Download or read book Spatial Discretization of the Shallow Water Equations in Spherical Geometry Using Osher Scheme written by D. Lanser and published by . This book was released on 1999 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Dispersive Shallow Water Waves written by Gayaz Khakimzyanov and published by Springer Nature. This book was released on 2020-09-15 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents cutting-edge research on dispersive wave modelling, and the numerical methods used to simulate the propagation and generation of long surface water waves. Including both an overview of existing dispersive models, as well as recent breakthroughs, the authors maintain an ideal balance between theory and applications. From modelling tsunami waves to smaller scale coastal processes, this book will be an indispensable resource for those looking to be brought up-to-date in this active area of scientific research. Beginning with an introduction to various dispersive long wave models on the flat space, the authors establish a foundation on which readers can confidently approach more advanced mathematical models and numerical techniques. The first two chapters of the book cover modelling and numerical simulation over globally flat spaces, including adaptive moving grid methods along with the operator splitting approach, which was historically proposed at the Institute of Computational Technologies at Novosibirsk. Later chapters build on this to explore high-end mathematical modelling of the fluid flow over deformed and rotating spheres using the operator splitting approach. The appendices that follow further elaborate by providing valuable insight into long wave models based on the potential flow assumption, and modified intermediate weakly nonlinear weakly dispersive equations. Dispersive Shallow Water Waves will be a valuable resource for researchers studying theoretical or applied oceanography, nonlinear waves as well as those more broadly interested in free surface flow dynamics.

Download or read book Adaptive Atmospheric Modeling written by Jr̲n Behrens and published by Springer Science & Business Media. This book was released on 2006-08-11 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gives an overview and guidance in the development of adaptive techniques for atmospheric modeling. This book covers paradigms of adaptive techniques, such as error estimation and adaptation criteria. Considering applications, it demonstrates several techniques for discretizing relevant conservation laws from atmospheric modeling.

Download or read book A Finite volume Discretization of the Shallow water Equations in Spherical Geometry written by Lars Pesch and published by . This book was released on 2002 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book High order Spatial Discretization Methods for the Shallow Water Equations written by Anita W. Tam and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We present new numerical methods for the shallow water equations on a sphere in spherical coordinates. In our implementation, the equations are discretized in time with the two-level semi-Lagrangian semi-implicit (SLSI) method, and in space on a staggered grid with the quadratic spline Galerkin (QSG) and the optimal quadratic spline collocation (OQSC) methods. When discretized on a uniform spatial grid, the solutions are shown through numerical experiments to be fourth-order in space locally at the nodes and midpoints of the spatial grids, and third-order globally. We also show that, when applied to a simplified version of the shallow water equations, each of our algorithms yields a neutrally stable solution for the meteorologically significant Rossby waves. Moreover, we demonstrate that the Helmholtz equation associated with the shallow water equations should be derived algebraically rather than analytically in order for the algorithms to be stable with respect to the Rossby waves. These results are verified numerically using Boyd's equatorial wave equations with initial conditions to generate a soliton. We then analyze the performance of our methods on various staggered grids--the A-, B-, and C-grids. From an eigenvalue analysis of our simplified version of the shallow water equations, we conclude that, when discretized on the C-grid, our algorithms faithfully capture the group velocity of inertia-gravity waves. Our analysis suggests that neither the A- nor B-grids will produce such good results. Our theoretical results are supported by numerical experiments, in which we discretize Boyd's equatorial wave equations using different staggered grids and set the initial conditions for the problem to generate gravitation modes instead of a soliton. With both the A- and B-grids, some waves are observed to travel in the wrong direction, whereas, with the C-grid, gravity waves of all wavelengths propagate in the correct direction.

Download or read book A Vorticity Divergence Global Semi Lagrangian Spectral Model for the Shallow Water Equations written by and published by . This book was released on 2001 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: The shallow water equations modeling flow on a sphere are useful for the development and testing of numerical algorithms for atmospheric climate and weather models. A new formulation of the shallow water equations is derived which exhibits an advective form for the vorticity and divergence. This form is particularly well suited for numerical computations using a semi-Lagrangian spectral discretization. A set of test problems, standard for the shallow water equations on a sphere, are solved and results compared with an Eulerian spectral model. The semi-Lagrangian transport method was introduced into atmospheric modeling by Robert, Henderson, and Turnbull. A formulation based on a three time level integration scheme in conjunction with a finite difference spatial discretization was studied by Ritchie. Two time level grid point schemes were derived by Bates et al. Staniforth and Cote survey developments of the application of semi-Lagrangian transport (SLT) methods for shallow water models and for numerical weather prediction. The spectral (or spherical harmonic transform) method when combined with a SLT method is particularly effective because it allows for long time steps avoiding the Courant-Friedrichs-Lewy (CFL) restriction of Eulerian methods, while retaining accurate (spectral) treatment of the spatial derivatives. A semi-implicit, semi-Lagrangian formulation with spectral spatial discretization is very effective because the Helmholz problem arising from the semi-implicit time integration can be solved cheaply in the course of the spherical harmonic transform. The combination of spectral, semi-Lagrangian transport with a semi-implicit time integration schemes was first proposed by Ritchie. A advective formulation using vorticity and divergence was introduced by Williamson and Olson. They introduce the vorticity and divergence after the application of the semi-Lagrangian discretization. The semi-Lagrangian formulation of Williamson and Olson and Bates et al. has the property that the metric terms of the advective form are treated discretely requiring a delicate spherical vector addition of terms at the departure point and arrival point. In their formulation, the metric terms associated with the advection operator do not appear explicitly. The spherical geometry associated with the combination of vector quantities at arrival and departure points treats the metric terms and is derived in Bates et al. The formulation derived in this paper avoids this vector addition. It is possible to do this because our formulation is based entirely on a scalar, advective form of the momentum equations. This new form is made possible by the generalization of a vector identity to spherical geometry. In Section 2 the standard form of the shallow water equations in spherical geometry are given. Section 3 presents the vector identities needed to derive an advective form of the vorticity and divergence equations. The semi-implicit time integration and semi-Lagrangian transport method are described in Section 4. The SLT interpolation scheme is described in Section 5. Section 6 completes the development of the discrete model with the description of the semi-implicit spectral equations. A discussion of results on several standard test problems is contained in Section 7.

