Numerical experiments with a multiple grid and a preconditioned Lanczos type method

When finite-difference methods are used to solve the benchmark problem of natural convection in a square cavity, a very fine grid is required to obtain predictions that are accurate to 1-2%. The derivation of the finite-difference equations requires the introduction of many approximations; this study systematically evaluates these approximations to establish which are mainly responsible for the fine-grid requirement. The poorest approximations are then improved one by one, resulting in a scheme that yields highly accurate predictions using a relatively coarse grid. The method of evaluating the accuracy of the approximations, the improved approximations themselves, and the solution method used all contain novel features. Storage and computing time requirements for a new sparse matrix solver, which was used in the current study to simultaneously solve for stream function and vorticity, are presented.

Improved finite-difference methods based on a critical evaluation of the approximation errors

Upwind difference, defect correction and central difference schemes for the solution of the convection-diffusion equation with small viscosity coefficient are compared. It is shown that central difference schemes and hence also standard Galerkin finite element methods are preferable above upwind and defect correction schemes, when Gaussian elimination is used for the solution of the resulting system of equations.When iterative solution methods are employed good results can be achieved by a defect-correction method, whereas upwind difference schemes are generally inaccurate.

Aspects of Numerical Methods for Elliptic Singular Perturbation Problems

A comparative study of eight discretization schemes for the equations describing convection-diffusion transport phenomena is presented The (differencing) schemes considered are the conventional central, upwind and hybrid difference schemes,1,2 together with the quadratic upstream,3,4 quadratic upstream extended4 and quadratic upstream extended revised difference4 schemes Also tested are the so called locally exact difference scheme5 and the power difference scheme6 In multi-dimensional problems errors arise from ‘false diffusion’ and function approximations It is asserted that false diffusion is essentially a multi-dimensional source of error Hence errors associated with false diffusion may be investigated only via two- and three-dimensional problems The above schemes have been tested for both one- and two-dimensional flows with sources, to distinguish between ‘discretization’ errors and ‘false diffusion’ errors7 The one-dimensional study is reported in Reference 7 For 2D flows, the quadratic upstream difference schemes are shown to be superior in accuracy to the others at all Peclet numbers, for the test cases considered The stability of the schemes and their CPU time requirements are also discussed

An evaluation of eight discretization schemes for two‐dimensional convection‐diffusion equations

A one-dimensional, unsteady numerical model for the prediction of the vertical distribution of suspended sediment concentration and rate of suspended sediment deposition in water stirred uniformly has been developed. The model is derived from the convective-diffusive transport equation and uses a fully implicit, exponential finite difference scheme. The model is unconditionally stable regardless of vertical diffusivities and particle fall velocities. Such a model has not previously been given in the sedimentation literature and is of interest for theory and applications. A generic study of vertical suspended sediment profiles and trap efficiencies has been made. The results, presented in dimensionless form, cover Peclet numbers Vh/D from zero to infinity. V is the particle fall velocity, h is the water depth, and D is the mean vertical exchange coefficient. In studies of suspended sediment behavior the parameters V and D are often unknown or not well determined. The generic results generated show what kind of variations in suspended sediment concentrations and trap efficiencies can be expected with different values of V and D. Sediment deposition is shown to occur through downward frontal movement when Vh/D is large (≥ 20) and through columnar precipitation when Vh/D is small (≤ 0.2). Residence time is shown theoretically to be far more important for estimates of sediment deposition rate than turbulence. This is in agreement with empirical findings for reservoir trap efficiency.

Unsteady one‐dimensional settling of suspended sediment

A general finite-difference formulation with application to Navier-Stokes equations

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A general finite-difference formulation with application to Navier-Stokes equations

A general finite-difference formulation with application to Navier-Stokes equations