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Title: Variational elliptic solver for atmospheric applications

Abstract

We discuss a conjugate gradient type method -- the conjugate residual -- suitable for solving linear elliptic equations that result from discretization of complex atmospheric dynamical problems. Rotation and irregular boundaries typically lead to nonself-adjoint elliptic operators whose matrix representation on the grid is definite but not symmetric. On the other hand, most established methods for solving large sparse matrix equations depend on the symmetry and definiteness of the matrix. Furthermore, the explicit construction of the matrix can be both difficult and computationally expensive. An attractive feature of conjugate gradient methods in general is that they do not require any knowledge of the matrix; and in particular, convergence of conjugate residual algorithms do not rely on symmetry for definite operators. We begin by reviewing some basic concepts of variational algorithms from the perspective of a physical analogy to the damped wave equation, which is a simple alternative to the traditional abstract framework of the Krylov subspace methods. We derive two conjugate residual schemes from variational principles, and prove that either definiteness or symmetry ensures their convergence. We discuss issues related to computational efficiency and illustrate our theoretical considerations with a test problem of the potential flow of a Boussinesq fluidmore » flow past a steep, three-dimensional obstacle.« less

Authors:
 [1];  [2]
  1. National Center for Atmospheric Research, Boulder, CO (United States)
  2. Los Alamos National Lab., NM (United States)
Publication Date:
Research Org.:
Los Alamos National Lab., NM (United States)
Sponsoring Org.:
USDOE, Washington, DC (United States)
OSTI Identifier:
10130964
Report Number(s):
LA-12712-MS
ON: DE94007725
DOE Contract Number:  
W-7405-ENG-36
Resource Type:
Technical Report
Resource Relation:
Other Information: PBD: Mar 1994
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 54 ENVIRONMENTAL SCIENCES; DIFFERENTIAL EQUATIONS; CALCULATION METHODS; METEOROLOGY; MATHEMATICAL MODELS; ATMOSPHERIC CIRCULATION; DYNAMICS; 580000; 990200; 540110; GEOSCIENCES; MATHEMATICS AND COMPUTERS; BASIC STUDIES

Citation Formats

Smolarkiewicz, P.K., and Margolin, L.G. Variational elliptic solver for atmospheric applications. United States: N. p., 1994. Web. doi:10.2172/10130964.
Smolarkiewicz, P.K., & Margolin, L.G. Variational elliptic solver for atmospheric applications. United States. doi:10.2172/10130964.
Smolarkiewicz, P.K., and Margolin, L.G. Tue . "Variational elliptic solver for atmospheric applications". United States. doi:10.2172/10130964. https://www.osti.gov/servlets/purl/10130964.
@article{osti_10130964,
title = {Variational elliptic solver for atmospheric applications},
author = {Smolarkiewicz, P.K. and Margolin, L.G.},
abstractNote = {We discuss a conjugate gradient type method -- the conjugate residual -- suitable for solving linear elliptic equations that result from discretization of complex atmospheric dynamical problems. Rotation and irregular boundaries typically lead to nonself-adjoint elliptic operators whose matrix representation on the grid is definite but not symmetric. On the other hand, most established methods for solving large sparse matrix equations depend on the symmetry and definiteness of the matrix. Furthermore, the explicit construction of the matrix can be both difficult and computationally expensive. An attractive feature of conjugate gradient methods in general is that they do not require any knowledge of the matrix; and in particular, convergence of conjugate residual algorithms do not rely on symmetry for definite operators. We begin by reviewing some basic concepts of variational algorithms from the perspective of a physical analogy to the damped wave equation, which is a simple alternative to the traditional abstract framework of the Krylov subspace methods. We derive two conjugate residual schemes from variational principles, and prove that either definiteness or symmetry ensures their convergence. We discuss issues related to computational efficiency and illustrate our theoretical considerations with a test problem of the potential flow of a Boussinesq fluid flow past a steep, three-dimensional obstacle.},
doi = {10.2172/10130964},
journal = {},
number = ,
volume = ,
place = {United States},
year = {1994},
month = {3}
}