This paper investigates the temporal accuracy of the velocity and pressure when explicit Runge–Kutta methods are applied to the incompressible Navier–Stokes equations. It is shown that, at least up to and including fourth order, the velocity attains the classical order of accuracy without further constraints. However, in case of a time-dependent gradient operator, which can appear in case of time-varying meshes, additional order conditions need to be satisfied to ensure the correct order of accuracy. Furthermore, the pressure is only first-order accurate unless additional order conditions are satisfied. Two new methods that lead to a second-order accurate pressure are proposed, which are applicable to a certain class of three- and four-stage methods. A special case appears when the boundary conditions for the continuity equation are independent of time, since in that case the pressure can be computed to the same accuracy as the velocity field, without additional cost. Relevant computations of decaying vortices and of an actuator disk in a time-dependent inflow support the analysis and the proposed methods.
Additional Metadata
Keywords Differential–algebraic equations, Incompressible Navier–Stokes equations, Temporal accuracy, Time integration, Runge–Kutta method, Moving meshes
MSC Multistep, Runge-Kutta and extrapolation methods (msc 65L06), Methods for differential-algebraic equations (msc 65L80)
THEME Life Sciences (theme 5), Energy (theme 4)
Publisher Elsevier
Journal Journal of Computational Physics
Sanderse, B, & Koren, B. (2012). Accuracy analysis of explicit Runge-Kutta methods applied to the incompressible Navier-Stokes equations. Journal of Computational Physics, 231(8), 3041–3063.