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A model for atmospheric pollutant transport is proposed considering an advection–diffusion–reaction equation. A splitting method is used to decouple the advection, diffusion and reaction parts. A scheme based on finite volume, finite difference and backward differentiation formula is used for solving an atmospheric transport-chemistry problem.

The derivation of a composite method for solving stiff ordinary differential equations is discussed. Combination of the harmonic and arithmetic means of the Runge–Kutta formulation has resulted in the introduction of a new formula for the numerical solution of stiff ordinary differential equations. The numerical results and the A-stability of this new formula are examined.

A simple and effective algorithm based on Haar wavelets is proposed to the solution of stiff problems in this article. It can integrate the stiff equation with very accurate results for any length of time. The simulation result shows that the single-term Haar wavelet method is better than the improved Runge–Kutta–Fehlberg method, while the terms of both expansions are the same.

In this article, we extended the existing explicit Taylor method and modified it to gain a new explicit Taylor-liked method in solving stiff differential equations. We also considered the stability property for this method since the stability property of the classical explicit fourth order Runge–Kutta (RK4) method is not adequate for the solution of stiff problems. Implicit methods could work well for stiff problems but have certain drawbacks especially when discussing about the cost. A comparison of the existing implicit Adam–Bashforth, the classical explicit (RK4) and the new explicit Taylor-liked method is presented.

We examine a single-step implicit-integration algorithm which is obtained by a modification of the well-known Simpson rule. The accuracy and stability properties of these methods are investigated. The obtained new method is a fourth-order numerical process and preserves the property of A-stability of the Simpson rule. Numerical results for the solution of certain stiff equations.

An algorithm for evaluation of time response to a time-invariant system based on Haar wavelets is proposed. The system is assumed to be governed by a high order linear differential equation with constant coefficients. The main objective of this study is to convert a differential equation into an algebraic-form equation and provide an elegant approach for computer programming. By the method, the computation complexity can be greatly reduced or much simplified in calculate the time response of a time-invariant system. The accurate and fastness of the method have been demonstrated by the examples of integration of the stiff systems

Numerical solutions to stiff problems have over the years been computational problems. A fourth order multi-derivative method is developed for stiff Initial Value Problems (IVPs). This scheme is in Predictor-Corrective mode. The proposed method is seen to be accurate and reliable for stiff problems which allowed for exponential fitting. The stability of the method is investigated and is shown to be suitable for stiff IVP's.

In this paper we propose a numerical method to integrate stiff ordinary differential systems of the form Y′ = f(t Y)t ∊ [t 0 t N ]Y ∊ R m m positive integer, with Y(t 0) = Y 0. The method is an explicit two-step scheme, with variable coefficients, depending on a stability parameter. We prove that the scheme is of the first order in accuracy and that it shows good stability properties. We conclude by giving some numerical results obtained solving known stiff systems of differential equations.

In this paper we present a numerical method which is suitable for the integration of chemical rate equations. These equations are normally extremely stiff due to large differences in the kinetic rate coefficients. The method takes advantage of the fact that the Jacobian matrix is readily obtainable for this type of problem. Stability analysis will also be discussed in a general framework.

This paper develops general criteria for third-order L-stable methods which require not more than three function evaluations and not more than four matrix-by-vector multiplications per step. The methods are required to remain third-order and provide a valid error estimate if the Jacobian matrix is replaced by an approximation with error of order h. Five specific types of methods are identified, and examples given of each type.

This paper deals with the efficient implementation of implicit methods for solving stiff ODEs, in the case of Jacobians with separable sets of eigenvalues. For solving the linear systems required we propose a method which is particularly suitable when the large eigenvalues of the Jacobian matrix are few and well separated from the small ones. It is based on a combination of an initial iterative procedure, which reduces the components of the vector error along to the nondominant directions of J and a projection Krylov method which reduces the components of the vector error along to the directions corresponding to the large eigenvalues. The technique solves accurately and cheaply the linear systems in the Newton's method, and computes the number of stiff eigenvalues of J when this information is not explicitly available. Numerical results are given as well as comparisons with the LSODE code.

A recent paper by Steihaug and Wolfbrandt [6] gives a second-order method for stiff differential equations, which includes an error estimate. We show how, in most practical circumstances, the error estimate can be obtained more simply, and derive two third-order methods along similar lines. All the methods described are single-step and linearly implicit, and they are designed to avoid frequent evaluations of the Jacobian and inversion of the associated matrix.

The second-order linearly implicit method of Steihaug and Wolfbrandt is found to give poor numerical accuracy if unsuitable error estimates are used. The criteria for satisfactory error estimates are discussed. A general procedure for deriving second-order linearly implicit methods is described. A typical method derived by this procedure is found to compare favourably with Gear's method when only moderate accuracy is required.

A numerical method applied to the differential equation y′=λy can lead to a solution of the form , where z and w = λh satisfy an equation P(w, z) = 0. In general P(w,z) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Runge-Kuta methods correspond to the cases M = 1 and N = 1 respectively. New methods are fottd by taking MS≧2N≧2. The approach here is first to find a suitable polynomial P(w,z) with the desired stability properties, and then to find a process which leads to this polynomial. Third- and fourth-order A- and L-stable processes are given of the semi-explicit arid linearly implicit types.