Numerical approximation to semi-linear stiff neutral equations via implicit–explicit general linear methods

Among the initial value problems of semi-linear neutral equations, there are a class of so-called stiff problems, where the classical explicit methods do not work as the methods have only bounded stability regions, which confine the computational stepsize to be excessively small and thus leads to an unsuccessful calculation. For resolving this difficult issue, ones turned to develop the implicit methods with unbounded stability regions to solve the stiff problems. Nevertheless, it is well-known that the implementation of an implicit method needs a large computational cost. In order to improve the computational efficiency, in Refs. Tan and Zhang (2018, 2020), the authors adopted the implicit–explicit (IMEX) splitting technique to derive the extended IMEX one-leg methods and IMEX Runge–Kutta methods, respectively. Unfortunately, these two methods have the serious order barrier. So far, for stiff neutral equations (SNEs), no IMEX method with order more than two has been found. To improve the computational accuracy and efficiency of IMEX methods, in the present paper, we construct a class of extended implicit–explicit general linear (EIEGL) methods for solving semi-linear SNEs. Under some suitable conditions, an EIEGL method is proved to be stable and convergent of order p whenever the underlying implicit–explicit general linear (IEGL) method has order p and stage order p. With applications to several concrete problems of SNEs, the computational accuracy of EIEGL methods are further illustrated, where we also verify that the convergence order of EIEGL methods can exceed two, namely, third- and fourth-order EIEGL methods can be obtained. Moreover, based on a numerical comparison with the extended implicit general linear (EIGL) methods, the advantage of EIEGL methods in computational efficiency is shown.