# Preprints

Filtering by Subject: Partial Differential Equations

## A study of extreme water waves using a hierarchy of models based on potential-flow theory

**Published**: 2023-12-02

**Subjects**: Applied Mathematics, Oceanography, Oceanography and Atmospheric Sciences and Meteorology, Partial Differential Equations, Physical Sciences and Mathematics

The formation of extreme waves arising from the interaction of three line-solitons with equal far-field amplitudes is examined through a hierarchy of water-wave models. The Kadomtsev-Petviashvili equation (KPE) is first used to prove analytically that its exact three-soliton solution has a ninefold maximum amplification that is achieved in the absence of spatial divergence. Reproducing this [...]

## A shallow approximation for ice streams sliding over strong beds

**Published**: 2023-03-29

**Subjects**: Glaciology, Partial Differential Equations

Ice streams are regions of rapid ice sheet flow characterised by a high degree of sliding over a deforming bed. The Shallow Shelf Approximation (SSA) provides a convenient way to obtain closed-form approximations of the velocity and flux in a rapidly-sliding ice stream when the basal drag is much less than the driving stress. However, the validity of the SSA approximation breaks down when the [...]

## Tidal Turbine Array Modelling using Goal-Oriented Mesh Adaptation

**Published**: 2022-04-30

**Subjects**: Civil and Environmental Engineering, Fluid Dynamics, Geophysics and Seismology, Numerical Analysis and Computation, Partial Differential Equations

Purpose: To examine the accuracy and sensitivity of tidal array performance assessment by numerical techniques applying goal-oriented mesh adaptation. Methods: The goal-oriented framework is designed to give rise to adaptive meshes upon which a given diagnostic quantity of interest (QoI) can be accurately captured, whilst maintaining a low overall computational cost. We seek to improve the [...]

## Spectral boundary integral method for simulating static and dynamic fields from a fault rupture in a poroelastodynamic solid

**Published**: 2021-08-24

**Subjects**: Applied Mechanics, Geophysics and Seismology, Partial Differential Equations, Tribology

The spectral boundary integral method is popular for simulating fault, fracture, and frictional processes at a planar interface. However, the method is less commonly used to simulate off-fault dynamic fields. Here we develop a spectral boundary integral method for poroelastodynamic solid. The method has two steps: first, a numerical approximation of a convolution kernel and second, an efficient [...]

## Calibration, inversion and sensitivity analysis for hydro-morphodynamic models

**Published**: 2021-08-03

**Subjects**: Applied Mathematics, Fluid Dynamics, Geomorphology, Numerical Analysis and Computation, Partial Differential Equations, Programming Languages and Compilers

The development of reliable, sophisticated hydro-morphodynamic models is essential for protecting the coastal environment against hazards such as flooding and erosion. There exists a high degree of uncertainty associated with the application of these models, in part due to incomplete knowledge of various physical, empirical and numerical closure related parameters in both the hydrodynamic and [...]

## Multi-scale hydro-morphodynamic modelling using mesh movement methods.

**Published**: 2020-10-22

**Subjects**: Applied Mathematics, Geomorphology, Mathematics, Numerical Analysis and Computation, Partial Differential Equations

Hydro-morphodynamic models are an important tool that can be used in the protection of coastal zones. They can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement [...]

## An analytical solution to the Navierâ€“Stokes equation for incompressible flow around a solid sphere

**Published**: 2020-08-25

**Subjects**: Applied Mathematics, Earth Sciences, Engineering, Fluid Dynamics, Mechanical Engineering, Other Mechanical Engineering, Partial Differential Equations, Physical Sciences and Mathematics, Physics, Special Functions

This paper is concerned with obtaining a formulation for the flow past a sphere in a viscous and incompressible fluid, building upon previously obtained well-known solutions that were limited to small Reynolds numbers. Using a method based on a summation of separation of variables, we develop a general analytical solution to the Navier--Stokes equation for the special case of axially symmetric [...]

## A mixed $RT_0 - P_0$ Raviart-Thomas finite element implementation of Darcy Equation in GNU Octave

**Published**: 2020-04-13

**Subjects**: Applied Mathematics, Bioresource and Agricultural Engineering, Chemical Engineering, Computational Engineering, Earth Sciences, Engineering, Environmental Sciences, Hydrology, Numerical Analysis and Computation, Partial Differential Equations, Physical Sciences and Mathematics, Water Resource Management

In this paper we shall describe mixed formulations -differential and variational- of Darcys flow equation, an important model of elliptic problem. We describe * Galerkin method with finite dimensional spaces; * Local matrices and assembling; * Raviart-Thomas $RT_0 - P_0$ elements; * Edge basis and local matrices for $RT_0 - P_0$ FEM; * Model problem with corresponding local matrices, right hand [...]

## Hydro-morphodynamics 2D modelling using a discontinuous Galerkin discretisation

**Published**: 2020-01-09

**Subjects**: Applied Mathematics, Computer Sciences, Earth Sciences, Geomorphology, Numerical Analysis and Computation, Partial Differential Equations, Physical Sciences and Mathematics, Sedimentology

The development of morphodynamic models to simulate sediment transport accurately is a challenging process that is becoming ever more important because of our increasing exploitation of the coastal zone, as well as sea-level rise and the potential increase in strength and frequency of storms due to a changing climate. Morphodynamic models are highly complex given the non-linear and coupled nature [...]

## Goal-Oriented Error Estimation and Mesh Adaptation for Shallow Water Modelling

**Published**: 2019-12-31

**Subjects**: Applied Mathematics, Computer Sciences, Engineering, Non-linear Dynamics, Numerical Analysis and Computation, Numerical Analysis and Scientific Computing, Partial Differential Equations, Physical Sciences and Mathematics

Numerical modelling frequently involves a diagnostic quantity of interest (QoI) - often of greater importance than the PDE solution - which we seek to accurately approximate. In the case of coastal ocean modelling the power output of a tidal turbine farm is one such example. Goal-oriented error estimation and mesh adaptation can be used to provide meshes which are well-suited to achieving this [...]

## Certified Reduced Basis Method in Geosciences Addressing the challenge of high dimensional problems

**Published**: 2019-06-28

**Subjects**: Applied Mathematics, Earth Sciences, Numerical Analysis and Computation, Partial Differential Equations, Physical Sciences and Mathematics

One of the biggest challenges in Computational Geosciences is finding ways of efficiently simulating high-dimensional problems. In this paper, we demonstrate how the RB method can be gainfully exploited to solve problems in the Geosciences. The reduced basis method constructs low-dimensional approximations to (high-dimensional) solutions of parametrized partial differential equations. In contrast [...]

## A type D breakdown of the Navier Stokes equation in d=3 spatial dimensions

**Published**: 2017-10-26

**Subjects**: Applied Mathematics, Partial Differential Equations, Physical Sciences and Mathematics

In this paper a type D breakdown of the Navier Stokes equation in d=3 spatial dimensions is demonstrated. The element of breakdown also occurs in the Euler equation.