3D Convection Diffusion Equation Research Articles

Sharp error estimates for discretizations of the 1D convection–diffusion equation with Dirac initial data

This paper derives sharp estimates of the error arising from explicit and implicit approximations of the constant-coefficient 1D convection‐diffusion equation with Dirac initial data. The error analysis is based on Fourier analysis and asymptotic approximation of the integrals resulting from the inverse Fourier transform. This research is motivated by applications in computational finance and the desire to prove convergence of approximations to adjoint partial differential equations.

The dynamic procedure for accuracy improvement of numerical discretizations in fluid mechanics

In CFD computations, discretization or truncation errors should be small providing an acceptable level of accuracy. In this paper, an extension is made of the recently proposed LES formalism based on sampling operators. It is shown that the sampling-based dynamic procedure, in combination with an appropriate truncation error model, can be used as a technique to increase the numerical accuracy of a discretization. The technique is resemblant to the well-known Richardson extrapolation. The procedure is tested on a 1D convection–diffusion equation and a 2D lid-driven cavity at Re=400, using a finite difference method. Promising results are found.

Study and solution of identification problems for nonstationary 2D and 3D convection–diffusion equations

In this paper we consider two nonstationary heat convection–diffusion models. The first model describes convection–diffusion processes in the 'surface' ocean layer. The second one is a 3D model and describes heat propagation processes in the whole ocean. We pose identification problems for the models and propose methods for solving the problems posed. In the case of the 2D model, the methods of its solution are theoretically justified.

Study and solution of identification problems for nonstationary 2D and 3D convection–diffusion equations

Study and solution of identification problems for nonstationary 2D and 3D convection–diffusion equations

Finite difference scheme for solving general 3D convection–diffusion equation

The effect of ergodisation (either by additional coils like in the TEXTOR-DED tokamak or by intrinsic plasma effects like in the W7-X stellarator) defines the need for transport models being able to describe this properly. A prerequisite for this is the concept of local magnetic coordinates allowing a correct discretization with mimimized numerical errors. In the present paper we employ the finite difference discretization method which allows the numerical simulation of energy transport in complex 3D edge geometries (in particular for W7-X) using a custom-tailored unstructured grid in local magnetic coordinates. This grid is generated by field-line tracing to guarantee an exact discretization of the dominant parallel transport. Therefore the radial and parallel fluxes are almost perfectly separated which minimizes the numerical diffusion connected with strong anisotropy of the system ( D ||/ D ⊥∼10 7). In addition, the parallel and radial directions can be treated independently in the numerical method. Along the magnetic field lines we use the standard Patankar concept to discretize the convection–diffusion equation. In order to solve a quasi-isotropic problem in a plane (toroidal cut), we modified the finite volume method to obtain its finite difference representation. The radial fluxes on plasma cross-sections are interpolated using a constrained Delaunay triangulation (keeping the structural information for magnetic surfaces if they exist). The general concept of an upwind scheme is used to discretize the convective terms. The first tests for W7-X geometry were successfully performed.

An efficient box-scheme for convection–diffusion equations with sharp contrast in the diffusion coefficients

In this paper, we introduce a box-scheme for time-dependent convection–diffusion equations, following principles previously introduced by Courbet in [Rech. Aérospatiale 4 (1990) 21] for hyperbolic problems. This scheme belongs to the category of mixed finite-volume schemes. This means that it works on irregular meshes (finite-volume scheme) and computes simultaneously the principal unknown and its gradient in all Peclet regimes, ranging from pure diffusion ( Pe=0) to pure convection ( Pe=+∞). The present paper focuses mainly on the design of the scheme, which is non-standard, in the case of the 1D convection–diffusion equation. The version of the scheme presented here is of first or second order depending on the local Peclet number. We extend the 1D scheme afterwards in 2D by an ADI like technique. Several numerical results on 1D and 2D test-cases of interest for flow simulation in porous media are presented, some of them exhibiting sharp contrasts in diffusion coefficients.

