High-order methods and numerical boundary conditions

In this paper we present high-order difference schemes for convection diffusion problems. When we apply high-order numerical methods to problems where physical boundary conditions are not periodic there is a need to choose adequate numerical boundary conditions in order to preserve the high-order accuracy. Next to the boundary we do not usually have enough discrete points to apply the high-order scheme and therefore at these nodes we must consider different approximations, named the numerical boundary conditions.The choice of numerical boundary conditions can influence the overall accuracy of the scheme and most of the times do influence the stability. Here, we discuss which orders of accuracy are reasonable to be considered at the numerical boundary conditions, such that we do not pay a high price in accuracy and stability.