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Supercomputer power has been doubling approximately every 14 months for several decades, increasing the capabilities of scientific modelling at a similar rate. However, to utilize these machines effectively for applications such as computational fluid dynamics, improvements to strong scalability are required. Here, the particular focus is on semi-implicit, viscous-flow CFD, where the largest bottleneck to strong scalability is the parallel solution of the linear pressure-correction equation — an elliptic Poisson equation. State-of-the-art linear solvers, such as Krylov subspace or multigrid methods, provide excellent numerical performance for elliptic equations, but do not scale efficiently due to frequent synchronization between processes. Complete desynchronization is possible for basic, Jacobi-like solvers using the theory of ‘chaotic relaxations’. These non-deterministic, chaotic solvers scale superbly, as demonstrated herein, but lack the numerical performance to converge elliptic equations — even with the relatively lax convergence requirements of the example CFD application. However, these chaotic principles can also be applied to multigrid solvers. In this paper, a ‘chaotic-cycle’ algebraic multigrid method is described and implemented as an open-source library. It is tested on a model Poisson equation, and also within the context of CFD. Two CFD test cases are used: the canonical lid-driven cavity flow and the flow simulation of a ship (KVLCC2). The chaotic-cycle multigrid shows good scalability and numerical performance compared to classical V-, W- and F-cycles. On 2048 cores the chaotic-cycle multigrid solver performs up to faster than Flexible-GMRES and faster than classical V-cycle multigrid. Further improvements to chaotic-cycle multigrid can be made, relating to coarse-grid communications and desynchronized residual computations. It is expected that the chaotic-cycle multigrid could be applied to other scientific fields, wherever a scalable elliptic-e

The iterative error can be an important part of the total numerical error of any Com- putational Fluid Dynamics simulation when the iterative convergence stagnates or when loose convergence criteria are used. In the quest for better iterative convergence of CFD simulations, we consider the design of iterative methods for the Reynolds-averaged Navier-Stokes equations, discretized by finite-volume methods with cell-centered, co-located variables. The central point is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation.

In this paper, CFD calculations for current loads on an LNG carrier and a semi-submersible are presented, both for model and full-scale situations, for current angles ranging from 180 to 0 degrees. MARIN’s in-house URANS code ReFRESCO is used. Numerical studies are carried out concerning iterative convergence and grid refinement. In total, more than 100 calculations have been performed. Detailed verification analysis is carried out using modern techniques, and numerical uncertainties are calculated. Afterwards, quantitative validation for model-scale Reynolds number is done taking into account numerical and experimental uncertainties. Scale effects on the current coefficients are investigated, having in mind the estimated numerical uncertainties, and unsteady effects are briefly studied. Good iterative convergence is obtained in most calculations, i.e. a decrease in residuals of more than 5 orders is achieved. The sensitivity to grid resolution has been investigated for both model and full scale using five consecutively refined grids and for 3 current headings. The differences in the solution between two consecutive refinements converge for all cases. The numerical uncertainties are larger for angles with small values of the loads. Comparison with experiments shows that ReFRESCO provides good quantitative prediction of the current loads at model scale: for angles with larger forces the CFD results are validated with 15% of uncertainty. To determine scale effects the numerical uncertainties must be considered in order to prevent wrong conclusions drawn on basis of numerical differences rather than on physical differences. For the full-scale results larger numerical uncertainties are found than for model scale and for absolute values for scale effects this uncertainty should be improved. For the LNG carrier significant scale effects, i.e. more than 40%, have been obtained for current angles where the friction component is dominant. For these cases the numerical uncert

In this article, we analyse the convergence of multigrid (MG) iteration for solving the algebraic equations arising from a space-time discontinuous Galerkin (DG) discretization of the advection-diffusion equation. To keep the MG method fully explicit, we consider Runge-Kutta smoothers that solve the algebraic equations by marching in pseudo-time to steady state. Depending on the P

An overview is given of a space-time discontinuous Galerkin finite element method for the compressible Navier-Stokes equations. This method is well suited for problems with moving (free) boundaries which require the use of deforming elements. In addition, due to the local discretization, the space-time discontinuous Galerkin method is well suited for mesh adaptation and parallel computing. The algorithm is demonstrated with computations of the unsteady flow field about a delta wing and a NACA0012 airfoil in rapid pitch up motion.