Breaking the Mold in Computer Modeling

To numerically model unsteady phenomena in space and time, scientists continually wrestle with tradeoffs between accuracy, speed, and economics.

In general, to improve the accuracy and speed of simulation, a numerical algorithm must increase in complexity, memory and CPU usage, and associated cost.

SAIC's Hong Luo has defied this conventional wisdom by creating a new computer algorithm that is fast, accurate, and economical. Luo's "matrix-free implicit method" conserves memory and CPU time — a welcome deviation from traditional CPU- and memory-intensive methods for simulating fluid dynamics.

Luo summarizes, "With the speed of the new matrix-free implicit algorithm, we can solve problems that before now were impractical to solve. And because the method uses only a fraction of the memory required by previous methods, the matrix-free approach is even cost effective for solving flow problems around complex, three dimensional configurations." While Luo tested the algorithm on several unsteady fluid dynamics problems, the method could be applied to virtually any engineering application that is governed by a set of partial differential equations.

The predecessor of Luo's matrix-free implicit method is the "fully implicit method." While the fully implicit method executes rapidly, it requires a vast amount of memory to store a Jacobian matrix at each time step, making it unrealistic to solve — even with modern supercomputers. An alternative algorithm, the explicit method, is more realistic in its memory requirements but is too slow to be useful in solving complex problems.

Together with Joseph Baum and Rainald Löhner, Luo has developed the matrix-free implicit approach, which uses only a small fraction (3%) of the original matrix of data from the implicit method. The matrix-free method boasts just one-fifth the required memory of the fully implicit method and one hundred times the speed of the explicit method, making it far more practical than previous algorithms.

And what about accuracy? In the team's prior research, the matrix-free method successfully approximated steady-state problems (at one point in time) but could not approximate unsteady, time-dependent problems (over a given time period). In their current research, the scientists have extended the accuracy of the matrix-free method to time-dependent problems by introducing a correction mechanism called the "pseudo time variable." The new pseudo time-marching scheme increases the matrix-free implicit method's accuracy to that of the fully implicit and explicit methods.

Luo's award-winning paper, "An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids," appeared in Computers & Fluids. The research was sponsored by the Defense Special Weapons Agency.

In its annual competition, SAIC's Executive Science & Technology Council recognizes some of the most innovative research and best written technical papers and books by SAIC scientists and engineers. This article is a summary from one of the award winners.