Mathematical modeling is an interdisciplinary project requiring close collaboration between scientists inside and outside ACMD. ACMD researchers cooperate with the outside scientists to develop a mathematical model, specific to a problem area, that incorporates the essence of the problem and that will be computationally tractable. Then we use mathematical analysis on the model; propose and analyze numerical algorithms; and develop computer programs, partly custom programming and partly drawing on libraries of mathematical software. The resulting program is run to give simulations, which have to be compared with experimental results to validate the whole process.
The end benefits are to provide cheaper, quicker, and better information than experimentation alone. This information allows the outside scientists to gain understanding or to predict behavior of a complex system, and to improve the performance of the system under study.
Continuum models of physical systems generally require partial differential equations. Some well known areas include electromagnetism and fluid flow. Much of the work here is in materials science. This group of researchers forms a coherent, tightly knit community, and benefits from the synergy of daily interactions.
Some problems, though fundamentally continuum problems, are modeled as
discrete problems because the simpler model is adequate. Other problems are
fundamentally discrete. Usually these problems are nonlinear and can be
described by time-dependent ordinary differential equations.