Space-time finite-element objects: Efficiently modeling physically complex flows

Accurate modeling of high-explosive systems requires detailed consideration of many different physical properties and processes: These diverse processes generally occur in localized regions of the problem. Thus the very partial differential equations used to mathematically model the problem change from one region of space and time to another. The numerical algorithms generally used to solve these equations are frequently conceived in terms of data values for physical field variables u{sup i} defined at a number of spatial points indexed by multi-integer subscripts x{sub J}, resulting in a number of discrete state variables u{sup i}{sub J}. Instead of using as the fundamental object a physical field, which naturally maps to an array, the authors imagine a small piece of space modeled for a small amount of time, a space-time ``element``. Within it, various physical processes occur at various times. Self-contained, it gives account of what happens within its borders. It cooperates with a set of neighbors that organize into meshes, which organize into problems. The authors achieve in the software model a decoupling between the where and the how and the what, lack of which historically has been the source of a great deal of the software overhead of modelling continuum systems, and which is a necessary consequence of writing down u{sup i}{sub J}. An efficient implementation of this idea requires a reformulation of the discretization and solution of systems of conservation laws, and careful class design. A working prototype for systems in one space dimension using Mathematica and C++ is provided.