Amount Of Numerical Diffusion Research Articles

Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

High-order spectral element methods (SEM) for large-eddy simulation (LES) are still very limited in industry. One of the main reasons behind this is the lack of robustness of SEM for under-resolved simulations, which can lead to the failure of the computation or to inaccurate results, aspects that are critical in an industrial setting. To help address this issue, we introduce a non-modal analysis technique that characterizes the numerical diffusion properties of spectral element methods for linear convection–diffusion problems, including the scales affected by numerical diffusion and the relationship between the amount of numerical diffusion and the level of under-resolution in the simulation. This framework differs from traditional eigenanalysis techniques in that all eigenmodes are taken into account with no need to differentiate them as physical or unphysical. While strictly speaking only valid for linear problems, the non-modal analysis is devised so that it can give critical insights for under-resolved nonlinear problems. For example, why do SEM sometimes suffer from numerical stability issues in LES? And, why do they at other times be robust and successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these questions in turn provides crucial guidelines to construct more robust and accurate schemes for LES.For illustration purposes, the non-modal analysis is applied to the hybridized discontinuous Galerkin methods as representatives of SEM. The effects of the polynomial order, the upwinding parameter and the Péclet number on the so-called short-term diffusion of the scheme are investigated. From a non-modal analysis point of view, and for the particular case of hybridized discontinuous Galerkin methods, polynomial orders between 2 and 4 with standard upwinding are found to be well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected due to low and non-monotonic numerical diffusion. The non-modal analysis results are tested against under-resolved turbulence simulations of the Burgers, Euler and Navier–Stokes equations. While devised in the linear setting, non-modal analysis successfully predicts the behavior of the scheme in the nonlinear problems considered. Although the focus of this paper is on LES, the non-modal analysis can be applied to other simulation fields characterized by under-resolved scales.

Analysis of upwind and high‐resolution schemes for solving convection dominated problems in porous media

Abstract The conservation laws governing the multiphase flows in porous media are often convection‐dominated and have a steep fronts that require accurate resolution. Standard discretization methods of the convection terms do not perform well for such problems. The main aim of this work is to analyze the use of upwind and high‐ resolution schemes in such cases. First, we use a first differential approximation method to perform a theoretical analysis of a standard upwind approximation and different time stepping schemes for the linear hyperbolic equations in 1‐ and 2D. Next, we present a popular approach to reduce the amount of numerical diffusion introduced by upwind approximation ‐ high‐resolution schemes. We compare our implementation of one of the recently proposed central‐upwind schemes against the upwind schemes on several test problems based on Buckley‐Leverett equation and discuss the results. Finally, a parallel version of central‐upwind scheme in 2D is presented. It was implemented using our C++ ...

ANALYSIS OF UPWIND AND HIGH‐RESOLUTION SCHEMES FOR SOLVING CONVECTION DOMINATED PROBLEMS IN POROUS MEDIA

The conservation laws governing the multiphase flows in porous media are often convection‐dominated and have a steep fronts that require accurate resolution. Standard discretization methods of the convection terms do not perform well for such problems. The main aim of this work is to analyze the use of upwind and high‐ resolution schemes in such cases. First, we use a first differential approximation method to perform a theoretical analysis of a standard upwind approximation and different time stepping schemes for the linear hyperbolic equations in 1‐ and 2D. Next, we present a popular approach to reduce the amount of numerical diffusion introduced by upwind approximation ‐ high‐resolution schemes. We compare our implementation of one of the recently proposed central‐upwind schemes against the upwind schemes on several test problems based on Buckley‐Leverett equation and discuss the results. Finally, a parallel version of central‐upwind scheme in 2D is presented. It was implemented using our C++ library of parallel arrays ‐ ParSol.

