An Interactive Forum for Convection-Diffusion Problems

Bradley C. Beck - Graduate Research Assistant
António M. Baptista - Associate Professor
Center for Coastal and Land-Margin Research and Department of Environmental Science & Engineering
Oregon Graduate Institute of Science & Technology

1 Abstract

Work is currently underway at the Center for Coastal and Land-Margin Research (CCALMR) to develop an interactive forum to address issues related to the solution of convection-diffusion (CD) problems. This forum is an electronic extension of the forums held during 1986 and 1988 which solicited solutions to a predefined set of CD problems. When completed, the Electronic CD Forum will provide an interactive environment in which participants can study and compare current techniques used to numerically solve CD problems.

Currently, the Electronic CD Forum provides participants with the definitions of the 1-D problems from the original CD forums. It also provides them with animations of solutions to some of these problems. These solutions are obtained using a range of finite difference (FD) or finite element (FE) techniques. An interface is provided which allows the participants to experiment with the input parameters (e.g. velocity and diffusion) and the solution method.

An important but only partially implementd feature of the Electronic CD forum is be the ability of participants to submit and retrieve solutions. This involves maintaining a database of solutions submitted to the CD forum, and providing the ability to search this database for several parameters including author, problem type, and solution method. This feature will allow developers of new solution techniques to compare their results with those contained in the CD forum solution database.

2 The Original CD Forum

The Convection-Diffusion Forum (CD Forum) was conducted in conjunction with the 6th International Conference on Finite Elements in Water Resources held in 1986 in Lisboa, Portugal, and the 7th International Conference on Computational Methods in Water Resources held in 1988 in Cambridge, Massachusetts, USA [1]. Prior to each conference, prospective participants were sent a set of five benchmark problems, which included analytical solutions, error measures, and output formats. Participants were asked to submit solutions (plus detailed summaries of the discrepancies between their numerical and exact solutions) to the forum conveners before the conference.

Problems ranged in complexity from the transport of Gauss and triangle-hills or advancing fronts in one-dimensional uniform flow to the transport of cone-hills in a two-dimensional multi-cell deformational flow. Error measures differed somewhat for each problem, but included integral and discrete norms (measures of overall square error), peak concentration error (point measure of artificial damping), maximum negative concentration (point measure of artificial dispersion or spurious oscillation), 0th spatial moment (integral measure of mass conservation), 1st spatial moment (integral measure of phase error), and 2nd spatial moment (integral measure of artificial diffusion).

Approximately twenty sets of solutions were submitted, with some contributors choosing to solve as few as one problem and others solving all five. Techniques varied widely and included centered and upwind Eulerian methods using both finite difference and finite element formulations, several types of Eulerian-Lagrangian methods, spectral methods and particle tracking methods. The disciplines of the contributors also varied widely, representing civil engineering, oceanography, atmospheric modeling, and groundwater hydrology.

3 Vision for an Electronic CD Forum (ECDF)

Like many others, our first introduction to the Web was through Mosaic from the National Center for Supercomputing Applications (NCSA). With the introduction of NCSA's Web browser (Mosaic) and Web server (HTTPD), it became clear that it might be possible to conduct the CD Forum electronically. Initially, the ECDF was to be a distributor of information gathered during the original CD forums, but as we became more experienced at using Mosaic & HTTPD, our concept of the electronic forum expanded.

We now picture the ECDF behaving more like a live forum while also functioning as an archive for information pertaining to CD problems. We want the ECDF to be interactive, allowing the body of knowledge contained in it to evolve dynamically through contributions by the participants themselves.

Several components will be required to support our vision of the ECDF. These include general information about CD problems, a database of solutions to CD Forum problems, and a bibliographic database of literature related to CD problems. General information will include anything that helps participants to become more familiar with CD problems (e.g. problem definitions, example solutions, and an interactive model). The solution database will allow participants to submit their own solutions and compare solutions contained within the database. Much like the solution database, the bibliographic database will allow participants to add relevant references.

4 Current Status of the ECDF

Over the past few months, our vision of the ECDF has begun to take on physical form. We currently have a set of example solutions that are available through the ECDF. We have also implemented a model interface which allows participants to solve some of the 1-D CD Forum problems. Finally, we have an implementation of the solution database which is not yet available to the entire Web.

4.1 Example Solutions

One of the main functions of the ECDF is to help individuals become familiar with CD problems and to gain an appreciation of some the difficulties encountered when trying to solve these problems numerically. A way to achieve this is to provide participants with example solutions of selected CD problems. These solutions are obtained using various numerical methods (e.g. finite elements). Since CD problems are time varying by nature, it makes sense to present the solutions as animations, which provide a clearer representation of the evolution of the solution.

