MATRIX-FREE MODIFIED EXTENDED BACKWARD DIFFERENTIATION FORMULAE APPLIED TO THE DISCONTINUOUS GALERKIN SOLUTION OF COMPRESSIBLE UNSTEADY VISCOUS FLOWS

In this work a matrix-free modified extended backward differentiation time integration method has been implemented in a high-order discontinuous Galerkin solver for the unsteady Navier-Stokes equations. The resulting non-linear systems at each time step are solved iteratively using a preconditioned inexact Newton/Krylov method. In order to speed-up the solution process a frozen preconditioner formulation and a polynomial extrapolation technique for computing a better initial guess for the Newton iterations have been considered. Numerical results for compressible inviscid and viscous test cases show the effectiveness of the proposed numerical strategies and the performance advantages of the matrix-free method compared to its matrix-explicit counterpart for this class of implicit multi-stage time schemes. Furthermore, the influence of some physical (low Mach) and space discretization (stretched grid) aspects is examined to highlight pros and cons of the proposed time integration algorithm and its potential in solving non-stiff and stiff systems with respect to the widely used explicit Runge-Kutta schemes.