Towards High-Order Boundary Procedures for Finite-Volume and Finite-Difference Schemes

In this paper, we present high-order boundary procedures for finite-volume and finite-difference schemes. The proposed procedures are deliberately constructed with pre-computed derivatives such that they can be implemented without directly accessing cells beyond neighbors and without explicitly setting up ghost cells. Therefore, they can be implemented in a practical unstructured-grid computational-fluid-dynamics solver relatively easily and are expected to serve also as an effective technique to preserve high-order accuracy for structured-grid schemes near physical as well as partition boundaries in a parallel-computing environment. The developed schemes are expected to be useful especially for practical applications allowing smoothly-varying quadrilateral/hexahedral grids. In this paper, we will focus on the basics of the proposed procedures for third-order accuracy, deriving and demonstrating the proposed boundary procedures for nonlinear conservation laws in one dimension.