Finite difference method numerical diffusion

Many diffusion problems cannot be solved anal3dically, such as concentration-dependent D, complicated initial and boundary conditions, and irregular boundary shape. In these cases, numerical methods can be used to solve the diffusion equation (Press et al., 1992). There are many different numerical algorithms to solve a diffusion equation. This section gives a very brief introduction to the finite difference method. In this method, the differentials are replaced by the finite differences ... 

Ion-Exchange Rate and Transient Concentration Profiles The numerically implicit finite difference method was used to solve the set of nonlinear differential equations (8) for a wide range of model parameters such as diffusivities, Dg, D,, and Dy, dissociation constants. Kg, exchanger capacity, ag, and bulk concentration, Cg, of the solution. 

Finite-Difference Methods. The numerical analysis literature abounds with finite difference methods for the numerical solution of partial differential equations. While these methods have been successfully applied in the solution of two-dimensional problems in fluid mechanics and diffusion (24, 25), there is little reported experience in the solution of three-dimensional, time-dependent, nonlinear problems. Application of these techniques, then, must proceed by extending methods successfully applied in two-dimensional formulations to the more complex problem of solving (7). The various types of finite-difference methods applicable in the solution of partial differential equations and their advantages and disadvantages are discussed by von Rosenberg (26), Forsythe and Wasow (27), and Ames (2S). 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

E(u,v) is the inner product of u and v. In conventional OCFE calculation methods, the exponent of Az in Eq. 10.112 is 6, hence the degree of convergence between the calculated and the true profiles is of the sixth degree with respect to the space increment. One expects the value of C I4 to be rather small in the type of problems dealt with here. The fourth-order Runge-Kutta method used in the OCFE algorithm discussed here introduces an error of the fifth order. Accordingly, we may anticipate that the numerical solutions of the system of partial differential equations of chromatography calculated by an OCFE method will be more accurate than those obtained with a finite difference method [48] or even with the controlled diffusion method [49,50]. 

Smith GD (1985) Numerical Solution of Partial Differential Equations Finite Difference Methods. Third edition. Clarendon Press, Oxford Smolarkiewicz PK (1983) A simple positive definite advection scheme with small implicit diffusion. Mon Wea Rev 11 479-486... 

ABSTRACT In order to investigate the effect of coal particle size on gas desorption and diffusion law at constant temperature, the constant temperature dynamic coal particle gas adsorption and desorption experiment with different particle sizes was conducted in the coal gas adsorption and desorption experiment system. The results suggest that gas desorption laws of different particle size of coal samples show a good consistency at different pressures, and the cumulative desorption of gas coal particle is linear with time. For the same particle, the higher the initial pressure, the more the maximum gas desorption the smaller the coal particle is, the more quickly the gas desorption rate is at the same initial pressure. Then, the gas spherical flow mathematical model is built based on Darcy law and is analysed with finite difference method. At last, the gas spherical flow mathematical model is constructed with Visual Basic. The contrast between numerical simulation and experimental results shows that the gas flow in the coal particle internal micropore accords with Darcy s law. 

In the previous chapter finite difference methods were introduced for one of the simplest situations from a theoretical point of view cyclic voltammetry of a reversible E mechanism (i.e., charge transfer without chemical complications) at planar electrodes and with equal diffusion coefficients for the electroactive species. However, electrochemical systems are typically more complex and some refinements must be introduced in the numerical methods for adequate modelling. 

Then, to analyze the obtained current, a Fourier transform is applied and the responses at the fundamental, co, and harmonic, 2m, 3m, 4m,..., frequencies are obtained. Next, the current responses at the fundamental and harmonic frequencies are extracted by an inverse Fourier transform. Harmonics up to the eighth order were obtained. Analysis of the kinetic parameters is carried out by comparison of the experimental and simulated data. Theoretical ac voltammograms were simulated using classical numerical simulations of the diffusion-kinetic process using an implicit finite-difference method [658, 659] with a subsequent Fourier analysis of the simulated data. An example of the comparison of the experimental and simulated data is shown in Fig. 15.5. In this case, oxidation of ferrocenmethanol appeared reversible, and a good agreement was found with the simulated data for the reversible process. 

The determination of the diffusion coefficient of al-kanethiol ink in PDMS stamps is possible by means of simple linear-diffusion experiments, in which the basic parameters of//CP (ink concentration, printing time, and stamp geometry) are taken into account. Ink transport is monitored by direct adsorption on gold substrates from consecutive prints. We showed that the ink transport through the PDMS slab follows Pick s law of diffusion. A simplified analytical model was found to be accurate for experiments with high initial concentrations (saturation) but is likely to become inaccurate at low initial concentrations. Therefore, a more precise one-dimensional, numerical model based on the finite-difference method was developed, which also proved to be accurate at low concentrations. 

The one-dimensional decoupled diffusion equation without the elimination of stress-dependent terms is easier to solve numerically. With li-ion concentration obtained, intercalation-induced stresses can be calculated using the analytical expressions of Equations 26.4 and 26.5. Governing equations for one-dimensional spherical particles can be solved using the finite difference method. Governing equations for three-dimensional elHpsoidal particles are solved using the finite element analysis package GOMSOL Multiphysics. 

This problem was solved semianalytically in terms of two-dimensional (2D) integral equations [2,3] and numerically by using Krylov integrator [4] and the alternating direction implicit (ADI) finite difference method [3,5]. Potentiostatic transients were computed for two limiting cases a diffusion-controlled process and totally irreversible kinetics [3-5]. The analysis of the simulation... 

Another approach to TG/SC experiments does not rely on the mediator feedback [56]. The reactant galvanostatically electrogenerated at the tip diffuses to the substrate and undergoes the reaction of interest at its surface. The substrate current is recorded as a function of either time or the tip/ substrate separation distance (approach cnrves). The theory for transient responses, steady-state TG/SC approach curves, and polarization cnrves (i.e., 4 vs. E ) was generated solving the diffnsion problem numerically (an explicit finite difference method was used). The substrate process was treated as a first-order irreversible reaction, and the effects of its rate constant and the experimental parameters were illnstrated by families of the dimensionless working curves (Figure 5.11). 

The basic idea of the finite differences method for solving partial differential equations is to replace spatial and time derivatives by suitable approximations, then to solve numerically the resulting difference equations. In other words, instead of solving for C(x,t) with x and t, it is solved for Cjj = C(Xi, tj), where Xj s iAx, tj = jAt. Thus, the concentrations,, of the diffusing species for the location step, i, and the time step, j -H1, are calculated from the neighbouring concentrations according to the implicit Crank-Nicolson solution for the diffusion differential equation (29.2) ... 

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