Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier–Stokes equations. In this paper implicit Runge–Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge–Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge–Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convection-dominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier–Stokes equations.
Journal of Computational Physics

Sanderse, B. (2013). Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations. Journal of Computational Physics, 233, 100–131. doi:10.1016/