A class of Rosenbrock methods for a reconstructed discontinuous Galerkin method for the unsteady compressible flows

A set of Rosenbrock-Wanner (ROW) time integration schemes for index-2 Differential Algebraic Equations (DAEs), are presented for a hierarchical WENO reconstruction-based discontinuous Galerkin (rDG) method for time-accurate solution of the unsteady compressible Navier-Stokes equations on three- dimensional hybrid grids, where the index refers to the minimum number of times that a DAEs system must be differentiated to obtain an ODEs system. A comparative study between the ROW for index-1 and index-2, and ESDIRK (explicit first stage, single diagonal, diagonally implicit Runge-Kutta) schemes is conducted in the context of rDG method through a variety of test cases. We found that the ROW schemes for index-2 DAEs not only achieved the designed formal order of temporal accuracy in benchmark test, but also require significantly less computing time than the equivalent ESDIRK to achieve the same level of temporal accuracy in all the flow simulations of our study, indicating a more efficient high-order DG solution for unsteady flows.