On an extended clarifier-thickener model with singular source and sink terms

A one-dimensional model of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. This model also includes a singular feed source. In this paper, the clarifier-thickener model studied in a previous paper (Numer. Math.97 (2004) 25–65) is extended by a singular sink through which material is extracted from the unit. A difficulty is that in contrast to the singular source, the sink term cannot be incorporated into the flux function; rather, the sink is represented by a new non-conservative transport term. To focus on the new analytical difficulties arising due to this non-conservative term, a reduced problem is formulated, which contains the new sink term of the extended clarifier-thickener model, but not the source term and flux discontinuities. The paper is concerned with numerical methods for both models (extended and reduced) and with the well-posedness analysis for the reduced problem. For the reduced problem, a definition of entropy solutions, based on Kružkov-type entropy functions and fluxes, is provided. Jump conditions are derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. Two variants of the numerical scheme are introduced. Numerical examples illustrate that all three variants converge to the entropy solution, but introduce different amounts of numerical diffusion.

Numerical diffusion in the FCT algorithm, revisited

Numerical diffusion in a flux-corrected transport (FCT) algorithm embedded in a Navier–Stokes solver (TINY3D) has been analytically and numerically studied for flows where density variations can be neglected. It is found that numerical diffusion can be analytically expressed in a form similar to that of viscous diffusion. The effective total viscosity can be written as an effective viscosity which is the sum of the physical and numerical viscosities. A low-Mach-number laminar boundary-layer flow is used to test the analytical model of numerical diffusion. A series of simulations, in which the amount of numerical diffusion is varied, show results consistent with predictions of boundary-layer theory when the effective total viscosity is used. The minimum required numerical viscosity to meet the linear stability condition and the lower and upper limits of the cell Reynolds number are also derived.

Potential Vorticity and the Quasigeostrophic and Semigeostrophic MesoscaleVertical Velocity

Results of a variety of numerical simulations are presented and the accuracy of quasigeostrophic (QG) and semigeostrophic (SG) vertical velocity estimates of the total vertical velocity is analyzed. The authors examine the dependence of the results on the potential vorticity (PV) anomaly of the flow, its time evolution, and the amount of numerical diffusion. The SG ω equation is solved in a novel way in the original physical coordinates rather than in geostrophic coordinates. A three-dimensional numerical model is used that explicitly conserves the PV on isopycnal surfaces through a contour-advective semi-Lagrangian (CASL) algorithm. The numerical simulations consist initially of one or two horizontal cylinders of anomalous PV: a shear zone that induces two or three counterflowing jets. These jets destabilize and break into cyclones or anticyclones. This is accompanied by enhanced vertical motion, which exhibits a dominantly balanced quadrupole pattern in horizontal cross sections, depending on the ellipticity of the gyres, together with weak second-order inertia–gravity waves. For flows containing only negative PV anomalies the magnitude of both the QG and SG vertical velocities are smaller than the magnitude of the total vertical velocity, while the opposite occurs for flows containing only positive PV anomalies. The reason for this behavior is that the QG ω equation misses a term proportional to the Laplacian of the horizontal velocity. A new, more accurate, ω equation is proposed to recover the vertical velocity when both experimental density and horizontal velocity data are available. The SG solution is nearly always more accurate than the QG solution, particularly for the largest vertical velocity values and when the flow has single-signed PV anomalies. For flows containing both positive and negative PV anomalies, for example, mushroomlike eddies, the QG vertical velocity is a better approximation to the total vertical velocity than the SG solution. The reason for this anomalous behavior lies in one additional assumption concerning the conservation of volume that is usually adopted to derive the SG ω equation.

Comparison of FCT with other numerical schemes for the burgers equation

The flux corrected transport (FCT) method is used to solve the linear and nonlinear Burgers Equations and to compare with other commonly used explicit methods such as the central difference (CD), the upwind (UW), and the predictor corrector or MacCormack (MC) schemes. The results show that the FCT scheme exhibits the best behavior for the linear as well as nonlinear cases. The amount of numerical diffusion is the least for the FCT scheme compared to the other schemes and more importantly there are no non-physical oscillations.

