Numerical methods for fully nonlinear second order partial differential equations

This dissertation concerns the numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). The numerical methods and analysis are based on a new concept of weak solutions called moment solutions, which unlike viscosity solutions, are defined by a constructive method called the vanishing moment method. The main idea of the vanishing moment method is to approximate fully nonlinear second order PDEs by a family of fourth order quasi-linear PDEs. Because the method is constructive, we can develop a wealth of convergent numerical discretization methods to approximate fully nonlinear second order PDEs. We first study the numerical approximation of the prototypical second order fully nonlilnear PDE, the Monge-Ampère equation, det(D²u) = f (\u3e 0), using C¹ finite element methods, spectral Galerkin methods, mixed finite element methods, and a nonconforming Morley finite element method. We then generalize the analysis to other fully nonlinear second order PDEs including the nonlinear balance equation, a nonlinear formulation of semigeostrophic flow equations, and the equation of prescribed Gauss curvature

Asymptotic limit analysis for numerical models of atmospheric frontogenesis

Accurate prediction of the future state of the atmosphere is important throughout society, ranging from the weather forecast in a few days time to modelling the effects of a changing climate over decades and generations. The equations which govern how the atmosphere evolves have long been known; these are the Navier-Stokes equations, the laws of thermodynamics and the equation of state. Unfortunately the nonlinearity of the equations prohibits analytic solutions, so simplified models of particular flow phenomena have historically been, and continue to be, used alongside numerical models of the full equations. In this thesis, the two-dimensional Eady model of shear-driven frontogenesis (the creation of atmospheric fronts) was used to investigate how errors made in a localised region can affect the global solution. Atmospheric fronts are the boundary of two different air masses, typically characterised by a sharp change in air temperature and wind direction. This occurs across a small length of O(10 km), whereas the extent of the front itself can be O(1000 km). Fronts are a prominent feature of mid-latitude weather systems and, despite their narrow width, are part of the large-scale, global solution. Any errors made locally in the treatment of fronts will therefore affect the global solution. This thesis uses the convergence of the Euler equations to the semigeostrophic equations, a simplified model which is representative of the large-scale flow, including fronts. The Euler equations were solved numerically using current operational techniques. It was shown that highly predictable solutions could be obtained, and the theoretical convergence rate maintained, even with the presence of near-discontinuous solutions given by intense fronts. Numerical solutions with successively increased resolution showed that the potential vorticity, which is a fundamental quantity in determining the large-scale, balanced flow, approached the semigeostrophic limit solution. Regions of negative potential vorticity, indicative of local areas of instability, were reduced at high resolution. In all cases, the width of the front reduced to the grid-scale. While qualitative features of the limit solution were reproduced, a stark contrast in amplitude was found. The results of this thesis were approximately half in amplitude of the limit solution. Some attempts were made at increasing the intensity of the front through spatial- and temporal-averaging. A scheme was proposed that conserves the potential vorticity within the Eady model.Open Acces

Rotating hydraulic control : 1997 summer study program in geophysical fluid dynamics

Rotating Hydraulic Control was the topic of the thirty ninth year of the Geophysical Fluid Dynamics (GFD) program at the Woods Hole Oceanographic Institution. This theme was principally centered about those nonlinear problems in which either a free surface or internal stratification is so modified by flow that it acts to choke off increased flux as the forcing is increased. It is a peculiar form of convection, which shares many constraints with more general buoyancy driven motion but which has its own internal limits. Lectures and seminars were given by GFD staff and visitors, most of whom are founders of this young field of study. This volume contains notes from the talks given by the principal lecturers and written reports on the research projects cared out by the ten student fellows. The volume, therefore, summarizes a sizable percentage of the present understanding of the topic of Rotating Hydraulic Control.Funding was provided by the National Science Foundation through Grant No. OCE-9314484 and the Office of Naval Research through Grant No. ONR-N00014-97-1-0934

Wave-turbulence interaction in shallow water numerical models: asymptotic limits, and subgrid interactions

