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Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 54 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
2D Shallow Water Equations by Composite Schemes
, 1997
"... Composite schemes are formed by global composition of several LaxWendroff steps followed by a diffusive LaxFriedrichs or WENO step which filters out the oscillations around shocks typical for the LaxWendroff scheme. These schemes are applied to the shallow water equations in 2D. The LaxFriedrich ..."
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Cited by 6 (2 self)
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Composite schemes are formed by global composition of several LaxWendroff steps followed by a diffusive LaxFriedrichs or WENO step which filters out the oscillations around shocks typical for the LaxWendroff scheme. These schemes are applied to the shallow water equations in 2D. The LaxFriedrichs composite is also formulated for a trapezoidal mesh, which is necessary in several example problems. The suitability of the composite schemes for the shallow water equations is demonstrated on several examples including the circular dam break problem, the shock focussing problem and supercritical channel flow problems. 1 Introduction The shallow water equations are a system of conservation laws. There are two simple wellknown finite difference schemes for conservation laws, LaxWendroff (LW) and LaxFriedrichs (LF). Both of these methods have some drawbacks. The LW scheme is second order accurate but oscillatory close to shocks. The LF scheme is nonoscillatory but also excessively diffu...
Analysis and Computation with Stratified Fluid Models
, 1997
"... Vertical averaging of the three dimensional incompressible Euler equations leads to several reduced dimension models of flow over topography, including the onelayer and twolayer classic shallow water equations, and the onelayer and twolayer nonhydrostatic GreenNaghdi equations. These equatio ..."
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Cited by 3 (2 self)
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Vertical averaging of the three dimensional incompressible Euler equations leads to several reduced dimension models of flow over topography, including the onelayer and twolayer classic shallow water equations, and the onelayer and twolayer nonhydrostatic GreenNaghdi equations. These equations are derived and their wellposedness is discussed. Several implicit and explicit finite difference approximations of both the shallow water and GreenNaghdi models are presented, but for GreenNaghdi these are obtained using automatic code generation software. Numerical results are given in both wellposed and illposed regimes and compared with computations obtained by others. 2 1 Introduction Vertically averaged models of incompressible flow have an obvious computational advantage over the full threedimensional Euler equations, provided that important features of the flow are retained. Single layer models such as the hyperbolic shallow water equations and the dispersive GreenNa...
The application of PVbased control variable transformations in variational data assimilation
 Department of Mathematics, University of Reading
, 2007
"... I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged. ..."
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Cited by 2 (0 self)
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I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged.
Composite Centered Schemes for Multidimensional Conservation Laws
"... . The oscillations of a centered second order finite difference scheme and the excessive diffusion of a first order centered scheme can be overcome by global composition of the two, that is by performing cycles consisting of several time steps of the second order method followed by one step of the d ..."
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Cited by 1 (1 self)
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. The oscillations of a centered second order finite difference scheme and the excessive diffusion of a first order centered scheme can be overcome by global composition of the two, that is by performing cycles consisting of several time steps of the second order method followed by one step of the diffusive method. We show the effectiveness of this approach on some test problems in two and three dimensions. 1. Introduction For a system of conservation laws U t = f x (U ), it is well known that the LaxWendroff (LW) finite difference scheme produces oscillations behind shock waves while the LaxFriedrichs (LF) method is excessively diffusive, smearing out the shocks more than is usually acceptable. Simple twostep versions of both schemes are defined as follows. For both schemes the first half step defines new values on a staggered dual grid as U n+1=2 i+1=2 = 1 2 [U n i + U n i+1 ] + \Deltat 2\Deltax [f(U n i+1 ) \Gamma f(U n i )]: (1) The second half step of the LF scheme is gi...
THE UNIVERSITY OF READING DEPARTMENTS OF MATHEMATICS AND METEOROLOGY Correlated observation errors
"... Data assimilation techniques combine observations and prior model forecasts to create initial conditions for numerical weather prediction (NWP). The relative weighting assigned to each observation in the analysis is determined by the error associated with its measurement. Remote sensing data often h ..."
