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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 modelling tasks at the 'macroscopic' length scales of interest.  The plan is to use the microscopic rules restricted to small patches of the domain, the 'teeth', followed by interpolation to estimate macroscopic fields in the 'gaps'. The challenge begun here is to find general boundary conditions for the patches of microscopic simulators that appropriately connect the widely separated 'teeth' to achieve high order accuracy over the macroscale.  Here we start exploring the issues in the simplest case when the microscopic simulator is the quintessential example of a partial differential equation. In this case analytic solutions provide comparisons.  We argue that classic high-order interpolation provides patch boundary conditions which achieve arbitrarily high-order consistency in the gap-tooth scheme, and with care are numerically stable.  The high-order consistency is demonstrated on a class of linear partial differential equations in two ways: firstly, using the dynamical systems approach of holistic discretisation; and secondly, through the eigenvalues of selected numerical problems.  When applied to patches of microscopic simulations these patch boundary conditions should achieve efficient macroscale simulation

Higher order accuracy in the gap-tooth scheme for large-scale dynamics using microscopic simulators