# Computational Methods in Transport

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## Book Preface

There exist a wide range of applications where a significant fraction of the momentum and energy present in a physical problem is carried by the transport of particles. Depending on the specific application, the particles involved may be photons, neutrons, neutrinos, or charged particles. Regardless of which phenomena is being described, at the heart of each application is the fact that a Boltzmann like transport equation has to be solved.

The complexity, and hence expense, involved in solving the transport problem can be understood by realizing that the general solution to the 3D Boltzmann transport equation is in fact really seven dimensional: 3 spatial coordinates, 2 angles, 1 time, and 1 for speed or energy. Low-order approximations to the transport equation are frequently used due in part to physical justification but many in cases, simply because a solution to the full transport problem is too computationally expensive. An example is the diffusion equation, which effectively drops the two angles in phase space by assuming that a linear representation in angle is adequate. Another approximation is the grey approximation, which drops the energy variable by averaging over it. If the grey approximation is applied to the diffusion equation, the expense of solving what amounts to the simplest possible description of transport is roughly equal to the cost of implicit computational fluid dynamics. It is clear therefore, that for those application areas needing some form of transport, fast, accurate and robust transport algorithms can lead to an increase in overall code performance and a decrease in time to solution.

Besides the multi-dimensional nature of the transport equation, because of the coupling of particle transport to other phenomena the transport equation can in fact be non-linear. Hence, except for a few simple benchmark answers, the transport problem is solvable only via numerical methods. These numerical methods have developed and grown over the years and with the advent of massively parallel architectures, new scalable methods are being sought. Unfortunately, it is still true that in most computer codes, transport is the largest consumer of computational resources. In application areas that use transport, the computational time is usually dominated by the transport calculation. Therefore, there is a potential for great synergy; progress in transport algorithms could help quicken the time to solution for many applications.