FOSLS makes no bones about improving computer models

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PDEs are the core of many computer simulations.  But computers can’t solve PDEs until the equations are converted to systems of algebraic equations.  In essence, mathematicians must convert models of continuous processes – such as waves – into equations describing a series of discrete points a digital computer can calculate.

PDEs involve derivatives – mathematical expressions describing the rate of change of a quantity.  “If I know the distance an object travels over time, its velocity is the first derivative,” Manteuffel says.  “Acceleration is the second derivative.”

“We try to write almost all these physical models as a system of equations only involving first derivatives,” he adds – thus, a first-order system.

Many applications start that way, but mathematicians looking for a simple way to write complex equations often used higher-order derivatives – second, third and beyond, Manteuffel says.  “Sometimes equations got all tied up in knots,” he adds.

“We’re unraveling them back.  We’re saying we’re going to write them back as first-order because we’re not interested in making a pretty equation on the wall.  We’re interested in solving something on the computer.”

Reducing PDEs to first-order systems makes them easier to solve using a fast solution technique called multigrid methods.

“What we’ve been doing is recasting partial differential equations in a way that the … systems that result are amenable to this very fast solution technique” on massively parallel high-performance computers, Manteuffel says.

“We’re changing things over here so when they come out the bottom you have a solution technique that will solve them rapidly.”

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