General tooth boundary conditions for equation free modeling

We are developing a framework for multiscale computation which enables models at a "microscopic" level of description, for example, lattice Boltzmann, Monte Carlo, or molecular dynamics simulators, to perform modeling tasks at "macroscopic" length scales of interest. The plan is to use the microscopic rules restricted to small "patches" of the domain, the "teeth," using interpolation to bridge the "gaps." Here we explore general boundary conditions coupling the widely separated "teeth" of the microscopic simulation that achieve high order accuracy over the macroscale. We present the simplest case when the microscopic simulator is the quintessential example of a PDE. We argue that classic high order interpolation of the macroscopic field provides the correct forcing in whatever boundary condition is required by the microsimulator. Such interpolation leads to tooth boundary conditions, which achieve arbitrarily high order consistency. The high order consistency is demonstrated on a class of linear PDEs in two ways: first through the eigenvalues of the scheme for selected numerical problems, and second using the dynamical systems approach of holistic discretization on a general class of linear PDEs. Analytic modeling shows that, for a wide class of microscopic systems, the subgrid fields and the effective macroscopic model are largely independent of the tooth size and the particular tooth boundary conditions. When applied to patches of microscopic simulations these tooth boundary conditions promise efficient macroscale simulation. We expect the same approach will also accurately couple patch simulations in higher spatial dimensions.