Numerical Solution Of Diffusion Problems Research Articles

The Truncation Error and Correction Algorithm of the Support-Operator

For the definition of heterogeneous and non-isotropic medium diffusion equation, we have established a finite-difference algorithm for the numerical solution of diffusion problems. This algorithm not only applies to the convex meshes, but also applies to the concave grid or self-AC grid. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, but it is superior to the problems with rough coefficients and no-skewed grids. We will discuss the truncation error and correction algorithm of the support-operator in this article.

Heterogeneous Multiscale FEM for Diffusion Problems on Rough Surfaces

We present a finite element method for the numerical solution of diffusion problems on rough surfaces. The problem is transformed to an elliptic homogenization problem in a two dimensional parameter domain with a rapidly oscillating diffusion tensor and source term. The finite element method is based on the heterogeneous multiscale methods of E and Engquist [Commun. Math. Sci., 1 (2003), pp. 87--132]. For periodic surface roughness of scale $\varepsilon$ and amplitude $O(\varepsilon)$, the method converges at a robust rate, i.e., independent of $\varepsilon$, to the homogenized solution.

Numerical solution of parabolic equations in high dimensions

We consider the numerical solution of diffusion problems in (0,T ) x Ω for and for T > 0 in dimension d d ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order p d ≥ 1, and hp discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2 (Ω)-error of O(N-p ) for u(x,T) where N is the total number of operations, provided that the initial data satisfies with e > 0 and that u(x,t) is smooth in x for t>0 . Numerical experiments in dimension d up to 25 confirm the theory.

A New Algorithm for the Numerical Solution of Diffusion Problems Related to the Smoluchowski Equation

Diffusion-influenced reactions can often be described with simple kinetic models, whose basic features are a spherically symmetric potential, a distance-dependent relative diffusion coefficient, and a distance-dependent first-order rate coefficient. A new algorithm for the solution of the corresponding Smoluchowski equation has been developed. Its peculiarities are: (1) A logarithmic increase of the radius; (2) the systematic use of numerical fundamental solutions w of the Smoluchowski equation; (3) the use of polynomials of up to the 8

A discontinuous hp finite element method for diffusion problems: 1-D analysis

This paper presents the mathematical analysis of a new variant of the discontinuous Galerkin method which is applicable to the numerical solution of diffusion problems, not requiring auxiliary variables such as those used in mixed methods. The focus of this study is on a class of linear second-order boundary value problems for which we prove stability and a priori error estimates in both the finite- and infinite-dimensional spaces.

A DiscontinuoushpFinite Element Method for Diffusion Problems

We present an extension of the discontinuous Galerkin method which is applicable to the numerical solution of diffusion problems. The method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries. Within each element, arbitrary spectral approximations can be constructed with different orderspin each element. We demonstrate that the method is elementwise conservative, a property uncharacteristic of high-order finite elements.For clarity, we focus on a model class of linear second-order boundary value problems, and we developpriorierror estimates, convergence proofs, and stability estimates. The results of numerical experiments onh- andp-convergence rates for representative two-dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence.

The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials

A new second-order finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non-isotropic media is constructed. On problems with rough coefficients or highly nonuniform grids, the new algorithm is superior to all other algorithms we have compared it with. For problems with smooth coefficients on smooth grids, the method is comparable with other second-order methods. The new algorithm is formulated for logically rectangular grids and is derived using thesupport-operators method. A key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator, defined as the material properties tensor times the gradient, rather than the gradient, as one of the basic first-order operators in the support-operators method. The discrete analog of the flux operator must also be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resulting method is conservative and the discrete analog of the variable coefficient Laplacian is symmetric and negative definite on nonuniform grids. In addition, on any grid, the discrete divergence is zero on constant vectors, the null space for the gradient is the constant functions, and, when the material properties are piecewise constant, the discrete flux operator is exact for piecewise linear functions. We compare the methods on some of the most difficult examples to be found in the literature.

Solving Diffusion Equations with Rough Coefficients in Rough Grids

A finite-difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and nonisotropic media is constructed for logically rectangular grids. The performance of this algorithm is comparable to other algorithms for problems with smooth coefficients and regular grids, and it is superior for problems with rough coefficients and/or skewed grids. The algorithm is derived using thesupport-operators method,which constructs discrete analogs of the divergence and flux operator that satisfy discrete analogs of the important integral identities relating the continuum operators. This paper gives the first application of this method to the solution of diffusion problems in heterogeneous an nonisotropic media.The support-operators method forces the discrete analog of the flux operator to be the negative adjoint of the discrete divergence in an inner product weighted by the conductivity, as in the differential case. Once this is accomplished, many other important properties follow; for example, the scheme is conservative and the discrete analog of the variable material Laplacian is symmetric and negative definite. In addition, on any grid, the discrete divergence is zero on constant vectors and the discrete flux operator is exact for linear functions in case whenKis piecewise constant. Moreover, the discrete gradient's null space is the constant functions, just as in the continuum. Because the algorithm is flux based, it has twice as many unknowns as more standard algorithms. However, the matrices that need to be inverted are symmetric and positive definite, so the most powerful linear solvers can be applied. Also, the scheme is second-order accurate so, all things considered, it is efficient. For rectangular grids, the discrete operators reduce to well-known discrete operators and the treatment of discontinuous conductivity coefficients in the case of isotropic media is equivalent to the well-known harmonic-averaging procedure. Comparison with standard schemes is presented. Numerical examples validate advantage of new method.

