Computation of some dispersive equations through their iterated linearisation
Guannan Chen, Arieh Iserles, Karolina Kropielnicka, Pranav Singh
TL;DR
The paper introduces an iterative linearisation framework for nonlinear dispersive Hamiltonian PDEs, recasting the problem as a sequence of linear Schrödinger-type solves advanced by a Magnus expansion with Hermite quadratures. It proves that each iteration increases the formal order and provides convergence bounds, while showing that key invariants such as the $L^2$ norm and momentum are preserved and the Hamiltonian is approximated with increasing accuracy as the number of iterations grows. The Magnus-Hermite (MH) method is then combined with splitting or Krylov techniques to compute the matrix exponential efficiently, and its efficacy is demonstrated through extensive numerical experiments on cubic NLS and GP models. The work highlights the method’s flexibility, invariant properties, and potential applicability to a broader class of PDEs beyond dispersive equations.
Abstract
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 Schrödinger or Gross--Pitaevskii equation, while the `easy' equation is linear Schrödinger 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 Schrödinger 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.