Abstract In this paper, a fully discrete θ -method with 0 ≤ θ ≤ 1 is suggested to solve the initial–boundary value problem of semi-linear reaction–diffusion equations with time-variable delay. Under some appropriate conditions, a novel global stability criterion of the method is derived and it is shown that this method has the computational accuracy O ( τ 2 + h 2 ) ( resp. O ( τ + h 2 ) ) when θ = 1 2 ( resp. θ ≠ 1 2 ) , where h and τ denote spatial and temporal stepsizes, respectively. Moreover, with some numerical experiments, the theoretical accuracy and global stability of the method are further illustrated.

中文翻译：

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求解具有时变延迟的半线性反应-扩散方程的完全离散 θ 方法

摘要 本文提出了一种0 ≤ θ ≤ 1 的全离散θ 方法来求解时变时滞半线性反应扩散方程的初边值问题。在适当的条件下，推导出了该方法新的全局稳定性判据，结果表明，当θ=1 2 时，该方法的计算精度为O ( τ 2 + h 2 ) ( resp. O ( τ + h 2 ) ) ( resp. θ ≠ 1 2 ) ，其中 h 和 τ 分别表示空间和时间步长。此外，通过一些数值实验，进一步说明了该方法的理论准确性和全局稳定性。