A fast solver for systems of reaction-diffusion equations.

PDF Version Also Available for Download.

Description

In this paper we present a fast algorithm for the numerical solution of systems of reaction-diffusion equations, {partial_derivative}{sub t} u + a {center_dot} {del}u = {Delta}u + f(x,t,u), and x element of {Omega} contained in R{sup 3}, t > 0. Here, u is a vector-valued function, u triple bond u(x,t) element of R{sup m} is large, and the corresponding system of ODEs, {partial_derivative}{sub t}u = F(x,t,u), is stiff. Typical examples arise in air pollution studies, where a is the given wind field and the nonlinear function F models the atmospheric chemistry. The time integration of Eq. (1) is best handled ... continued below

Physical Description

8 pages

Creation Information

Garbey, M.; Kaper, H. G. & Romanyukha, N. April 20, 2001.

Context

This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided by UNT Libraries Government Documents Department to Digital Library, a digital repository hosted by the UNT Libraries. More information about this article can be viewed below.

Who

People and organizations associated with either the creation of this article or its content.

Sponsor

Publisher

Provided By

UNT Libraries Government Documents Department

Serving as both a federal and a state depository library, the UNT Libraries Government Documents Department maintains millions of items in a variety of formats. The department is a member of the FDLP Content Partnerships Program and an Affiliated Archive of the National Archives.

Contact Us

What

Descriptive information to help identify this article. Follow the links below to find similar items on the Digital Library.

Description

In this paper we present a fast algorithm for the numerical solution of systems of reaction-diffusion equations, {partial_derivative}{sub t} u + a {center_dot} {del}u = {Delta}u + f(x,t,u), and x element of {Omega} contained in R{sup 3}, t > 0. Here, u is a vector-valued function, u triple bond u(x,t) element of R{sup m} is large, and the corresponding system of ODEs, {partial_derivative}{sub t}u = F(x,t,u), is stiff. Typical examples arise in air pollution studies, where a is the given wind field and the nonlinear function F models the atmospheric chemistry. The time integration of Eq. (1) is best handled by the method of characteristics. The problem is thus reduced to designing for the reaction-diffusion part a fast solver that has good stability properties for the given time step and does not require the computation of the full Jacobi matrix. An operator-splitting technique, even a high-order one, combining a fast nonlinear ODE solver with an efficient solver for the diffusion operator is less effective when the reaction term is stiff. In fact, the classical Strang splitting method may underperform a first-order source splitting method. The algorithm we propose in this paper uses an a posteriori filtering technique to stabilize the computation of the diffusion term. The algorithm parallelizes well, because the solution of the large system of ODEs is done pointwise; however, the integration of the chemistry may lead to load-balancing problems. The Tchebycheff acceleration technique proposed in offers an alternative that complements the approach presented here. To facilitate the presentation, we limit the discussion to domains {Omega} that either admit a regular discretization grid or decompose into subdomains that admit regular discretization grids. We describe the algorithm for one-dimensional domains in Section 2 and for multidimensional domains in Section 3. Section 4 briefly outlines future work.

Physical Description

8 pages

Source

  • 13th International Conference on Domain Decomposition Methods, Lyon (FR), 10/09/2000--10/12/2000

Language

Item Type

Identifier

Unique identifying numbers for this article in the Digital Library or other systems.

  • Report No.: ANL/MCS/CP-104638
  • Grant Number: W-31-109-ENG-38
  • Office of Scientific & Technical Information Report Number: 786924
  • Archival Resource Key: ark:/67531/metadc721040

Collections

This article is part of the following collection of related materials.

Office of Scientific & Technical Information Technical Reports

Reports, articles and other documents harvested from the Office of Scientific and Technical Information.

Office of Scientific and Technical Information (OSTI) is the Department of Energy (DOE) office that collects, preserves, and disseminates DOE-sponsored research and development (R&D) results that are the outcomes of R&D projects or other funded activities at DOE labs and facilities nationwide and grantees at universities and other institutions.

What responsibilities do I have when using this article?

When

Dates and time periods associated with this article.

Creation Date

  • April 20, 2001

Added to The UNT Digital Library

  • Sept. 29, 2015, 5:31 a.m.

Description Last Updated

  • March 24, 2016, 9:24 p.m.

Usage Statistics

When was this article last used?

Yesterday: 0
Past 30 days: 0
Total Uses: 6

Interact With This Article

Here are some suggestions for what to do next.

Start Reading

PDF Version Also Available for Download.

Citations, Rights, Re-Use

Garbey, M.; Kaper, H. G. & Romanyukha, N. A fast solver for systems of reaction-diffusion equations., article, April 20, 2001; Illinois. (digital.library.unt.edu/ark:/67531/metadc721040/: accessed May 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.