Computational codes for direct numerical simulations of Rayleigh-Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra = 10 8 and Pr = 1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code (AFID/RBflow) and a specialized fourth-order finite-volume code (Goldfish) are compared with a general purpose finite-volume approach (OpenFOAM) and a general purpose spectral-element code (Nek5000). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID/RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes.

Abstract. A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear PDEs the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as a source term, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection-diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection-diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.Key words. parallel computing, exponential integrators, partial differential equations, parallel in time, block Krylov subspace.AMS subject classifications. 65F60, 65L05, 65Y051. Introduction. Recent developments in hardware architecture, bringing to practice computers with hundreds of thousands of cores, urge the creation of new, as well as a major revision of existing numerical algorithms [14]. To be efficient on such massively parallel platforms, the algorithms need to employ all possible means to parallelize the computations. When solving partial differential equations (PDEs) with a time-dependent solution, an important way to parallelize the computations is, next to the parallelization across space, parallelization across time. This adds a new dimension of parallelism with which the simulations can be implemented. In this paper we present a new time-parallel integration method extending the Paraexp method [21] to nonlinear partial differential equations using Krylov methods and waveform relaxation.Several approaches to parallelize the simulation of time-dependent solutions in time can be distinguished. The first important class of the methods are the waveform relaxation methods [6,38,44,56], including the space-time multigrid methods for parabolic PDEs [8,24,30,37]. The key idea is to start with an approximation to the numerical solution for the whole time interval of interest and update the solution, solving an easier-to-solve approximate system in time. The Parareal method [36], which attracted significant attention recently, is a prime example related to the class of waveform relaxation methods [19].Parallel Runge-Kutta methods and general linear methods, where the parallelism is determined and restricted by the number of stages or steps, form another class of the time-parallel methods [8,12,52]. Time-parallel schemes can also be obtained

We present an exponential time integration method for the incompressible Navier-Stokes equation. An essential step in our procedure is the treatment of the pressure by applying a divergence-free projection to the momentum equation. The differential-algebraic equation for the discrete velocity and pressure is then reduced to a conventional ordinary differential equation that can be solved with the proposed exponential integrator. A promising feature of exponential time integration is its potential time parallelism within the Paraexp algorithm. We demonstrate that our approach leads to parallel speedup assuming negligible parallel communication.

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