The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.

Error analysis and estimation for the finite volume method with applications to fluid flows

Modeling of highly advective transport is embarrassingly difficult, even in the superficially simple case of one-dimensional constant-velocity flow. In this paper, a fresh approach is taken, based on an explicit conservative control-volume formulation that makes use of a universal limiter for transient interpolation modeling of the advective transport equations. This ‘ULTIMATE’ conservative difference scheme is applied to unsteady, one-dimensional scalar pure advection at constant velocity, using three representative test profiles: a discontinuous step, an isolated sine-squared wave, and a semi-ellipse. The goal is to devise a single robust scheme which achieves sharp monotonic resolution of the step without corrupting the other profiles. The semi-ellipse is particularly challenging because of its combination of sudden and gradual changes in gradient. The ULTIMATE strategy can be applied to explicit conservative advection schemes of any order of accuracy. Second-order methods are shown to be unsatisfactory because of steepening and clipping typical of currently popular so-called ‘high resolution’ shock-capturing or TVD schemes. The ULTIMATE third-order upwind scheme is highly satisfactory for most flows of practical importance. Higher order methods give predictably better step resolution, although even-order schemes generate a (monotonic) waviness in the difficult semi-ellipse simulation. By using adaptive stencil expansion to introduce (in principle arbitrarily) higher order resolution locally in isolated regions of high curvature or high gradient, extremely accurate coarse-grid results can be obtained with very little additional cost above that of the base (third-order) scheme.

The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection

SUMMARY The issue of boundedness in the discretisation of the convection term of transport equations has been widely discussed. A large number of local adjustment practices has been proposed, including the well-known total variation diminishing (TVD) and normalised variable diagram (NVD) families of differencing schemes. All of these use some sort of an ‘unboundedness indicator’ in order to determine the parts of the domain where intervention in the discretisation practice is needed. These, however, all use the ‘far upwind’ value for each face under consideration, which is not appropriate for unstructured meshes. This paper proposes a modification of the NVD criterion that localises it and thus makes it applicable irrespective of the mesh structure, facilitating the implementation of ‘standard’ bounded differencing schemes on unstructured meshes. Based on this strategy, a new bounded version of central differencing constructed on the compact computational molecule is proposed and its performance is compared with other popular differencing schemes on several model problems. Copyright © 1999 John Wiley & Sons, Ltd.

High resolution NVD differencing scheme for arbitrarily unstructured meshes

A prognostic cloud fraction and prognostic condensate scheme has been developed for the Met Office Unified Model. This is designed to replace the scheme currently used in weather forecast and climate simulations, in which cloud fraction and liquid water content are calculated diagnostically. Such a scheme overprescribes links between cloud fraction, condensate and water vapour contents. By contrast, our new prognostic cloud fraction and prognostic condensate scheme (PC2) calculates increments to prognostic variables of liquid, ice and total cloud fractions, water vapour and liquid condensate as a result of each physical process represented in the model. (Ice condensate is already represented prognostically.) This paper provides a summary of the PC2 scheme, describes how it is implemented, and discusses its relationship with other existing cloud schemes. Key aspects of the PC2 formulation are: the consistent derivation of prognostic terms, the reversible nature of the scheme under idealised forcing scenarios, the well-behaved performance in the limit of very low and very high cloud fraction, the inclusion of ice microphysical processes, and the improved representation of cloud erosion processes. A companion paper presents the results from the scheme. © Crown Copyright 2008. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.

PC2: A prognostic cloud fraction and condensation scheme. I: Scheme description

https://ntrs.nasa.gov/api/citations/19850015928/downloads/19850015928.pdf

Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics

Part 1: Semi-Lagrangian advective schemes and their use in meteorological modeling by J. R. Bates Adaptive mesh refinement for hyperbolic equations by M. J. Berger Averaged multivalued solutions and time discretization for conservation laws by Y. Brenier Computing with high-resolution upwind schemes for hyperbolic equations by S. R. Chakravarthy and S. Osher Mathematical modeling of the steam-water condensation in a condenser by C. Conca Some theoretical and computational considerations of atmospheric fronts by M. J. P. Cullen and R. J. Purser Numerical solution of Euler's equation by perturbed functionals by S. K. Dey A mathematical model and numerical study of platelet aggregation during blood clotting by A. L. Fogelson Computational methods for discontinuities in fluids by O. A. McBryan Problems and numerical methods of the incorporation of mountains in atmospheric models by F. Mesinger and Z. I. Janjic Discrete shocks for systems of conservation laws by D. Michelson Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems by D. Michelson A mixed finite element method for $3d$ Navier-Stokes equations by J. C. Nedelec Stability of the flow around a circular cylinder to forced disturbances by V. A. Patel Some contributions to the modelling of discontinuous flows by P. Roe Techniques for numerical simulation of large-scale eddies in geophysical fluid dynamics by R. Sadourny Finite difference techniques for nonlinear hyperbolic conservation laws by R. Sanders Conforming finite element methods for incompressible and nearly incompressible continua by L. R. Scott and M. Vogelius Vortex methods and turbulent combustion by J. A. Sethian Numerical solutions of rotating internal flows by C. G. Speziale High resolution TVD schemes using flux limiters by P. Sweby Stability of hyperbolic finite-difference models with one or two boundaries by L. N. Trefethen Matching the Navier-Stokes equations with observations by T. Gal-Chen Dynamics of flame propagation in a turbulent field by A. F. Ghoniem New stability criteria for difference approximations of hyperbolic initial-boundary value problems by M. Goldberg and E. Tadmor A modified finite element method for solving the incompressible Navier-Stokes equations by P. M. Gresho Computational fusion magnetohydrodynamics by R. C. Grimm A finite amplitude eigenmode technique to solve and analyze nonlinear equations by R. Grotjahn Numerical boundary conditions by B. Gustafsson Numerical simulation in three space dimensions of time-dependent thermal convection in a rotating fluid by D. H. Hathaway and R. C. J. Somerville Static rezone methods for tensor-product grids by J. M. Hyman and M. J. Naughton A nonoscillatory shock capturing scheme using flux limited dissipation by A. Jameson Part 2: The use of spectral techniques in numerical weather prediction by M. Jarraud and A. P. M. Baede Improved flux calculations for viscous incompressible flow by the variable penalty method by H. Kheshgi and M. Luskin TVD schemes in one and two space dimensions by R. J. LeVeque and J. B. Goodman Upwind-difference methods for aerodynamic problems governed by the Euler equations by B. van Leer An MHD model of the earth's magnetosphere by C. C. Wu Application of TVD schemes for the Euler equations of gas dynamics by H. C. Yee, R. F. Warming, and A. Harten Recent applications of spectral methods in fluid dynamics by T. A. Zang and M. Y. Hussaini.

Björn Engquist

Papers

Large-scale computations in fluid mechanics