Numerical simulation of random solutions to nonlinear differential equations

Kuchemann-Scales, Moe and Blower, Gordon and Khaleghi, Azadeh (2025) Numerical simulation of random solutions to nonlinear differential equations. PhD thesis, Lancaster University.

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Abstract

This thesis focusses on simulation of weak solutions to nonlinear differential equations. Methods employed include reformulation into stochastic differential equations, and transport of measures via pushforward maps and an action principle. The common thread to approaches discussed is to consider an ensemble of solutions and more generally the distribution that this ensemble will take, rather than specific solutions themselves. A second focus of the work is to restrict solutions to the surface of the sphere $\mathbb{S}^2$, simplifying the use of Fourier series but posing difficulties for the long term stability of numerical algorithms. Numerical simulations of random solutions to the nonlinear Schr\"odinger equation (NLSE) and the isentropic Euler equations of fluid motion are carried out. For the NLSE, in the case of $\beta = 0$, the empirical distribution is compared with the theoretic distribution of solutions (the Gibbs measure) statistically. Formulating the problem on the sphere involves a Lax pair, one of which is the Frenet-Serret matrix and the second a result of the Hasimoto transform applied to the NLSE. In the case of the Euler equations, the numerical simulation is compared with evolution of a known closed form solution in the one dimensional dam break problem. The proposed algorithm uses optimal transport theory and convexity over a space of absolutely continuous measures in order to allow weak solutions to the differential equations that are distributions in the sense of Schwartz. The simulation is found to closely follow the well established Ritter solutions to the dam break problem.