Direct numerical solution of diffusion problems with intrinsic randomness

A hybrid numerical-symbolic manipulation scheme is developed for the analysis of diffusion problems with intrinsic randomness. The scheme is applied in the study of one-dimensional heat conduction in fully-saturated packed beds in order to study the effects of packing disorder on the medium effective conductivity. Conduction is modeled by a parabolic partial differential equation with random local conductivity. Randomness originates from the spatial fluctuation of porosity near solid walls and in the bulk region. Assuming a certain porosity statistical distribution, the steady and unsteady heat conduction problems are solved. The steady-state solution is used to obtain both the mean value and the standard deviation of the effective conductivity for a range of fluid to solid conductivity ratios. The mean and standard deviation are used to interpret the scatter of experimental results found in the literature. The unsteady heat conduction equation is discretized on a finite spatial grid and an explicit integration in time is carried out symbolically for each time step.

TLM: a technique with application in the numerical solution of diffusion problems

Transmission Line Modelling (TLM) is a technique which allows the numerical solution, for voltage or current, of networks of transmission lines. The basic equation describing the behaviour of such networks in space and time is an attenuating wave equation. Under appropriate conditions this equation simplifies to the standard diffusion equation. Therefore TLM can be applied to any system where diffusion is to be simulated. The technique has advantages over the finite-element type of approach as it is simple to apply and robust in operation. In this paper the basic principles of the technique are outlined, and some examples of TLM presented.

MULTIGRID TECHNIQUES FOR THE NUMERICAL SOLUTION OF THE DIFFUSION EQUATION

An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multi-grid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved Independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.

Implementation of the “Hopscotch” technique in the numerical solution of diffusion problems

An algorithm is given for solving the time-dependent diffusion equation by the “hopscotch” method in cylindrical and Cartesian co-ordinates. Boundaries between media having different diffusion coefficients, solubilities and continuous “sinks” for the diffusing substances may be either along or between lines of grid points.

Some Computational Problemls In the Simulation of Stack Gas Dispersion

Abstract This paper examines some aspects of the numerical simulation of plume dispersion. Several difficulties associated with numerical solutions are considered, and practical methods for the handling of these problems are provided. Some analytical solutions which are extremely useful for the testing of numerical methods are also presented. It is shown that, with adequate care, it is possible to simulate numerically the dispersion of pollutants from isolated stacks. Introduction THERE HAS BEEN a growing concern amongst the public over the quality of air it breathes. This has resulted in a number of air pollution studies to determine where an effluent goes. In particular, there have been a number of numerical studies of pollutant dispersion from stacks: Ragland (1973), Carr (1973), Egan and Mahoney (1972), Lantz at or. (1972), Randerson 11970), and Shir (1970). Previous investigators have concerned with problems varying complexity, but with the exception of Carr (1973) they do not discuss computational problems. Consequently, one is led to believe that there are no serious problems associated with the solution air pollution dispersion problems; all one needs to do is to set up the governing equations, find appropriate coefficients, define boundary conditions and integrate the problem to find the solution. Although no single part of the problem is actually easy, it is the integration that causes the greatest difficulty. Some of the studies referenced above report the maximum allowed integration interval for explicit numerical techniques (e.g. Randerson, 1970), but fail to discuss the discretization error caused by their differencing schemes and the allied problem of grid spacing and its effect on the error. Also they do not discuss the method used for the introduction of a point source into the numerical calculations. In this paper, a practical look is taken at these problems. In addition, the use of characteristic curves for the solution of transient pollutant dispersion problems is considered. A grid spacing guide related to stack height has been developed as an aid in the numerical solution of diffusion problems. Some of the results should also be useful in allied engineering applications. (equations in full paper) Numerical Techniques Analytical solutions are difficult or impossible to obtain when irregular terrain and diffusivity variation with position must be accounted for. Numerical techniques are ideally suited to the handling of these problems. Before using these methods, it is important to have some understanding of their limitations. Much insight can be gained by comparing the results predicted by numerical procedures with available analytical solutions. In the following discussion, some of the problems associated with numerical computation are considered. All finite differences used, excepting the integration procedure, are second-order central-difference approximations to the governing equation. Wallis and Aziz (1975) outline some advantages of explicit methods over implicit methods for the numerical solution of steady-state plume dispersion problems. For this reason, only explicit methods are considered in this paper, Because Equation 5 contains first-order derivatives with respect to time and space, the solution may be obtained by marching either in time or in the downwind direction.