Numerical Methods for Solving Convection-Diffusion Problems

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For different problems, a convection-diffusion equation may be be written in various forms. The most popular formulation of convective transport employs the divergent (conservative) form. In some cases, the nondivergent (characteristic) form seems to be preferable. The so-called skew-symmetric form of convective transport operators that is the half-sum of the operators in the divergent and nondivergent forms is of great interest in some applications. Here we discuss the basic classes of discretization in space: finite difference schemes on rectangular grids, approximations on general polyhedra (the finite volume method), and finite element procedures. The key properties of discrete operators are studied for convective and diffusive transport. We emphasize the problems of constructing approximations for convection and diffusion operators that satisfy the maximum principle at the discrete level --- they are called monotone approximations. Two- and three-level schemes are investigated for transient problems. Unconditionally stable explicit-implicit schemes are developed for convection-diffusion problems. Stability conditions are obtained both in finite-dimensional Hilbert spaces and in Banach spaces depending on the form in which the convection-diffusion equation is written

Semidiscrete least squares methods for linear convection-diffusion problem

AbstractIn this paper, some approximate methods for solving linear convection-diffusion problems are presented. The methods consist in discretizing with respect to time and solving the resulting convection dominated elliptic problem for fixed time by least squares finite element methods. An analysis of least squares approximations is given, including optimal order estimates for piecewise polynomial approximation spaces. The model problem considered is the time-dependent convection dominated linear convection-diffusion equation. Numerical results for the Burgers' equation will also be presented

Hybrid algorithms for cyclically reduced convection-diffusion problems

We consider hybrid and adaptive iterative algorithms for cyclically-reduced discrete convection-diffusion problems. Hybrid algorithms combine via a two phase algorithm, iterative methods which require no a priori information about the coefficient matrix in the first phase with Chebyshev or Richardson iteration in the second phase. For two-dimensional convection-diffusion problems, central difference discretization is considered and the resulting linear system is reduced to approximately half its size by applying one step of cyclic reduction. We examine the numerical performance of the hybrid methods for solving the reduced systems. Our numerical experiments show that for the class of problems considered, an adaptive Chebyshev algorithm that uses modified moments to approximate the eigenvalues requires less work in most cases than the hybrid algorithms based on GMRES/Richardson methods

Modelling of impulse loading in high-temperature superconductors. Assessment of accuracy and performance of computational techniques.

Purpose – The aim of this paper is to access performance of existing computational techniques to model strongly non-linear field diffusion problems. Design/methodology/approach – Multidimensional application of a finite volume front-fixing method to various front-type problems with moving boundaries and non-linear material properties is discussed. Advantages and implementation problems of the technique are highlighted by comparing the front-fixing method with computations using fixed grids. Particular attention is focused on conservation properties of the algorithm and accurate solutions close to the moving boundaries. The algorithm is tested using analytical solutions of diffusion problems with cylindrical symmetry with both spatial and temporal accuracy analysed. Findings – Several advantages are identified in using a front-fixing method for modelling of impulse phenomena in high-temperature superconductors (HTS), namely high accuracy can be obtained with a small number of grid points, and standard numerical methods for convection problems with diffusion can be utilised. Approximately, first order of spatial accuracy is found for all methods (stationary or mobile grids) for 2D problems with impulse events. Nevertheless, errors resulting from a front-fixing technique are much smaller in comparison with fixed grids. Fractional steps method is proved to be an effective algorithm for solving the equations obtained. A symmetrisation procedure has to be introduced to eliminate a directional bias for a standard asymmetric split in diffusion processes. Originality/value – This paper for the first time compares in detail advantages and implementation complications of a front-fixing method when applied to the front-type field diffusion problems common to HTS. Particular attention is paid to accurate solutions in the region close to the moving front where rapid changes in material properties are responsible for large computational errors. Keywords - Modelling, Numerical analysis, Diffusion, High temperatures, Superconductors Paper type - Research pape

A coupled finite volume and discontinuous Galerkin method for convection-diffusion problems

This work formulates and analyzes a new coupled finite volume (FV) and discontinuous Galerkin (DG) method for convection-diffusion problems. DG methods, though costly, have proved to be accurate for solving convection-diffusion problems and capable of handling discontinuous and tensor coefficients. FV methods have proved to be very efficient but they are only of first order accurate and they become ineffective for tensor coefficient problems. The coupled method takes advantage of both the accuracy of DG methods in the regions containing heterogeneous coefficients and the efficiency of FV methods in other regions. Numerical results demonstrate that this coupled method is able to resolve complicated coefficient problems with a decreased computational cost compared to DG methods. This work can be applied to problems such as the transport of contaminant underground, the CO 2 sequestration and the transport of cells in the body

Numerics of singularly perturbed differential equations

The main purpose of this report is to carry out the effect of the various numerical methods for solving singular perturbation problems on non-uniform meshes. When a small parameter epsilon known as the singular perturbation parameter is multiplied with the higher order terms of the differential equation, then the differential equation becomes singularly perturbed. In this type of problems, there are regions where the solution varies very rapidly known as boundary layers and the region where the solution varies uniformly known as the outer region. Standard finite difference/element methods are applied on the singularly perturbed differential equation on uniform mesh give unsatisfactory result as epsilon tends to zero. Due to presence of boundary layer, standard difference schemes unable to capture the layer behaviour until the mesh parameter and perturbation parameter are of the same size which results vast computational cost. In order to overcome this difficulty, we adapt non-uniform meshes. The Shishkin mesh and the adaptive mesh are two widely used special type of non-uniform meshes for solving singularly perturbed problem. Here, in this report singularly perturbed problems namely convection-diffusion and reaction-diffusion problems are considered and solved by various numerical techniques. The numerical solution of the problems are compared with the exact solution and the results are shown in the shape of tables and graphs to validate the theoretical bounds

Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems

Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology