L. Sanmartin et Y. Oono, PHYSICS-MOTIVATED NUMERICAL SOLVERS FOR PARTIAL-DIFFERENTIAL EQUATIONS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 57(4), 1998, pp. 4795-4810
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.