To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard-Lindel\"of iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, three-dimensional Liouville-Bratu-Gelfand, and three-dimensional nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators.

翻译：针对大规模非线性微分方程组的时间积分问题，我们考虑一种非线性波形松弛法（亦称动态迭代或皮卡-林德勒夫迭代）的变体形式，该变体在每次迭代中需求解一个线性非齐次微分方程组。我们采用指数块Krylov子空间（EBK）方法实现这一求解过程。由此构成一种内外迭代法，其在特定时间区间内确定迭代近似解，无需逐步时间推进。近期研究表明，该方法作为PARAEXP框架中的时间并行积分器具有高效性。本文从理论与实际两个层面评估该方法的收敛特性，并通过求解非线性Burgers方程、三维Liouville-Bratu-Gelfand方程及三维非线性热传导方程，与传统时间步进积分器进行性能对比，检验该方法效率。