Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations

In this paper a high-order implicit multi-step method, known in the literature as Two Implicit Advanced Step-point (TIAS) method, has been implemented in a high-order Discontinuous Galerkin (DG) solver for the unsteady Euler and Navier–Stokes equations. Application of the absolute stability condition to this class of multi-step multi-stage time discretization methods allowed to determine formulae coefficients which ensure A-stability up to order 6. The stability properties of such schemes have been verified by considering linear model problems. The dispersion and dissipation errors introduced by TIAS method have been investigated by looking at the analytical solution of the oscillation equation. The performance of the high-order accurate, both in space and time, TIAS-DG scheme has been evaluated by computing three test cases: an isentropic convecting vortex under two different testing conditions and a laminar vortex shedding behind a circular cylinder. To illustrate the effectiveness and the advantages of the proposed high-order time discretization, the results of the fourth-and sixth-order accurate TIAS schemes have been compared with the results obtained using the standard second-order accurate Backward Differentiation Formula, BDF2, and the five stage fourth-order accurate Strong Stability Preserving Runge–Kutta scheme, SSPRK4.