Stability Of Exponential Methods For Solving Stiff Differential Equations

In many high-tech field and realistic problems, stiff ordinary differentialequations (ODEs) are often used to describe the chemical or physical processes,while another important source is ODEs deduced from the semi-discretization ofpartial differential equations, which makes the numerical solution unquestionableimportant. Highly stable implicit methods are often applied to solve stiffproblems instead of explicit methods to avoid the rigid of step. It is well knownthat there are some problems in using A-stable methods to deal with the stiffnonlinear systems, the numerical solutions obtained from some A-stable methodsmay be highly unstable and theirs accuracy often happened to be disrelated to theorder of these methods. This led Prothero and Robinson to define a new form ofstability, i.e., S-stability. The explicit exponential integrators are superior to theimplicit methods because of the large computational cost of the implicit methods.This paper focuses on the stability of exponential integrators for stiff ODEs.First of all, we introduce the research background of stiff differential andillustrate the application of the stiff problems in real life, then we brifelyintroduce the research of stiff differential equations and exponential intergrator.Secondly, we research the stability of exponential one-step methods for stiffODEs. We define the concepts and discuss S-stability and stiff consistency ofexponential one-step methods. Then we shall present some numerical results ofexponential Runge-Kutta (RK) methods of order one, two and three, study theirS-stability, and then apply them to two specific euqations, to prove the results.Then, we research the stability of exponential multistep methods (EMMs)for initial value problems of stiff ODEs. We define the concepts of S-stability ofEMMs and discuss the S-stability properties for exponential Adams methods. Wealso analyze a class of explicit exponential general linear methods (EGLMs) andgive the concepts of S-stability. Then we discuss the S-stability properties forEGLMs. Finally, we give numerical examples that illustrate the methods’properties. We shall analyze the stability of two class two order EMMs and threeorder EGLM with two stage two steps, and then apply them to two specificequations to prove the results, and these results also clearly indicate that S-stableEMMs and EGLMs are superior to the S-stable exponential one-step methods.Finally, summarize the paper and show the future research work.