Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

TL;DR

This paper introduces the LCBC method, a novel approach for deriving boundary conditions in high-order finite-difference schemes, improving accuracy and stability by using local polynomials based on governing equations.

Contribution

The paper presents the LCBC method, enabling automatic, local boundary condition approximations for high-order finite-difference schemes, enhancing accuracy and stability over traditional methods.

Findings

LCBC provides more accurate boundary approximations.

The method is stable for high-order schemes up to sixth order.

Applicable to various PDEs in two dimensions.

Abstract

We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.