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Structure-preserving variational schemes for fourth order nonlinear partial di?erential equations with a Wasserstein gradient ?ow structure

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posted on 2023-06-09, 21:42 authored by Blake Ashworth
There is a growing interest in studying nonlinear partial di?erential equations which constitute gradient ?ows 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 ?lm 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 di?erence 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 veri?ed to be monotonic (it might be), yet with additional assumptions, convergence is obtained as well as other basic properties of gradient ?ows. We propose fully discrete schemes which preserve positivity for the DLSS equation, the Thin Film equation and other nonlinear partial di?erential equations. We present results of numerical experiments con?rming improved rates of convergence for higher order schemes. Furthermore, numerical results with non-constant time steps are presented, improving the e?ciency of the proposed schemes.

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  • Published version

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142.0

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  • Mathematics Theses

Qualification level

  • doctoral

Qualification name

  • phd

Language

  • eng

Institution

University of Sussex

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  • Yes

Legacy Posted Date

2020-09-28

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