Family of Runge–Kutta-Based Algorithms for Unsteady Incompressible Flows

Abstract

A family of algorithms based on implicit Runge–Kutta time integration suitable for unsteady incompressible flow is developed. Spatially discretized momentum equations are integrated in time using implicit Runge–Kutta methods. An equation for pressure is formed by combining the discretized continuity equation with the discretized momentum equations. This family of algorithms, called IRK-SIMPLER, uses only exact discretized mass and momentum equations and requires no relaxation to converge. Two distinct IRK-SIMPLER variants are analyzed. The first variant solves the pressure equation once per time step and has the advantage of fewer computations per time step but is limited to first-order accuracy in time. The second variant centers on reformulating a pressure equation each Runge–Kutta stage by manipulating the momentum stage equations, and this pressure equation is solved multiple times within a time step. The second variant is shown to achieve a higher temporal order of accuracy. By analyzing the largest time step allowable and run time required to achieve time-accurate solutions, the most efficient algorithms in this family are delineated. For the chosen problems, IRK-SIMPLER is able to achieve as high as 101 times speedup compared to the traditional SIMPLER algorithm with Crank–Nicolson time integration using a finite volume discretization scheme.