The ability to directly simulate all atmospheric motion is currently well beyond the limits of the computers available to us. As such techniques must be developed that accurately model important processes in an affordable manner. Large-scale balanced motion is well understood, but as affordable resolution increases, models are able to resolve scales where large-scale turbulence and small-scale waves are important. This requires a new set of techniques that respect the interactions between these different kinds of motion. In this thesis we look at two ways of assessing the accuracy of models capable of representing the scales at which these interactions occur. The first approach uses asymptotic limit solutions to derive a set of terms whose scale is known. These terms can then be evaluated as the model approaches a relevant asymptotic regime, and a `good' model should reproduce the expected rate of scaling. We apply this method of asymptotic limit solutions to an Eulerian and a Lagrangian shallow water model. The former is based upon ENDGame, the model currently in use at the Met Office, and the latter is based upon a candidate model from GungHo which is seeking a replacement for ENDGame. In addition, the Eulerian model is evaluated with both small and large timesteps and the results confirm the ability of the semi-implicit scheme to retain accuracy at large timesteps. Errors in the higher-order diagnostics used in this section highlight the need to make these analytic diagnostics consistent with the discretisations of the model in question. The second method involves looking at the exchanges of energy in a spectral shallow water model in order to inform the design of subgrid models. By running a high-resolution simulation and truncating the energy at a certain wavenumber, comparing the result to a run without truncation shows the contribution of the scales below the truncation limit. We extend this by separating the total energy into separate components that may be truncated and evaluated individually in order to give a more complete picture of energy exchanges at the subgrid scale

The dynamics of unsteady strait and still flow

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Meteorology and Physical Oceanography, 1982.Microfiche copy available in Archives and ScienceBibliography: leaves 108-109.by Lawrence J. Pratt.Ph.D

Physics–Dynamics Coupling in weather, climate and Earth system models: Challenges and recent progress

This is the final version. Available from American Meteorological Society via the DOI in this record.Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these components can be classified as the resolved fluid dynamics, unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical aspects such as radiation and microphysical processes. Typically, each component is developed, at least initially, independently. Once development is mature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can have a significant impact on the model. As the error associated with each component decreases, the errors introduced by the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performance of the overall system. The challenges associated with combining the components to create a coherent model are here termed physics–dynamics coupling. The issue goes beyond the coupling between the parameterizations and the resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives: to illustrate the phenomenology of the coupling problem with references to examples in the literature, to show how the problem can be analyzed, and to create awareness of the issue across the disciplines and specializations. The topics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components, and emerging issues such as those that arise as model resolutions increase and/or models use variable resolutions.Natural Environment Research Council (NERC)National Science FoundationDepartment of Energy Office of Biological and Environmental ResearchPacific Northwest National Laboratory (PNNL)DOE Office of Scienc

An Efficient Operator-Splitting Method for the Eigenvalue Problem of the Monge-Amp\`{e}re Equation

We develop an efficient operator-splitting method for the eigenvalue problem of the Monge-Amp\`{e}re operator in the Aleksandrov sense. The backbone of our method relies on a convergent Rayleigh inverse iterative formulation proposed by Abedin and Kitagawa (Inverse iteration for the {M}onge-{A}mp{\`e}re eigenvalue problem, {\it Proceedings of the American Mathematical Society}, 148 (2020), no. 11, 4975-4886). Modifying the theoretical formulation, we develop an efficient algorithm for computing the eigenvalue and eigenfunction of the Monge-Amp\`{e}re operator by solving a constrained Monge-Amp\`{e}re equation during each iteration. Our method consists of four essential steps: (i) Formulate the Monge-Amp\`{e}re eigenvalue problem as an optimization problem with a constraint; (ii) Adopt an indicator function to treat the constraint; (iii) Introduce an auxiliary variable to decouple the original constrained optimization problem into simpler optimization subproblems and associate the resulting new optimization problem with an initial value problem; and (iv) Discretize the resulting initial-value problem by an operator-splitting method in time and a mixed finite element method in space. The performance of our method is demonstrated by several experiments. Compared to existing methods, the new method is more efficient in terms of computational cost and has a comparable rate of convergence in terms of accuracy

Rotating convection : 1995 Summer Study Program in Geophysical Fluid Dynamics

The 1995 program in Geophysical Fluid Dynamics addressed "Rotating Convection," with particular emphasis on high-Rayleigh-number convection and on convection in the ocean.Funding was provided by the National Science Foundation under Grant No. OCE-8901012