A novel strategy for deriving high-order stable boundary closures based on global conservation, I: Basic formulas

It is well known that for initial boundary value problems one may need to decrease the order of boundary closures to ensure strict stability (also called time stability) of finite difference schemes. For high-order schemes, this often reduces the global convergence rate significantly. To cure this issue, we derive high-order accurate boundary closures that satisfy a global conservation property. We show that the boundary closures cannot be both high-order accurate and conservative in the l2-inner product, if the grid is equidistant. Therefore, we derive particular sets of nonuniform solution points and quadrature weights near the domain boundaries. The new schemes are high-order accurate on the modified grids and conservative in the derived quadrature rules. Although we have no theoretical stability proof, numerical eigenvalue analysis suggests that the new schemes are strictly stable up to eleventh-order global accuracy for scalar linear advection equations. Numerical experiments with one- and two-dimensional linear equations corroborate the design orders of the schemes.