Computing rough solutions of the KdV equation below ${\bf L^2}$

TL;DR

This paper introduces a novel numerical and analytical framework for solving the KdV equation in negative Sobolev spaces, overcoming limitations of classical methods at low regularities.

Contribution

It develops the first numerical method capable of solving the KdV equation in negative Sobolev spaces, with nearly optimal convergence rates.

Findings

Achieved nearly optimal-order convergence in $H^{-rac{1}{2}}$ norm.

Bridged the gap between numerical analysis and well-posedness in low regularity spaces.

Established a rescaling strategy to handle small initial data.

Abstract

We establish a novel numerical and analytical framework for solving the Korteweg--de Vries (KdV) equation in the negative Sobolev spaces, where classical numerical methods fail due to their reliance on high regularity and inability to control nonlinear interactions at low regularities. Numerical analysis is established by combining a continuous reformulation of the numerical scheme, the Bourgain-space estimates for the continuous reformulation, and a rescaling strategy that reduces the reformulated problem to a small initial value problem, which allow us to bridge a critical gap between numerical analysis and theoretical well-posedness by designing the first numerical method capable of solving the KdV equation in the negative Sobolev spaces. The numerical scheme is proved to have nearly optimal-order convergence with respect to the spatial degrees of freedom in the $H^{-\frac{1}{2}}$ norm for initial data in $H^s$, with $-\frac{1}{2} < s \leq 0$, a result unattainable by existing numerical methods.