PHYSICS-MOTIVATED NUMERICAL SOLVERS FOR PARTIAL-DIFFERENTIAL EQUATIONS

Trying to capture the essential physics of a natural phenomenon direct ly on computers may lead us to useful numerical schemes to solve the p artial differential equation describing the phenomenon. Here we try to capture the consequences of space-time translational symmetry such as advection in fluids or Huygens' principle in wave propagation. Effici ent modeling of these phenomena becomes possible with the aid of Hermi te polynomial interpolations to realize a continuum on discrete lattic es. To illustrate these ideas, we present a new method to derive wave equation solvers that are high order but local (the computational cell or stencil includes nearest neighbors only), a clear advantage over s tandard high-order algorithms of the finite-difference or finite-eleme nt families. The purpose of the paper is to demonstrate our methodolog y. Therefore, in two-and three-spaces, details are given only for the lowest-order algorithms, a preview of a more optimal higher-order sche me is also included.