The Role of Mathematics, Computations and Simulations in Nuclear and Environmental Applications

The role of computer-assisted modeling, analysis and simulation of physical processes has increased tremendously in the last decade. This increase is driven by, and is in turn driving, increases in computing speed, capacity, and versatility in both hardware and software. Since computers operate on mathematical models of physical reality, the computed results must be compared to experimental measurements whenever possible. Discrepancies between measurements and calculations arise, though, because of uncertainties both in experimental measurements and in mathematical models. A mathematical model comprises independent variables, dependent variables, and a system of relationships (e.g., equations, look-up tables, etc.) that relate the dependent to the independent variables. Even if a physical process is faithfully modeled mathematically, the numerical methods used to solve the equations underlying the mathematical model introduce themselves numerical errors. The situation is even more complicated if the numerical calculations are affected by chaotic attractors. Furthermore, mathematical models also include parameters (such as material properties, constitutive relations) whose actual values are not known precisely, but may vary within some ranges of uncertainty. The effects of such parameter variations must be quantified in order to validate the respective model; moreover, the effects of uncertainties in the model's parameters on the uncertainty in the calculated results must also be quantified. It is thus obvious that computations and simulations of physical systems must go hand in hand with experiments; neither computations nor experiments could by themselves provide the full understanding of a physical phenomenon. This symbiotic role played by computations/simulations and experiments is particularly evident in nuclear and environmental applications. Computations and numerical simulations, together with their experimental counterparts, constitute the indispensable yin and yang for nuclear and environmental applications, just as for any other large-scale, complex human undertakings. Ideally, numerical simulations should reach the stage where they can replace full-scale experiments, once the underlying physical, chemical, or biological mechanisms are understood. Once reached, such a stage would provide a platform for further explorations of the remaining scientific terra incognita, thus perpetuating progress