Abstract
In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.
Highlights
In many engineering and scientific problems, unsteady compressible fluid dynamics play a key role
For an explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) method, the first stage becomes explicit, whereas for a ROW method, the nonlinear solve is replaced by a linear solve only
There is a feedback loop between the nonlinear iterations and the time step size, which in a way creates an upper bound on the time step
Summary
In many engineering and scientific problems, unsteady compressible fluid dynamics play a key role. The use of these schemes in the context of compressible Navier-Stokes equations was analyzed in [2] where they were demonstrated to be more efficient than implicit Euler (BDF-1) and BDF-2 methods for moderately accurate solutions This result about the comparative efficiency of BDF-2 and ESDIRK4 was later confirmed for unsteady Euler flow by [15] and for a discontinuous Galerkin discretization by [44]. We compare time adaptive SDIRK, ESDIRK and Rosenbrock methods in the context of finite volume discretizations of the compressible Navier-Stokes equations based on their work error ratio for realistic problems. We present results for the nonlinear convection-diffusion equation and Navier-Stokes simulations in Sections 4 and 5
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