Lawrence Livermore National Laboratory

Modeling an Earth-Shaking Event

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To develop their simulations, each of the participating groups began with a USGS-created geological model of the Greater San Francisco Bay area.  The model characterized rock and soil properties and was developed from years of study of seismic data, drilling, and tomography up to a depth of 50 kilometers.  The data was crucial to creating a computerized picture of the earthquake in progress because different types of earth have dramatically different responses to the spreading shockwaves.

To create its model, the SWP team used two LLNL-based supercomputers:

• MCR incorporates 2,300 processors and has a peak performance of about seven teraflops (trillion floating point operations per second).
• Thunder has 4,096 processors and a peak performance of about 21 teraflops.

Doing the Math

One reason creating an earthquake model is difficult is that “a continuous elastic body, such as the earth, has an infinite number of degrees of freedom corresponding to motion at each point in space,” Petersson, the SWP team leader, says.  “So before the motion of a system with an infinite number of degrees of freedom can be calculated in a computer, we need to reduce the number of degrees of freedom to a finite, large number through a process called discretization.”

The group used a finite difference discretization method.  It broke the area being modeled, known as the computational domain, into a series of equally-spaced grid points.

“In our calculations we used a grid spacing of 125 meters, which corresponds to something like 2.3 billion grid points,” Petersson says.  “At each of these grid points you have three degrees of freedom, so you get about 6.8 billion degrees of freedom. You reduce your infinite number of the degrees of freedom to 6.8 billion,” which “is still fairly large.”

The earthquake’s motion also is integrated in time, which also must be discretized into a manageable number.  The group took equal steps in time to simulate the first 300 seconds of earthquake motion. “Each of these steps is 0.01 seconds, so you take about 30,000 time steps in the calculation,” Petersson says.

In discretization, “All the derivatives in the partial differential equations are replaced by what are called divided differences, and that converts the original mathematical equation to a set of algebraic equations, and those can be solved in the computer,” Petersson says.

Without discretization, a computer cannot solve the original equations because they are too complicated — even for a supercomputer.  “The computer can only deal with simple operations like addition, subtraction, multiplication and division.  The partial differential equation involves very complicated relations between how the solution varies in space and time,” says Petersson.

The Freedom is in the Details

But there is another factor to complicate the discretization process: stability.  “If the method is unstable then perturbations due to round-off errors in the computer will accumulate during simulations, and they can make the computed result completely useless,” Petersson says.

That’s where the mathematician’s expertise comes in.  The key is to develop a mathematical theory that guarantees the stability of the process “so you know before you start your calculation that it is not going to go unstable, or ‘blow up,’ as we also say,” Petersson adds.  To do so, “Mathematically, you study how perturbations propagate through such a calculation without actually computing the solution.  You can then estimate how large these perturbations can become in a calculation.”

This is accomplished the old-fashioned way.  “With pen and paper you can analyze the properties of your numerical method.  And there are various ways you can modify your finite difference method so there’s not just one prescription, there’s a lot of freedom in how you do all the details.  So you’ve got to figure out the way to deal with all the details such that you can guarantee that it is stable.  And that,” Petersson says, “is the challenge.”

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