A Method For Stiff Ordinary Differrential Equations

In many scientific and technical fields and practical engineerings such as aviation,aerospace,thermonuclear reaction,automatic control,electronic network and chemical dynamics,physical or chemical processes,researchers frequently encounter physical or chemical processes that can be described by ordinary differential equations.These engineerings often contain multiple sub-processes that interact with varying speeds,which lead to the solution of ordinary differential equations contain both very fast decaying components and relatively slow-changing variables.People call the above process ”Stiffness ”,and the initial value problem of ordinary differential equations that describing such problems is called ”stiff problems”.The paper analyzes a splitting technique into fast and slow dynamical components of ordinary differential equations as suggested.The technique is based on a real block-Schur decomposition of the Jacobian matrix of the right hand side of the ordinary differntial equations.As a consequence,the original ordinary differntial equation problems are transformed into the differential algebra equation problems.There are several different classifications of differential algebra equations here.In this paper,they are transformed into index-1 differential algebra equations.As for the analysis,a computationally cheap monitor for the possible necessary recovering of the splitting is derived by singular perturbation theory.In addition,the differential variables y can be got by the way of the extrapolation method,considerating the LU factorization in terms of the inverse of several matrices.Besides,Newton iteration method upon on the algebraic variables z.Meanwhile,the next step is going on or not depend on the relationship between convergence rates and spectral radius.In 2009,Conte proposed the multi-step collocation method,and successifully applied this method in Volterra integral equations.Here,multi-step collocation method could be used to deal with the index-1 differntial algebra equations.We give the idea about the multi-step collocation method but without the experiment examples.At the final,we give the numerical examples mainly from chemical combustions.Besides,giving the numerical results by MATLAB codes.