UNIFORMLY HIGH-ORDER PDE SOLVERS BY RECOVERING THE LOST BOUNDARY PHYSICS

In numerically solving a physical problem governed by partial differential equations with proper boundary condition, central-kind difference schemes can yield high-order accuracy in the interior of the computational domain. But near the boundary the stencil structure is limited to only upwind-kind schemes. This has led to a reduction of the accuracy that in turn pollutes the entire interior solution as a long-standing critical obstacle in developing high-order accuracy numerical PDE solvers. The root of this difficulty lies in the ignorance of the use of (discrete) PDE at boundary, which is essentially an ignorance of the key role of the boundary physics in the determination of the solution. In this paper we recover the boundary physics by applying the original PDE all the way to the boundary along with the original boundary condition, which can yield uniformly high-order discretisation.