Download or read book Shallow Water Hydrodynamics written by W.Y. Tan and published by Elsevier. This book was released on 1992-08-17 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: Within this monograph a comprehensive and systematic knowledge on shallow-water hydrodynamics is presented. A two-dimensional system of shallow-water equations is analyzed, including the mathematical and mechanical backgrounds, the properties of the system and its solution. Also featured is a new mathematical simulation of shallow-water flows by compressible plane flows of a special virtual perfect gas, as well as practical algorithms such as FDM, FEM, and FVM. Some of these algorithms have been utilized in solving the system, while others have been utilized in various applied fields. An emphasis has been placed on several classes of high-performance difference schemes and boundary procedures which have found wide uses recently for solving the Euler equations of gas dynamics in aeronautical and aerospatial engineering. This book is constructed so that it may serve as a handbook for practicians. It will be of interest to scientists, designers, teachers, postgraduates and professionals in hydraulic, marine, and environmental engineering; especially those involved in the mathematical modelling of shallow-water bodies.

Download or read book Computational Algorithms for Shallow Water Equations written by Eleuterio F. Toro and published by Springer Nature. This book was released on with total page 413 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Mechanics written by and published by . This book was released on 2001 with total page 422 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Adaptive Grids in Weather and Climate Modeling written by Christiane Jablonowski and published by . This book was released on 2004 with total page 572 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book High order Spatial Discretization Methods for the Shallow Water Equations microform written by Anita W. (Anita Wing-Yan) Tam and published by National Library of Canada = Bibliothèque nationale du Canada. This book was released on 2001 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Standard Test Set for Numerical Approximations to the Shallow Water Equations in Spherical Geometry written by and published by . This book was released on 1991 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: A suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry. The shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth. These cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade-offs which must always be made in numerical modeling. Before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods. The cases are presented in order of complexity. They consist of advection across the poles, steady state geostrophically balanced flow of both global and local scales, forced nonlinear advection of an isolated low, zonal flow impinging on an isolated mountain, Rossby-Haurwitz waves and observed atmospheric states. One of the cases is also identified as a computer performance/algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers. 31 refs.

Download or read book High order Spatial Discretization Methods for the Shallow Water Equations written by and published by . This book was released on 2001 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Spectral Element Shallow Water Model on Spherical Geodesic Grids written by and published by . This book was released on 2001 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt: The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagrange-Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336 368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully three-dimensional equations.

Download or read book An Adaptive Multiblock High order Finite volume Method for Solving the Shallow water Equations on the Sphere written by and published by . This book was released on 2015 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: We present a high-order finite-volume approach for solving the shallow-water equations on the sphere, using multiblock grids on the cubed-sphere. This approach combines a Runge--Kutta time discretization with a fourth-order accurate spatial discretization, and includes adaptive mesh refinement and refinement in time. Results of tests show fourth-order convergence for the shallow-water equations as well as for advection in a highly deformational flow. Hierarchical adaptive mesh refinement allows solution error to be achieved that is comparable to that obtained with uniform resolution of the most refined level of the hierarchy, but with many fewer operations.

Download or read book A Spline Collocation Scheme for the Spherical Shallow Water Equations written by Jochen Göttelmann and published by . This book was released on 1998 with total page 48 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Variational Integrators for the Rotating Shallow Water Equations written by Rüdiger Brecht and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The numerical simulation of the Earth's atmosphere plays an important role in developing our understanding of climate change. The atmosphere and ocean can be seen as a shallow fluid on the globe; here, we use the shallow water equations as a first step to approximate these geophysical flows. Then, the numerical model can only be accurate if it has good conservation properties, e.g. without conserving mass the simulation can not be physical. Obtaining such a numerical model can be achieved using numerical variational integration. Here, we have derived a numerical variational integrator for the rotating shallow water equations on the sphere using the Euler-Poincaré framework. First, the continuous Lagrangian is discretized; then, the numerical scheme is obtained by computing the discrete variational principle. The conservational properties and accuracy of the model are verified with standard test cases. However, in order to obtain more realistic simulations, the shallow water equations need to include physical parametrizations. Thus, we introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. Then, benchmarks are carried out to demonstrate that the spatial part of the stochastic scheme preserves the total energy. The proposed random model better captures the structure of a large-scale flow than a comparable deterministic model. Furthermore, to be able to carry out long term simulations we extend the discrete Euler-Poincaré framework with a selective decay. The selective decay dissipates an otherwise conserved quantity while conserving energy. We apply the new framework to the shallow water equations to dissipate the potential enstrophy. Then, we carry out standard benchmarks to demonstrate the conservation properties. We show that the selective decay resolves more small scales compared to a standard dissipation.