On coupling resolved and unresolved physical processes in finite element discretisations of geophysical fluids

At the heart of modern numerical weather forecasting and climate modelling lie simulations of two geophysical fluids: the atmosphere and the ocean. These endeavours rely on numerically solving the equations that describe these fluids. A key challenge is that the fluids contain motions spanning a range of scales. As the small-scale processes (unresolved by the numerical model) affect the resolved motions, they need to be described in the model, which is known as parametrisation. One major class of methods for numerically solving such partial differential equations is the finite element method. This thesis focuses on the coupling of such parametrised processes to the resolved flow within finite element discretisations. Four sets of research are presented, falling under two main categories. The first is the development of a compatible finite element discretisation for use in numerical weather prediction models, so as to avoid the bottleneck in computational scalability associated with the convergence at the poles of latitude-longitude grids. We present a transport scheme for use with the lowest-order function spaces in such a compatible finite element method, which is motivated by the coupling of the resolved and unresolved processes within the model. This then facilitates the use of the lower-order spaces within Gusto, a toolkit for studying such compatible finite element discretisations. Then, we present a compatible finite element discretisation of the moist compressible Euler equations, parametrising the unresolved moist processes. This is a major step in the development of Gusto, extending it to describe its first unresolved processes. The second category with which this thesis is concerned is the stochastic variational framework presented by Holm [Variational principles for stochastic fluid dynamics, P. Roy. Soc. A-Math. Phy. 471 (2176), (2015)]. In this framework, the effect of the unresolved processes and their uncertainty is expressed through a stochastic component to the advecting velocity. This framework ensures the circulation theorem is preserved by the stochastic equations. We consider the application of this formulation to two simple geophysical fluid models. First, we discuss the statistical properties of an enstrophy-preserving finite element discretisation of the stochastic quasi-geostrophic equation. We find that the choice of discretisation and the properties that it preserves affects the statistics of the solution. The final research presented is a finite element discretisation of the stochastic Camassa-Holm equation, which is used to numerically investigate the formation of ‘peakons’ within this set-up, finding that they do still always form despite the noise’s presence.Open Acces

Hyperbolic Conservation Laws

[no abstract available

Numerical superposition of Gaussian beams over propagating domain for high frequency waves and high-order invariant-preserving methods for dispersive waves

This thesis is devoted to efficient numerical methods and their implementations for two classes of wave equations. The first class is linear wave equations in very high frequency regime, for which one has to use some asymptotic approach to address the computational challenges. We focus on the use of the Gaussian beam superposition to compute the semi--classical limit of the Schr {o}dinger equation. The second class is dispersive wave equations arising in modeling water waves. For the Whitham equation, so-called the Burgers--Poisson equation, we design, analyze, and implement local discontinuous Galerkin methods to compute the energy conservative solutions with high-order of accuracy. Our Gaussian beam (GB) approach is based on the domain-propagation GB superposition algorithm introduced by Liu and Ralston [Multiscale Model. Simul., 8(2), 2010, 622--644]. We construct an efficient numerical realization of the domain propagation-based Gaussian beam superposition for solving the Schr odinger equation. The method consists of several significant steps: a semi-Lagrangian tracking of the Hamiltonian trajectory using the level set representation, a fast search algorithm for the effective indices associated with the non-trivial grid points that contribute to the approximation, an accurate approximation of the delta function evaluated on the Hamiltonian manifold, as well as efficient computation of Gaussian beam components over the effective grid points. Numerical examples in one and two dimensions demonstrate the efficiency and accuracy of the proposed algorithms. For the Burgers--Poisson equation, we design, analyze and test a class of local discontinuous Galerkin methods. This model, proposed by Whitham [Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974] as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. The proposed numerical method is high order accurate and preserves two invariants, hence producing solutions with satisfying long time behavior. The $L^2$-stability of the scheme for general solutions is a consequence of the energy preserving property. The optimal order of accuracy for polynomial elements of even degree is proven. A series of numerical tests is provided to illustrate both accuracy and capability of the method

A numerical study of variational discretizations of the Camassa-Holm equation

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa-Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare the variational discretizations to existing methods over several numerical examples.Comment: 45 pages, 14 figure

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学