Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations

Abstract

a b s t r a c tEnergy-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 convectiondominated 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.

Boundary treatment for fourth-order staggered mesh discretizations of the incompressible Navier–Stokes equations

“…This second order method is an example of a Runge-Kutta method based on Gauss quadrature, which conserves energy in time. They are discussed in more detail in [18]. We will use the implicit midpoint method to demonstrate energy conservation in both time and space in the numerical simulations that will be presented later.…”

“…The velocity field is discontinuous in the corners. Furthermore, in contrast to [3], we use the implicit midpoint method for time integration instead of the Crank-Nicolson method, since the latter is not truly energy-conserving [18].…”

“…As was the case for the convective operator, the fluxes on both the small and large volume are approximated to fourth order: 18) and similarly for ( ∂u ∂x ) i+3/2, j . In short, the diffusive discretization reads .…”

“…Recently, symmetry-preserving and energy-conserving discretization methods have gained popularity for the solution of the incompressible Navier-Stokes equations [26,18]. In a symmetry-preserving discretization, the discrete difference operators mimic certain symmetry properties of the continuous differential operators.…”

A Novel Scheme for Liouville’s Equation with a Discontinuous Hamiltonian and Applications to Geometrical Optics

“…Such integrators play an important role in for instance astrophysics [14,15]. One remarkable application is the usage of a symplectic integrator to obtain an energy-preserving computational fluid dynamics scheme [16].…”