Our current implementation of the CD Forum allows a participant to choose a particular 1-D problem for which they would like to see a solution. This choice is made from an HyperText Markup Language (HTML) form ([2] is an example of one such tool for X11)

Mosaic can be configured to recognize files with MIME type video/x-flc by adding the line "video/x-flc; xanim %s" to a file called .mailcap in the participants home directory. This assumes that you have the XAnim software already installed. (For further information on multimedia configuration, see [3] and has a unique MIME type (application/x-xmgr-pipe). If you have ACE/gr installed and your browser is configured correctly, the solution should appear as an animation.

Since the model actually runs on the server itself, only one participant is allowed to be running the model at any given moment. The interface form provides instructions on how to configure a browser to correctly handle MIME type application/x-xmgr-pipe, and how to obtain a copy of ACE/gr, if you don't already have one.

4.3 Solution Database

A solution database plays a key role in helping the ECDF to achieve its full potential. Our current implementation allows a participant to submit their solution using an HTML form ([4] format and stores the file on local disk with a unique name.

For security reasons, the solution database is currently not available to the entire Web. As currently implemented the solution database will eventually require a large commitment of disk space. This will become even more of a problem when we start accepting 2-D solutions. A better long term solution would be to have participants submit URLs (Universal Resource Locators) to their solutions. One potential drawback is that this requires the participants to have a mechanism for making their solutions available to the Web (e.g. HTTPD, anonymous FTP, or Gopher). However, the benefits of this approach, in our opinion, far outweigh the potential drawbacks.

The ECDF also provides the participant with a listing of solutions contained in the database. Each solution in this list is distinguished by its meta-data. The participant can select a set of solutions from the list, by clicking on the checkboxes next to the appropriate items, and receive a comparison of the solutions selected.

5 Closing Remarks

The ECDF is currently in its humble infancy, but hopefully as it matures it will become a useful tool for both education and research. The ECDF can be reached via the Web at http://www.ccalmr.ogi.edu/CDF.

6 References

[1] Baptista A.M., E.E. Adams, and P. Gresho, "Benchmarks for the transport equation: the convection-diffusion forum and beyond", to Quantitative Skill Assessment for Coastal Ocean Models (Lynch and Davies, Eds.) , AGU Monograph Series, in press (1994).

[3] Turner P. J., "ACE/gr User's Manual: Graphics for exploratory data analsys", Software Documentation Series, SD93, 91-3, Center for Coastal and Land-Margin Research, Oregon Graduate Institute of Science & Technology, P.O. Box 91000, Portland, OR 97291-1000, USA. (ftp://ftp.ccalmr.ogi.edu/CCALMR/pub/acegr/)

[4] Rew R. K. and G.P. Davis, "The Unidata netCDF: Software for Scientific Data Access", 6th International Conference on Interactive Information and Processing Systems, Anaheim, CA, February, 1990.

Bradley C. Beck

Oregon Graduate Institute of Science & Technology
20000 N.W. Walker Rd., P.O. Box 91000, Portland, Oregon 97291-1000
Phone: (503) 690-1296 -- Fax: (503) 690-1273 -- Internet: beck@ccalmr.ogi.edu
RESEARCH INTERESTS: Numerical modeling: finite element methods, techniques for solving sparse systems of equations, and parallel computing.
SELECTED DEGREES: 1989 B.S. in Aerospace Engineering, Iowa State University. 1992 M.S. in Aerospace Engineering, Iowa State University, Thesis title: A dynamically adaptive grid scheme applied to a 1-D shock tube.
CURRENT POSITION: Graduate Research Assistant, Center for Coastal and Land-Margin Research.

António M. Baptista

Oregon Graduate Institute of Science & Technology
20000 N.W. Walker Rd., P.O. Box 91000, Portland, Oregon 97291-1000
Phone: (503) 690-1147 -- Fax: (503) 690-1273 -- Internet: baptista@ccalmr.ogi.edu
RESEARCH INTERESTS: Coasts and land-margins: integrated, interdisciplinary analysis and modeling of coastal hazards and environmental pathways at regional scales; tides and tsunamis; hydraulic transport; estuarine geochemical modeling; land-water interactions. Computational techniques: numerical methods for flow and transport; automatic grid generation; scientific visualization.
SELECTED DEGREES: 1987 Doctor of Philosophy in Civil Engineering, Massachusetts Institute of Technology Thesis title: Solution of Advection-dominated Transport by Eulerian-Lagrangian Methods Based on the Backwards Method of Characteristics.
CURRENT POSITION: Associate Professor, Department of Environmental Science and Engineering - since 6/93. Director, Center for Coastal and Land-Margin Research - since 7/91.

Bradley C. Beck <beck@ccalmr.ogi.edu>

António M. Baptista <beck@ccalmr.ogi.edu>