Collaborative Research: Bridging short and long time scales in global plate tectonics through solver adaptivity

The underlying geophysical connections linking great earthquakes with plate tectonics will be addressed using sophisticated models. These seismic events are the largest earthquakes with magnitudes of nine or higher and are among largest sources of natural hazard affecting many countries, including the United States. Scientists have long known that these earthquakes are linked to plate tectonics and occur where an oceanic plate dives into the earth’s interior at a subduction zone, like the one along the coasts of Washington State, Oregon and northern California.

Unfortunately, fundamental issues remain as to the reasons where and when they occur. This interdisciplinary team will link the long-term physics driving and resisting plate tectonics with that of great earthquakes. The outcome of the models will be a deeper understanding of earthquake occurrence.

Currently, the ability to solve the set of equations describing the physics of this coupled problem is beyond the ability of mathematical methods. Consequently, this project brings together mathematical scientists with earth scientists to attempt to solve this problem collaboratively. Mathematically, problems like this are solved on the largest supercomputers and are described by many equations.

For the plate tectonics problem by itself, the equations change only a small amount from one moment of time to the next in the computer model, but in this coupled problem only a set of the equations change when an earthquake happens and so the mathematicians will discover and implement new ways to solve such large sets of equations. Meanwhile, the earth scientists will use the new methods to understand the basic physics of the coupled earthquake and plate tectonics problem, ultimately tailoring the methods to models of individual plates and faults, such as the Juan de Fuca Plate which subducts below the Pacific northwest.

The algorithms are expected to efficiently use the largest supercomputers now in the planning stage, including the NSF-planned LCCF (Leadership-Class Computing Facility). Moreover, the computer software, called Rhea, will be distributed open-source and will be well-engineered and documented. The team will collaborate with the Computational Infrastructure for Geodynamics, supported by the NSF, for the distribution of Rhea to the broader scientific community.

The PIs will train graduate students at Caltech, Virginia Tech, and NYU at the boundary between the mathematical sciences and science applications. The team will participate in outreach programs: In California through a program that brings geophysical science to local Title I schools; in Virginia, through one that provides outreach projects for local high schools; and in New York City, through a program increases diversity in graduate programs by exposing undergraduates to mathematical research.

The forces controlling plate tectonics and the conditions leading to great earthquakes are currently treated as separate, fundamental problems, but in this project, they will be linked with a focused effort to develop and apply a new generation of finite element methods with solver adaptivity that will scale on the largest computers. The activity will involve major advances in mathematical and computational algorithms for multi-physics problems, the team will bridge the space–time divide and self-consistently compute the long-term motions of tectonic plates and the intervening space–time evolution of stress within and adjacent to plate boundaries.

This undertaking is beyond currently available methods and mathematically requires new concepts to allow tracking the shifting—but localized—regions of enormous computational need during earthquakes. The team will expand the notion of space and time discretization adaptivity towards solver adaptivity. Solver adaptivity will use equation residuals to focus computing resources towards the most efficient solution of large linear and nonlinear systems of equations.

Since the system arising by discretizing the equations in the earthquake–plate tectonic problem typically has tens and hundreds of millions of unknowns, solvers based on matrix factorization are out of question and one must rely on iterative solvers that also allow parallelization. The algorithms are expected to scale on the largest anticipated supercomputers with distributed memory and computational elements, such as the NSF-planned LCCF (Leadership-Class Computing Facility).

As such, the scalable algorithms will fill an important need and demonstrate the efficient use of future LCCF machines. The methods will be incorporated into the highly scalable Stokes solver, finite element package Rhea. Visco-elasticity and frictional material models will be implemented into Rhea.

The science and mathematical challenges will be addressed with an interdisciplinary team consisting of a geophysicist who works on the dynamics of plate tectonics, a mechanician who works on the physics of earthquakes, and applied mathematicians who work on linear and nonlinear scalable PDE solvers. The team will apply the methods to understand the coupled physics generically, first in two dimensions and then in three dimensions.

Then, using models regionally tailored by the explicit incorporation of seismic, geologic and fault structure, they will simulate Cascadia and the northwestern Pacific subduction systems.

This project is jointly supported by the Computational and Data-Enabled Science and Engineering in mathematical and Statistical Sciences program in the Division of Mathematical Sciences and the Geophysics program in the Division of Earth Sciences.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.