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 and Liouville-Bratu-Gelfand equations and comparing its performance with that of conventional time-stepping integrators.

翻译：为了对非线性微分方程大型系统进行时间积分，我们考虑了一种非线性波形松弛（也称为动态迭代或Picard-Lindelöf迭代）的变体，其中每次迭代需要求解一个线性非齐次微分方程组。这一步通过指数块Krylov子空间（EBK）方法完成。因此，我们得到了一种内外迭代方法，其中迭代近似是在特定时间区间内确定的，不涉及时间步进。最近的研究表明，该方法在PARAEXP框架内作为一种时间并行积分器是高效的。本文从理论和实践上评估了该方法的收敛行为。我们通过测试非线性Burgers方程和Liouville-Bratu-Gelfand方程，并将其性能与传统时间步进积分器进行比较，考察了该方法的效率。