Variational elliptic solver for atmospheric applications
- National Center for Atmospheric Research, Boulder, CO (United States)
- Los Alamos National Lab., NM (United States)
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.
- Research Organization:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE, Washington, DC (United States)
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 10130964
- Report Number(s):
- LA-12712-MS; ON: DE94007725
- Resource Relation:
- Other Information: PBD: Mar 1994
- Country of Publication:
- United States
- Language:
- English
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