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Data assimilation techniques combine observations and prior model forecasts to create initial conditions for numerical weather prediction (NWP). The relative weighting assigned to each observation in the analysis is determined by the error associated with its measurement. Remote sensing data often have correlated errors, but the correlations are typically ignored in NWP. As operational centres move towards highresolution forecasting, the assumption of uncorrelated errors becomes impractical. This thesis provides new evidence that including observation error correlations in data assimilation schemes is both feasible and beneficial. We study the dual problem of quantifying and modelling observation error correlation structure. Firstly, in original work using statistics from the Met Office 4DVar assimilation system, we diagnose strong crosschannel error covariances for the IASI satellite instrument. We then see how in a 3DVar framework, information content is degraded under the assumption of uncorrelated errors, while retention of an approximate correlation gives clear benefits. These novel results motivate further study. We conclude by modelling observation error correlation structure in the framework of a onedimensional shallow water model. Using an incremental 4DVar assimilation system we observe that analysis errors are smallest when correlated error covariance matrix approximations are used over diagonal approximations. The new results reinforce earlier conclusions on the benefits of including some error correlation structure. i Declaration I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged.
GRAVITY WAVES IN MULTILAYER SYSTEMS
"... fulfilment of the requirements for the Degree of Master of Science. I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged. i Acknowledgements I would like to thank my supervisor, Maarten Ambaum, for his help and encouragement throu ..."
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fulfilment of the requirements for the Degree of Master of Science. I confirm that this is my own work and the use of all material from other sources has been properly and fully acknowledged. i Acknowledgements I would like to thank my supervisor, Maarten Ambaum, for his help and encouragement throughout this project. I am extremely grateful for his guidance and useful suggestions. Thanks also to my family for their support, and to all the other M.Sc. students for making this year so enjoyable. Special thanks to Dan for cooking me numerous dinners and picking me up from the department whenever I’ve worked here after dark. This dissertation and my year of study at Reading University has been financed by the National Environmental Research Council. ii The generation of gravity waves by topography is examined in this study. These waves are important in the atmosphere on all scales. Their interaction with the mean flow has implications for global atmospheric circulation. They also feature prominently in localised weather in mountainous or hilly regions. The equations of motion for an homogeneous layer of fluid flowing over a symmetric, one dimensional, isolated mountain are studied and it is found that there is a critical mountain height above which the solution becomes discontinuous. An expression for this critical height is derived. A numerical model is developed to solve the nonlinear shallow water equations in a homogeneous layer and the results it produces are compared with established results. The theory of stratified flow is presented. The effect of approximating continuous vertical profiles of buoyancy frequency and velocity by a finite set of discrete layers is discussed and this multilayer approach is further investigated with the aid of an extension of the single layer numerical model written by the author. The results are compared to established solutions and suggestions are put forward for further work. Contents 1
School of Mathematical and Physical Sciences
, 2012
"... Data assimilation with correlated observation errors: analysis accuracy with approximate error covariance matrices by ..."
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Data assimilation with correlated observation errors: analysis accuracy with approximate error covariance matrices by
606 MONTHLY WEATHER REVIEW vel. 99, No. 8 UDC 661.W.313 PROPAGATION OF SYSTEMATIC ERRORS IN A ONE LAYER PRIMITIVEEQUATION A4 FOR SYNOPTIC SCALE MOTION
"... A onelayer, midlatitudeJ betaplane channel model of an incompressible homogeneous fluid is constructed to study the propagation of systematic errors on a nearly stationary synoptic scale wave. A time and spacecentered difference scheme is used to evaluate the governing primitive equations. Data ..."
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A onelayer, midlatitudeJ betaplane channel model of an incompressible homogeneous fluid is constructed to study the propagation of systematic errors on a nearly stationary synoptic scale wave. A time and spacecentered difference scheme is used to evaluate the governing primitive equations. Data fields resulting from height field perturbations injected at various locations in the synoptic wave are compared to the unperturbed synoptic wave at 3hr intervals for 5 model days. Results show that the lowfrequency or quasigeostrophic component of the error tends to move toward the core of maximum velocity in the basic state and that, after 5 days, these maximum height errors are in the core regardless of the location of the initial perturbation. 1.