•   PIC codes. Required to study the dynamics of weakly collisional, dilute plasmas. The mathematical model consists of the relativistic equations of motion for particles, plus Maxwell’s equations for the electric and magnetic fields they induce (a set of coupled first-order PDEs). Standard techniques are based on particle-in-cell (PIC) algorithms, in which Maxwell’s equations are solved on a grid using finite-difference methods and the particle motion is calculated by standard ODE integrators.

•   Fluid dynamics. Required for strongly collisional plasmas. The mathematical model comprises the standard equations of compressible fluid dynamics (the Euler equations, a set of hyperbolic PDEs), supplemented by Poisson’s equation for self-gravity (an elliptic PDE), Maxwell’s equation for magnetic fields (an additional set of hyperbolic PDEs), and the radiative transfer equation for photon or neutrino transport (a high-dimensional parabolic PDE). A wide variety of algorithms for fluid dynamics are used, including finite-difference, finite-volume, and operator-splitting methods on orthogonal grids, as well as particle methods that are unique to astrophysics—for example, SPH. To improve resolution across a broad range of length scales, grid-based methods often rely on static and adaptive mesh refinement (AMR). The AMR methods greatly increase the complexity of the algorithm, reduce the scalability, and complicate effective load-balancing yet are absolutely essential for some problems.

•   Transport problems. Required to calculate the effect of transport of energy and momentum by photons or neutrinos in a plasma. The mathematical model is a parabolic PDE in seven dimensions. Both grid-based (characteristic) and particle-based (Monte Carlo) methods are used. The high dimensionality of the problem makes first-principles calculations difficult, and so simplifying assumptions (for example, frequency-independent transport, or the diffusion approximation) are usually required.

•   Microphysics. Necessary to incorporate nuclear reactions, chemistry, and ionization/recombination reactions into fluid and plasma simulations. The mathematical model is a set of coupled nonlinear, stiff ODEs (or algebraic equations if steady-state abundances are assumed) representing the reaction network. Implicit methods are generally required if the ODEs are solved. Implicit finite-difference methods for integrating realistic networks with dozens of constituent species are extremely costly.4

In its look at atmospheric sciences, that same report identified the following necessary computational advances:

[Advancing the state of atmospheric science research requires] the development of (1) scalable implementations of uniform-grid methods aimed at


4 NRC, 2008. The Potential Impact of High-End Capability Computing on Four Illustrative Fields of Sciences and Engineering. The National Academies Press, Washington, D.C., p. 31.

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