of their accuracy and CPU-to-accuracy ratio, especially in ranges of space and time resolution parameters of practical use. It is shown in particular that the proposed second order Runge-Kutta schemes present the best CPU-to-accuracy compromise among the tested schemes.