Interim:
Problem Definition:

For our project, we will create an Eulerian 3-dimensional model of a fluid. This type of model can be used in many different areas. It can model many fluid problems such as shockwaves. We will be creating a tool that will run on a small number of processors and can model reasonable size problems. We will investigate how certain variables such as mesh size affect the accuracy and runtime of our model.
We will do 1 to 3 different problems using the final version of the code.

Problem Solution:

Numerical methods for modeling fluid dynamics are based on the Ideal Gas Law.
The mathematical equations are:

Conservation of mass:
rhot+ (rho*v)x=0

Conservation of momentum:
(rho*v)t + (rho*v2 + rho)x = 0

Conservation of energy:
E=0.5*rho*v2 + rho*e

We will be using the two-step (predictor-corrector) Lax-Wendroff equations which use the data at half of the time step and half of the spatial step to extrapolate the data at the complete time step. The simulation is based on the fundamental principles of physics.

Progress to date:

Currently, a program running a one-dimensional version of the simulation has been built. There are some problems due to the Lax-Wendroff wiggle, a tendency to fluctuate before and after a shock. These problems were to be expected and can be fixed with the addition of a TVD scheme. With the inclusion of a TVD scheme, Lax-Wendroff provides accurate results without the problem of trying to find which way the data is coming from.

Depending on time restraints other methods may or may not be able to be included in the final comparison. In lieu of the comparisons between methods we will compare the change in the time needed for the same simulation to the number of processors used.

Because of the high speed of the computers currently in use the graphics are displayed so quickly that in order to view it taking place at a reasonable speed one must slow it down. this line of code would not be included in the code used to compare times.

Expected results:

We expect to have at least a working two dimensional version of a stable Lax-Wendroff simulation with TVD.

References:

S. F. Davis, “A Simplified TVD Finite Difference Scheme via Artificial Viscosity”, SIAM J. Sci. Stat. Computing, Vol. 8, No. 1, January 1987, pp. 1-18

H. C. Yee, “Construction of Explicit and Implicit Symmetric TVD Schemes and Their Applications, Journal of Computational Physics”, Vol. 68, 1987, pp 151-179.

Gary A. Sod, “A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws”, Journal of Computational Physics, Vol 27, 1978, pp 1-30.

Evan Scannapieco and Francis H Harlow, Introduction to Finite-Difference Methods for Numerical Fluid Dynamics, LANL, 1995

Randall J LeVeque, Numerical Methods for Conservation Laws, Birkhauser, 1992

Team Members:

Jonathan Robey
Dov Shlachter

Sponsoring Teacher: Diane Medford

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