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2022 | Book

Mathematics of the Weather

Polygonal Spline Local-Galerkin Methods on Spheres

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About this book

"Mathematics of the Weather” details the mathematical techniques used to create numerical models of the atmosphere. It explains methods which are currently considered for practical use in models for the exaflop computers (10**19 operations per seconds). This book is a guide to developing and modifying the mathematical methods used in such models. This includes Implementations in spherical geometry. The books also concentrates on elements of Numerical Weather Predication (NWP) and Computational Fluid Dynamics (CFD).

Table of Contents

Frontmatter
Introduction
Abstract
This chapter defines the subject “numerics” of this book, as opposed to other subjects important for numerical weather prediction, such as data assimilation or radiation. The practical engineering approach prevalent in this book is exemplified, as opposed to a mainly theoretical approach. An overview of the different subjects treated is given.
Jürgen Steppeler, Jinxi Li
Simple Finite Difference Procedures
Abstract
This chapter gives a few definitions of standard numerical schemes to be used for comparison or as part of the schemes to be described in later chapters. As this book is mainly about spatial discretization, the fourth-order Runge–Kutta scheme is used in tests for time discretization. A rather comprehensive set of possibilities is given in Durran (Numerical methods for fluid dynamics: with applications to geophysics, 2nd edn. Springer, New York, pp. 35–146, 2010), along with an analysis of many standard numerical temporal schemes. This information is given here to make this book self-contained. It is readable with basic knowledge of mathematics and without previous knowledge of numerics and this chapter is overlapping with Durran (Numerical methods for fluid dynamics: with applications to geophysics, 2nd edn. Springer, New York, pp. 35–146, 2010) for the convenience of the reader.
Jürgen Steppeler, Jinxi Li
Local- Galerkin Schemes in 1D
Abstract
This chapter describes Local-Galerkin (L-Galerkin) methods, which are generalizations of the classic Galerkin procedure. L-Galerkin methods are local and therefore more practical on multiprocessing computers. A well-known example is the spectral element (SE) method, but there exist a large number of alternatives, named onom or the third-degree method. onom approximates the fields in order n and the fluxes in order m. All methods described here use piecewise continuous polynomial spaces and therefore are continuous Galerkin (CG) methods. An important consideration with onom and the classic Galerkin methods is the sparseness of the grid, which is alternatively called the serendipity grid which was already used with the classic Galerkin method. This means that not all points of a regular grid are used and this offers a considerable potential saving of computer time.
Jürgen Steppeler, Jinxi Li
2D Basis Functions for Triangular and Rectangular Meshes
Abstract
This chapter defines 2D field representations used for L-Galerkin methods, such as SE, FE, and onom. For convenience and ease of understanding, some formulas are given for regular cell structures. A generalization to irregular cells is possible. More general cell structures, such as quadrilaterals or hexagons, can be derived from the triangular representation by treating some of the amplitudes as diagnostic. Even if applicable to irregular meshes, the meshes will be presented as structured. For research in new numerical methods, structured test models have the advantage of being easy to create. Unstructured programming is conveniently done by using a neighborhood management system, which is present in the MPAS, COSMO, and Fluidity models.
Jürgen Steppeler, Jinxi Li
Finite Difference Schemes on Sparse and Full Grids
Abstract
The notion of FDs on sparse grids is introduced. It means that from a regular distribution of grid points not all are used, which offers the opportunity to make models more efficient for the same resolution. This section aims at transferring some of the 1D schemes defined in Chapter “Local-Galerkin Schemes in 1D” to two dimensions. There is no way the most general L-Galerkin scheme or a class of such schemes can be presented. The number of possibilities is too large. So just examples are presented to show how the schemes work. In particular, most examples are 2D. While the complexity of computer programs normally increases substantially when going to 3D, it is often obvious how to proceed from 2D to 3D.
Jürgen Steppeler, Jinxi Li
Full and Sparse Hexagonal Grids in the Plane
Abstract
Hexagonal grids are constructed by combining triangular patches. On the sphere, a few pentagonal cells are used.
Jürgen Steppeler, Jinxi Li
Platonic and Semi- Platonic Solids
Abstract
A number of options are given for constructing Platonic and Semi-Platonic solids for spherical discretization. This chapter provides just spherical grids and a simple non-conserving toy model. However, the tools and the example provided in Chapters “Finite Difference Schemes on Sparse and Full Grids” and “Full and Sparse Hexagonal Grids in the Plane” allow to create L-Galerkin sparse grids on the sphere. Further L-Galerkin methods will be presented in Chapter “Numerical Tests.”
Jürgen Steppeler, Jinxi Li
Numerical Tests
Abstract
A few tests are proposed being based on toy models.
Jürgen Steppeler, Jinxi Li
Summary and Outlook
Abstract
This section deals with potential applications and important research areas of numerical models of the atmosphere.
Jürgen Steppeler, Jinxi Li
Examples of Program
Abstract
This chapter provides programs in Matlab to perform some of the tests described in previous chapters.
Jürgen Steppeler, Jinxi Li
Backmatter
Metadata
Title
Mathematics of the Weather
Authors
Jürgen Steppeler
Jinxi Li
Copyright Year
2022
Electronic ISBN
978-3-031-07238-3
Print ISBN
978-3-031-07237-6
DOI
https://doi.org/10.1007/978-3-031-07238-3