Preprints

Filtering by Subject: Partial Differential Equations

On the use of mesh movement methods to help overcome the multi-scale challenges associated with hydro-morphodynamic modelling

Mariana C A Clare, Joseph Wallwork, Stephan C Kramer, Hilary Weller, Colin J Cotter, Matthew Piggott

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 Navier–Stokes equation for incompressible flow around a solid sphere

Ahmad Talaei, Timothy J. Garrett

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

Agah D. Garnadi, Corinna Bahriawati

Published: 2020-04-14
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

Mariana C A Clare, James Percival, Athanasios Angeloudis, Colin Cotter, Matthew Piggott

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

Joseph Wallwork, Nicolas Barral, Stephan C. Kramer, David Ham, Matthew Piggott

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

Denise Degen, Karen Veroy, Florian Wellmann

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

Han Geurdes

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.

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