Exponential Multistep Methods for Stiff Delay Differential Equations

Abstract

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.