Physics-motivated numerical solvers for partial differential equations

Trying to capture the essential physics of a natural phenomenon directly on computers may lead us to useful numerical schemes to solve the partial 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. Efficient modeling of these phenomena becomes possible with the aid of Hermite polynomial interpolations to realize a continuum on discrete lattices. 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 standard high-order algorithms of the finite-difference or finite-element families. The purpose of the paper is to demonstrate our methodology. Therefore, in two- and three-spaces, details are given only for the lowest-order algorithms, a preview of a more optimal higher-order scheme is also included.