Later, that same report identified the following additional challenges that can only be addressed through related advances in the mathematical sciences:
In the coming decade, the deployment of clusters of satellites and large arrays of ground-based instruments will provide a wealth of data over a very broad range of spatial scales. Theory and computational models will play a central role, hand in hand with data analysis, in integrating these data into first-principles models of plasma behavior. . . . The Coupling Complexity research initiative will address multiprocess coupling, nonlinearity, and multiscale and multiregional feedback in space physics. The program advocates both the development of coupled global models and the synergistic investigation of well-chosen, distinct theoretical problems. For major advances to be made in understanding coupling complexity in space physics, sophisticated computational tools, fundamental theoretical analysis, and state-of-the-art data analysis must all be integrated under a single umbrella program.2
The coming decade will see the availability of large space physics databases that will have to be integrated into physics-based numerical models. . . . The solar and space physics community has not until recently had to address the issue of data assimilation as seriously as have the meteorologists. However, this situation is changing rapidly, particularly in the ionospheric arena.3
Another example comes from the 2008 NRC report The Potential Impact of High-End Capability Computing on Four Illustrative Fields of Science and Engineering. This report identified the major research challenges in four disparate fields and the subset that depend on advances in computing. Those advances in computing are innately tied to research in the mathematical sciences. In the case of astrophysics, the report identified the following essential needs:
• N-body codes. Required to investigate the dynamics of collisionless dark matter, or to study stellar or planetary dynamics. The mathematical model is a set of first-order ODEs for each particle, with acceleration computed from the gravitational interaction of each particle with all the others. Integrating particle orbits requires standard methods for ODEs, with variable time stepping for close encounters. For the gravitational acceleration (the major computational challenge), direct summation, tree algorithms, and grid-based methods are all used to compute the gravitational potential from Poisson’s equations.
2 Ibid., pp. 64-66.
3 Ibid., pp. 89-90.