The Growth of Rounding Errors in LES

Abstract

Previous chapters have shown the power of Large Eddy Simulation (LES) to predict non-reacting [352; 318] as well as reacting turbulent flows [340; 338; 276; 275; 355]. The main strength of LES compared to classical Reynolds Averaged (RANS) methods is that, like Direct Numerical Simulation (DNS) [342; 326; 372], LES explicitly captures large scale unsteady motions due to turbulence instead of modeling them. LES also captures the multiple instability modes found in reacting flows. An often ignored aspect of this feature is that like DNS, LES is also submitted to a well-known feature of turbulent flows: the exponential separation of trajectories [371] implies that the flow solution exhibited by LES is very sensitive to any “small perturbations”. These small perturbations which can induce instabilities can have different sources: Rounding errors are the first source of random noise in any finite precision computation: they constitute an unavoidable forcing for the Navier-Stokes equations and may lead to LES variability. The study of error growth in finite precision computations is an important topic in applied mathematics [367; 264] but has found few applications in multidimensional fluid mechanics because of the complexity of the codes used in CFD. Initial conditions are a second source of LES results variability: these conditions are often unknown and any small change in initial conditions may trigger significant changes in the LES solution. Boundary conditions, in particular the unsteady velocity profiles imposed at inlets and outlets, can have the same effect as initial conditions but are not studied here. Due to its large computational resource requirements, modern LES heavily relies on parallel computing. However, in codes using domain decomposition, it is also an additional “noise” source in the Navier-Stokes equations especially at partition interfaces. Even in explicit codes, where the algorithm is independent of the number of processors, the different summation orders with which a nodal value is reconstructed at partition interfaces, may induce non-associativity errors. For example, in explicit codes on unstructured meshes using cell vertex methods [356], the residual at one node is obtained by adding the weighted residuals of the surrounding cells. Additions of only two summands are perfectly associative. Moreover, it must be noted that not all additions of more than two summands generate non-associativity errors. However, in some cases, summation may yield distinct results for floating-point accumulation: the rounding errors in (a+b)+c and in a+(b+c) may be different, in particular if there are large differences in orders of magnitude between the terms [294]. After thousands of iterations, the LES result may be affected. Since these rounding errors are induced by non deterministic message arrival at partition interfaces, it is believed that such behaviour may occur for any unstructured parallel CFD code, regardless of the numerical schemes used. As a consequence, the simulation output might change when run on a different number of processors. The case of implicit codes [318; 282] or in space (such as compact schemes) [312; 250; 363] is not considered here: for such schemes, the methods used to solve the linear system appearing at each iteration [351; 281] depend on the number of processors. Therefore, rounding errors are not the only reason why solutions obtained with different numbers of processors differ. Even on a single processor computation, internal parameters of the partitioning algorithm may couple with rounding errors to force the LES solution. For example, a different reordering of nodes using the Cuthill-McKee (CM) or the reverse Cuthill-McKee (RCM) algorithm [271; 314] may produce the same effect as a simple perturbation and can be the source of solution divergence.