In this work, we analyze the Ghost-Fluid method (GFM) with cubic extrapolation for a two-point boundary value problem, where the boundary points are not on the uniform mesh. This scheme employs the standard central difference to approximate the second derivative and the solution on the ghost point is extrapolated with four points (one boundary point and three nearest mesh points). We prove the local truncation error is second order convergent, the solution is second order accurate in the whole domain and third order accurate near the boundary, and the derivative is second order accurate. We also identify a range of the eigenvalues of the resulting matrix. Through comprehensive comparisons, we find the GFM with cubic extrapolation is comparable to the well-known Shortley-Weller scheme.

Ghost-Fluid method, cubic extrapolation, two-point boundary value problem, non-fitting mesh, Collatz scheme, Shortley-Weller scheme, superconvergence, discrete maximum principle, inverse-positive matrix.