Low-cost energy-preserving RK schemes for turbulent simulations

Energy-conserving numerical methods are widely employed in direct and large eddy simulation of turbulent flows. Full conservation of energy is usually obtained by employing suitable choiches for the discretization of both spatial and temporal operators. The construction of conservative spatial discretizations faces with the fact that convective terms can be expressed in several different forms which are not equivalent from a numerical viewpoint. The usually encountered advective and divergence forms are known to be not energy-preserving, and their use is usually ruled out for numerical simulation of turbulence. On the other hand, the skew-symmetric splitting, defined as a suitable average of the divergence and advective forms, generally allows kinetic energy to be locally conserved by convection. This last choice, however, has the drawback of being roughly twice as expensive as standard divergence or advective forms alone. Exact conservation of spatial discretizations is usually lost when non conservative temporal integration schemes are used. As regards temporal integration, in the framework of Runge Kutta schemes implicit time advancement is required, which further charges the cost of a single time step in both direct and large eddy simulations. A novel time-advancement strategy that retains the conservation properties of skew-symmetric-based schemes at a reduced computational cost has been developed in the framework of explicit Runge Kutta schemes. It is found that sub-optimal energy-conservation can be achieved by properly constructed Runge-Kutta schemes in which only divergence and advective forms for the convective term are adopted. These schemes can be considerably faster than skew-symmetric-based techniques. The new time advancement strategy consists in a procedure in which a given sequence of advective and divergence forms is employed within the stages of an explicit Runge-Kutta method. The error on energy conservation introduced at each time-step can be then expressed as a Taylor series in the integration interval size. By nullifying the successive terms of the expansion, one obtains a nonlinear system of equations for the Runge-Kutta coefficients, which can be coupled to the classical order conditions to obtain new Runge-Kutta schemes with prescribed accuracy on energy conservation. The procedure has proven to be able to produce new methods with a specified order of accuracy on both solution and energy, and to reveal interesting conservation properties of some classical Runge Kutta schemes. The effectiveness of the method is demonstrated by numerical simulation of Burgers' equation.