It is often the case that, while the numerical solution of the non-linear dispersive equation $\mathrm{i}\partial_t u(t)=\mathcal{H}(u(t),t)u(t)$ represents a formidable challenge, it is fairly easy and cheap to solve closely related linear equations of the form $\mathrm{i}\partial_t u(t)=\mathcal{H}_1(t)u(t)+\widetilde{\mathcal H}_2(t)u(t)$, where $\mathcal{H}_1(t)+\mathcal{H}_2(v,t)=\mathcal{H}(v,t)$. In that case we advocate an iterative linearisation procedure that involves fixed-point iteration of the latter equation to solve the former. A typical case is when the original problem is a nonlinear Schr\"odinger or Gross--Pitaevskii equation, while the `easy' equation is linear Schr\"odinger with time-dependent potential. We analyse in detail the iterative scheme and its practical implementation, prove that each iteration increases the order, derive upper bounds on the speed of convergence and discuss in the case of nonlinear Schr\"odinger equation with cubic potential the preservation of structural features of the underlying equation: the $\mathrm{L}_2$ norm, momentum and Hamiltonian energy. A key ingredient in our approach is the use of the Magnus expansion in conjunction with Hermite quadratures, which allows effective solutions of the linearised but non-autonomous equations in an iterative fashion. The resulting Magnus--Hermite methods can be combined with a wide range of numerical approximations to the matrix exponential. The paper concludes with a number of numerical experiments, demonstrating the power of the proposed approach.

翻译：通常，非线性色散方程 $\mathrm{i}\partial_t u(t)=\mathcal{H}(u(t),t)u(t)$ 的数值求解是一项艰巨挑战，而求解形式为 $\mathrm{i}\partial_t u(t)=\mathcal{H}_1(t)u(t)+\widetilde{\mathcal H}_2(t)u(t)$ 的密切相关的线性方程却相对简单且计算成本低廉，其中 $\mathcal{H}_1(t)+\mathcal{H}_2(v,t)=\mathcal{H}(v,t)$。在此情形下，我们提出一种迭代线性化方法，通过对后者方程进行不动点迭代来求解前者方程。典型案例包括原始问题为非线性薛定谔方程或格罗斯–皮塔耶夫斯基方程，而“简单”方程为含时势的线性薛定谔方程。我们详细分析了该迭代格式及其实际实现，证明每次迭代均可提升精度阶数，推导了收敛速度的上界，并针对含三次势的非线性薛定谔方程讨论了底层方程结构特征（$\mathrm{L}_2$ 范数、动量和哈密顿能量）的保持性。方法的关键要素在于将马格努斯展开与埃尔米特求积相结合，从而能够以迭代方式有效求解线性化但非自治的方程。由此产生的马格努斯–埃尔米特方法可兼容多种矩阵指数的数值逼近方案。本文通过一系列数值实验验证了所提方法的有效性。