A Comparative Study of Rosenbrock-Type and Implicit Runge-Kutta Time Integration for Discontinuous Galerkin Method for Unsteady 3D Compressible Navier-Stokes equations (Communications in Computational Physics, 2018)
A comparative study of two classes of third-order implicit time integrationschemes is presented for a third-order hierarchical WENO reconstructed discontinuousGalerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokesequations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3)scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differentialalgebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme,a remarkable feature of the ROW schemes is that, they only require one approximateJacobian matrix calculation every time step, thus considerably reducing the overallcomputational cost. A variety of test cases, ranging from inviscid flows to DNS ofturbulent flows, are presented to assess the performance of these schemes. Numericalexperiments demonstrate that the third-order ROW scheme for the DAEs of index-2can not only achieve the designed formal order of temporal convergence accuracy ina benchmark test, but also require significantly less computing time than its ESDIRK3counterpart to converge to the same level of discretization errors in all of the flowsimulations in this study, indicating that the ROW methods provide an attractive alternativefor the higher-order time-accurate integration of the unsteady compressibleNavier-Stokes equations.