Matrix‐free modified extended BDF applied to the discontinuous Galerkin solution of unsteady compressible viscous flows

Abstract

Summary In this work, a time‐accurate integration of the discontinuous Galerkin space‐discretized Navier‐Stokes equations is performed exploiting the matrix‐free (MF) approach to speed up the solution process of the modified extended backward differentiation formulae (MEBDF) schemes. MEBDF are high‐order accurate implicit multistep schemes composed by three nonlinear stages. The proposed algorithm consists in solving the resulting nonlinear system of each stage with a preconditioned MF Newton/Krylov method using a frozen preconditioner strategy to improve its efficiency. Numerical results for compressible inviscid and viscous test cases, both with a known analytical solution, aim at assessing the performance of the proposed MF‐MEBDF algorithm, by comparing it with the one obtained by using its matrix‐explicit counterpart or the explicit strong stability preserving Runge‐Kutta scheme. In particular, the influence of some relevant physical (low‐speed flows) and discretization (aspect ratio, polynomial degree) aspects on the performance of the different time integration schemes are investigated, highlighting the pros and cons of the proposed algorithm and its effectiveness in solving nonstiff and stiff systems. Furthermore, the scalability in parallel computations of the proposed algorithm is investigated and, finally, its potential for efficient long‐time simulations is demonstrated computing a laminar vortex shedding behind a circular cylinder at different Reynolds numbers and by comparing its effectiveness with that of the strong stability preserving Runge‐Kutta scheme.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

“…Nevertheless, starting from implicit "classical" methods, like the Backward Differentiation Formulae (BDF) [5] or the second-order accurate Crank-Nicolson scheme (CN2) [6], many other variants have been developed in the past with the aim to improve their performance. Some examples of methods that have been developed to improve the stability region of the BDF schemes are the MEBDF [2,7,8] or the TIAS [3,9] schemes, that are in fact multi-stage methods A-stable up to order 4 and 6, respectively, or the second derivative method proposed by Enright [1,10] that has obtained a single-stage method A-stable up to order 4 by using a higher derivative of the solution. Another way to improve the stability properties of the BDF schemes was found with the Composite-Backward Differentiation Formulae (C-BDF) [4,11,12], which are schemes that inherit the L-stability property of the second-order accurate BDF scheme and that, to the knowledge of the author, have been mainly used until now for electromagnetic transient simulations [13] and for thermal radiative diffusion problems [14], and only recently has their potential in solving structural mechanics [15] and fluid dynamics [16] problems been investigated.…”

“…It is well suited for stiff problems thanks to its stability properties; • Even if it is composed of two stages, the coefficient that multiplies the two Jacobian matrices is the same, i.e., γ∆t, therefore, in particular when it is coupled with a Matrix-Free approach and a Frozen Preconditioner strategy [8], the CPU time and memory required to advance the solution in time can be greatly reduced; • Extrapolated values can be easily computed, exploiting the previous known solutions, and can be used as very good initial guesses for both the stages, thus accelerating the convergence of Newton's method.…”