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Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
Adaptive modeling in atmospheric sciences has evolved to a state of maturity that it seems to be the right time to summarize what has been achieved so far and to sketch the near future of research directions. This work gives an overview of current approaches to adaptive atmospheric modeling. The author has included material and results cited from other sources in order to give a broader overview of the different approaches. It is clear that his own work is described in more detail, even if this might not in all cases be the mainstream in technological development. Many of the achievements of the author’s group are reported in a way that tries to be as understandable as possible, yet detailed enough to be reproducible. However, if in doubt, readability was given the preference.
Jörn Behrens

2. Principles of Adaptive Atmospheric Modeling

Abstract
In this chapter, major principles of adaptivity and especially adaptive atmospheric modeling will be discussed. We start with reconsidering the paradigms of adaptivity in the mathematical and in the meteorological communities respectively in order to clarify the basic notations. After this clarification, and a description of principle challenges in adaptivity, we introduce concepts of adaptive refinement techniques and discuss refinement criteria.
Jörn Behrens

3. Grid Generation

Abstract
We start the more formal description of adaptive atmospheric modeling methods with an introduction to mesh generation. In contrast to non-adaptive methods, where a fixed grid is given, in adaptive methods, the grid generating parts of a modeling software play a fundamental role. While in fixed grid applications, the software design is often derived from a computational stencil, in adaptive methods, the grid management forms the underlying basis for other derived data structures. We will discuss efficient data structures later in chap. 4 and will concentrate only on grid generation issues here.
Jörn Behrens

4. Data Structures for Computational Efficiency

Abstract
Each adaptive algorithm for solving atmospheric flow problems can be separated into two basic parts:
Jörn Behrens

5. Issues in Parallelization of Irregularly Structured Problems

Abstract
Problems in atmospheric modeling were counted among the grand challenges a few years ago. For almost a decade there has been extensive research in order to develop modeling software that is capable of utilizing modern high performance computing architectures (see e.g. [3, 103, 118, 154, 176, 181, 186, 201, 235, 237, 302, 333, 342, 418]). High performance inevitably means parallel computing. When dealing with adaptive mesh refinement, parallelization becomes a non-trivial issue for several reasons.
Jörn Behrens

6. Numerical Treatment of Differential Operators on Adaptive Grids

Abstract
This chapter is concerned with the realization of differential operators, mainly on unstructured and nonuniform grids. While a finite difference approximation to a differential operator can be easily derived for orthogonal and quadrilateral grids, it is not straight forward to do the same for unstructured and nonorthogonal grids. We will focus the presentation on the gradient operator
Jörn Behrens

7. Discretization of Conservation Laws

Abstract
Many of the basic equations in atmospheric modeling are based on conservation laws. Conservation of mass constitutes the continuity equation, and conservation of momentum establishes the momentum equations. When conservation properties are present in the continuous equations, the numerical (discrete) counterparts should also have conservative properties. Examples for numerical conservation of vorticity or other state variables can be found in [46, 350]. More generally we want a numerical method to adhere to a structure preservation property. To achieve conservation is paramount for all kinds of numerical methods that try to discretize conservation laws. However, for adaptive methods this often poses an additional challenge.
Jörn Behrens

8. Example Applications

Abstract
This chapter is devoted to some example applications in adaptive atmospheric (and oceanic) modeling. Of course, there are many more examples of the application of adaptivity in atmospheric modeling. These examples represent projects that were realized by or in collaboration with the author. The first two examples in sections 8.1 and 8.2 are linear tracer transport examples, the third and fourth examples (section 8.3) are extracted from the research project PLASMA in which dynamical cores for a simplified climate model are under development. While these four examples share semi-Lagrangian timediscretization schemes which are well suited for adaptive mesh refinement (see [34]), the fifth example, demonstrating adaptive wave dispersion in section 8.4, employs a finite volume type discretization method for the shallow water equations.
Jörn Behrens

9. Conclusions

Abstract
Adaptive atmospheric modeling is a truly interdisciplinary approach in scienti fic computing. We have seen several tools from applied mathematics, theoretical physics, computer science and even pure mathematics interacting in order to achieve efficiency, accuracy and robustness for adaptive algorithms in atmospheric modeling. This chapter aims in reviewing the methods and evaluating their respective usefulness. From this we try to derive a path for future research directions.
Jörn Behrens

A. Some Basic Mathematical Tools

Abstract
This appendix contains a brief section on basic mathematical tools that are used throughout this book. We denote by x = xx0,t0 (t) the position at time t of a fluid particle that started at position x0 at time t0.
Jörn Behrens

B. Metrics for Parallelizing Irregularly Structured Problems

Abstract
Naturally, all the metrics for measuring the parallel performance of a nonadaptive program apply similarly to an adaptive program. For parallel metrics one may consult the standard literature (e.g. [205]). We recall the most important performance metrics for use in this book:
Jörn Behrens

C. Rotating Shallow Water Equations in Spherical Geometries

Abstract
In sect. 7.1 we derived the shallow water equations from basic conservation principles. Here we extend the equations to be valid on the rotating sphere. Not all applications in atmospheric modeling need to account for the rotation, but since our emphasis is on global modeling, we have to consider it.
Jörn Behrens

Backmatter

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