Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

In this work we investigate the use of linearly implicit Rosenbrock-type Runge-Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier-Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier-Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil.