Ghost nodes (theory)

Figure 1: Ghost node configuration at the domain boundary.

Ghost nodes are a technique for discretizing (mostly) Neumann boundary conditions in PDEs.

Introduction

Ghost are a technique used for disretizing Neumann boundary conditions in FDM. To be able to use the central difference for first derivative, additional point, called ghost point, is introduced outside the domain boudanry. The unknown function value at the ghost node is added as a variable. At the boundary node, the Neumann condition is enforced, as well as the equation itself (two equations for two unknowns, the ghost and the boundary function value).

Meshless setting (implicit)

Consider the problem $\mathcal{L}u = f$ with Neumann boundary conditions $\frac{\partial u}{\partial n} = g$ on some portion of the boundary.

Explicit version

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