Comparison of Leapfrog, Smolarkiewicz, and MacCormack Schemes Applied to Nonlinear Equations

The MacCormack scheme is a finite-difference scheme widely used in the aerospace simulations. It is a two-step algorithm, and contains a small amount of implicit numerical diffusion that makes it numerically stable without having to use any explicit filtering. It uses a nonstaggered grid. A detailed comparison with the leapfrog and Smolarkiewicz schemes is presented using the nonlinear advection equation and the Euler equations for a variety of conditions at different Courant numbers. Of the schemes tested, the unfiltered leapfrog is the least acceptable for the solution of nonlinear equations. Although it is numerically stable for linear problems, when used to solve nonlinear equations (without using any explicit filtering) it becomes numerically unstable or nonlinearly unstable. Furthermore, it introduces large phase errors, and produces better results with small Courant numbers. The MacCormack scheme is nonlinearly stable, produces modest amounts of numerical dispersion and diffusion, has no phase speed error, and works best at Courant numbers close to 1. When applied to nonlinear equations, the Smolarkiewicz scheme exhibits the least amount of numerical diffusion but more numerical dispersion than the MacCormack scheme. For stability it requires Courant numbers equal to or smaller than 0.5. In practical applications, we recommend the MacCormack scheme for the solution of the nonlinear equations, and either the Smolarkiewicz or the MacCormack scheme for equations involving conservation of a passive scalar.

Modern Finite-Volume Advection Schemes: Numerical Experiments in a One-Dimensional Test-Bed

The numerical solution of the transport (i.e. the continuity) equation for trace species, in particular in three-dimensional circulation models, has recently received much attention. Two conceptually different approaches are common. Firstly, classical numerical methods employed in circulation models for the solution of other continuity equations are also used for the. transport problem. Secondly, methods have been developed, that are especially designed to obviate problems arising in the numerical solution of the transport equation. In a recent study (Muller [1]) both approaches were investigated and compared in a simple one-dimensional test-bed. It was concluded that it is questionable whether classical numerical methods are able to control undesirable numerical artefacts such as dispersive ripples, negative concentrations, phase errors and numerical diffusion. The modern advection schemes, in contrast, perform considerably better, where computationaly relatively inexpensive schemes (Bott [2], Smolarkiewicz, and Grabowski [3]) are preferable if a certain amount of numerical diffusion can be tolerated. Moreover, numerical diffusion can be controled much better if more sophisticated methods (Prather [4], Carpenter et al. [5]) are employed. This, however, increases the computational effort considerably. Here, some hitherto undis-cussed aspects of Eulerian finite-volume schemes are elucidated. In particular, it is shown that interpolation based schemes can advect multiple peak structures properly, provided the resolution is sufficient. Further, advection schemes, originally designed for transporting shock fronts, are shown to possess problematic features if applied to other (in this case triangular) structures.

An investigation into the numerical dispersion problem of the skew upwind finite difference scheme

An assessment of the numerical dispersion (over- and undershoot) problem for the skew upwind finite difference scheme in practical flow simulation has been undertaken. In an earlier study by the authors, the skew upwind scheme was found to have the least amount of numerical diffusion. The scheme, howeve,r produced unacceptable dispersion in the solution. This study examines the unboundedness in the solution for three test cases and the use of bounding schemes to eliminate the oscillations. The three test cases are a two-dimensional jet, two parallel interacting streams, and transport of four scalar profiles. The bounding techniques used are simple bounding and flux corrected transport. A time-dependent two-dimensional explicit prediction procedure based on the marker and cell method is utilized. For pure convection flow, the study shows that the scheme consistently produces significant numerical dispersion for profiles that possess sharp discontinuity. The dispersion is less significant for other profiles. Bounding is found to be very effective in alleviating the problem. Simple bounding is found to be convenient, economical, and effective as compared to the other bounding scheme.