Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.817531
Title: Structure-preserving variational schemes for fourth order nonlinear partial differential equations with a Wasserstein gradient flow structure
Author: Ashworth, Blake
Awarding Body: University of Sussex
Current Institution: University of Sussex
Date of Award: 2020
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.817531  DOI: Not available
Keywords: QA0299 Analysis. Including analytical methods connected with physical problems
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