Stability conditions for the Leapfrog‐Euler scheme with central spatial discretization of any order

AbstractIn this paper a sufficient condition (Theorem 2.3) for the von Neumann stability of the Leapfrog‐Euler scheme, which uses central spatial discretization of any order for 3D convection‐diffusion equation, is derived in terms of the Courant and the diffusion numbers and the coefficients of approximation schemes. In the case of the second order differencing this condition becomes the necessary condition for the stability. Some particular sufficient conditions for the stability of the second and the fourth order schemes are also derived. A comparison of the results, which were obtained applying the derived stability conditions to compute the time step in the direct numerical simulations (DNS) of turbulent pipe flow with the help of the second and the fourth order schemes, is presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modelling of the Ebro River plume. Validation with field observations

The spreading of the plume induced by the freshwater discharge from the Ebro River into northwestern Mediterranean coastal waters was modelled using two numerical codes. The coastal current field was obtained with a finite difference hydrodynamic model based on a steady-state version of the shallow-water equations, whereas the freshwater dispersion was calculated with a Lagrangian code that solves the 3D convection-diffusion equation and reproduces turbulent diffusion using a random-walk algorithm. The agreement obtained between numerical results, satellite observations and field measurements allows an analysis of the more relevant physical mechanisms and the corresponding tuning of the two models. The results show that local hydrodynamics near the river mouth, and consequently the spreading of the river plume, are highly dependent on the driving river discharge and wind field characteristics.

A fourth-order compact difference scheme on face centered cubic grids with multigrid method for solving 2D convection diffusion equation

We present a fourth-order compact finite difference scheme on the face centered cubic (FCC) grids for the numerical solution of the two-dimensional convection diffusion equation. The seven-point formula is defined on a regular hexagon, where the strategy of directional derivative is employed to make the derivation procedure straightforward, efficient, and concise. A corresponding multigrid method is developed to solve the resulting sparse linear system. Numerical experiments are conducted to verify the fourth-order convergence rate of the derived discretization scheme and to show that the fourth-order compact difference scheme is computationally more efficient than the standard second-order central difference scheme.

A two colorable fourth-order compact difference scheme and parallel iterative solution of the 3D convection diffusion equation

A new fourth-order compact difference scheme for the three-dimensional (3D) convection diffusion equation with variable coefficients is presented. The novelty of this new difference scheme is that it only requires 15 grid points and that it can be decoupled with two colors. The entire computational grid can be updated in two parallel subsweeps with a Gauss–Seidel type iterative method. This is compared with the known 19-point fourth-order compact difference scheme which requires four colors to decouple the computational grid. Numerical results, with multigrid methods implemented on a shared memory parallel computer, are presented to compare the 15- and 19-point fourth-order compact schemes.

High accuracy multigrid solution of the 3D convection–diffusion equation

We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional (3D) convection–diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelization-oriented multigrid method for fast solution of the resulting linear system using a four-color Gauss–Seidel relaxation technique for robustness and efficiency, and a scaled residual injection operator to reduce the cost of multigrid inter-grid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convection–diffusion equations are obtained for small to medium values of the grid Reynolds number. Effects of using different residual projection operators are compared on both vector and serial computers.

Explicitly preconditioned conjugate gradient method for the solution of unsymmetric linear systems

For the iterative solution of nonsingular linear system Ay = f of order n the preconditioned system Mx = b is considered, where M = GAF T b = Gf y = F T x, and the matrices G and F are sparse lower triangular. Their nonzero elements are obtained by alternate minimization of β (M T M) with respect to F and G, where β (M T M) = n − 1 tr M T M/(det M T M)1/n . Such preconditioning makes the matrix M closer to an orthogonal one, which improves the efficiency of the standard CGNE-type algorithms. This conclusion is confirmed by the estimate k ≤ ⌈ n log2 β (M T M) + log2(ε −1 )⌉ for the iteration number k needed for the ε times reduction of the error norm ‖ G(f − Ay k )‖. Numerical examples are given to show the efficiency of the resulting solution algorithms for the indefinite and strongly nonsymmetric linear systems arising from finite difference discretizations of the 2D convection-diffusion equation.