Abstract

Exponential time differencing methods provide instability-free explicit schemes for systems with fast decaying or oscillating modes (stiff systems), without limitation on the time step size. Moreover, with these methods, one can drastically diminish the error accumulation rate for numerical simulation of conservative systems. The methods yield an especially large performance gain for PDEs with high order of spatial derivatives. Simultaneously, the problem of analytical calculation of coefficients of exponential time differencing schemes becomes laborious or unsolvable in the case of a nondiagonal form of the principal linear part of equations. We introduce an approach, where the scheme coefficients are obtained from the direct numerical integration of certain auxiliary problems over a short time interval—one scheme step size. The approach is universal and its implementation is illustrated with four examples: analytically solvable system of two first-order ODEs, one-dimensional reaction-diffusion system under time-dependent conditions, two-dimensional reaction-diffusion system under time-independent and time-dependent conditions, and one-dimensional Cahn–Hilliard equation with constant coefficients. The employment of an exponential time differencing method of the two-step Runge–Kutta type yields a simulation performance gain for the diffusion-type equation, with program optimization made. Without program optimization, the performance gain increases by one order with respect to the spatial step size for each order of the highest spatial derivative, and appears starting from the third order of the derivative. With the tested method, one can extend the study of an analogue of the Anderson localization to two- and three-dimensional active media and achieve an acceptable performance for the direct numerical simulations of dynamics of the probability density function for active Brownian particles.

中文翻译：

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具有非对角线性零件的刚性系统的指数时差。

摘要

指数时差方法为具有快速衰减或振荡模式的系统（刚性系统）提供了不失稳的显式方案，而没有时间步长的限制。此外，通过这些方法，可以大大降低保守系统数值模拟的错误累积率。该方法对于具有高阶空间导数的PDE产生了特别大的性能增益。同时，在方程的主要线性部分为非对角形式的情况下，指数时差方案的系数的解析计算问题变得费力或无法解决。我们介绍了一种方法，其中方案系数是在较短的时间间隔内（一种方案步长）从某些辅助问题的直接数值积分中获得的。该方法是通用的，并通过四个示例说明了其实现：两个可解ODE的解析可解系统，与时间有关的条件下的一维反应扩散系统，与时间无关和在时间下的二维反应扩散系统。相依条件，以及具有恒定系数的一维Cahn-Hilliard方程。采用两步Runge–Kutta类型的指数时间微分方法可以使扩散类型方程具有仿真性能，并且可以进行程序优化。如果不进行程序优化，则相对于最高空间导数的每个阶的空间步长，性能增益将增加一个阶次，并且从导数的三阶开始出现。使用经过测试的方法