Structure-preserving variational schemes for fourth order nonlinear partial differential equations with a Wasserstein gradient flow structure

Ashworth, Blake (2020) Structure-preserving variational schemes for fourth order nonlinear partial differential equations with a Wasserstein gradient flow structure. Doctoral thesis (PhD), University of Sussex.

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Abstract

There is a growing interest in studying nonlinear partial differential equations which constitute gradient flows in the Wasserstein metric and related structure preserving variational discretisations. In this thesis, we focus on the fourth order Derrida-Lebowitz-Speer-Spohn (DLSS) equation, the thin film equation, as well as other fourth order examples. We adapt the minimising movement schemes from implicit Euler (BDF1) to higher order schemes, i.e. backward difference formulae and diagonally implicit Runge-Kutta (DIRK) methods.

We prove numerical convergence of discrete solutions of the DIRK2 scheme using a comparison principle type approach with semi-convex based conditions. With basic assumptions including semi-convexity of our energy, verifying that the energy is monotonic in time normally yields convergence of its discrete solution for decreasing time step. However, as in the BDF2 example, for the DIRK2 scheme considered here the energy was not verified to be monotonic (it might be), yet with additional assumptions, convergence is obtained as well as other basic properties of gradient flows.

We propose fully discrete schemes which preserve positivity for the DLSS equation, the Thin Film equation and other nonlinear partial differential equations. We present results of numerical experiments confirming improved rates of convergence for higher order schemes. Furthermore, numerical results with non-constant time steps are presented, improving the efficiency of the proposed schemes.

Item Type: Thesis (Doctoral)
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Library Cataloguing
Date Deposited: 28 Sep 2020 10:39
Last Modified: 28 Sep 2020 10:39
URI: http://sro.sussex.ac.uk/id/eprint/94049

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