Ross Hall, Room 202
1000 E. University Ave.
Laramie, WY 82071
Ph.D., Princeton University, 1983
M.A., Princeton University, 1978
A.B., Dartmouth College, 1976
Myron Allen's mathematical interests include numerical analysis, mathematical modeling, and fluid mechanics in porous media. Applications of these areas include the analysis and prediction of contaminant flows in groundwater aquifers and flows of native and injected fluids in oil and gas reservoirs.
Flows of this type obey a remarkable variety of partial differential equations, including equations of elliptic, parabolic, and hyperbolic type as well as nonlinear, coupled systems having mixed type. The equations generally require numerical solution. Numerical methods of interest include mixed finite elements, cell-centered finite-differences, Eulerian- Lagrangian methods such as the modified method of characteristics, streamline diffusion methods, and finite-volume methods.
All of these discrete approximation schemes yield large, sparse matrix equations. Solving them involves numerical linear algebra, including gradient-based methods, preconditioning, multigrid methods, and domain decomposition. Often accompanying the linear algebraic issues is the question of how best to exploit advanced and emerging computer architectures, such as massively parallel machines.
To model porous-media flows found in nature, one must address some physical issues that cause severe numerical difficulties. First, The equations are often nonlinear, so one must devise sensitive iterative schemes based, for example, on Newton's method. Second, the equations often have solutions with steep gradients, near which most numerical methods yield poor approximate solutions. Third, the equations sometimes include chemical or biological reactions, which involve complicated thermodynamic constraints or reaction-diffusion structures, such as traveling waves. Fourth, to accommodate natural geologic heterogeneity, one must account for differences in scale between the measurements used to determine model inputs and the computational grids used to solve the problems.