Boundary stability is a mesh-uniform estimate for a finite-difference initial-boundary problem with forced boundary data. Stability of the whole-line Cauchy scheme is necessary but may fail to control boundary-supported numerical modes.
A boundary closure supplies the boundary rows or ghost values needed by a spatial finite difference method. It must reflect the PDE boundary data and preserve a mesh-uniform stability estimate. Interior von Neumann stability analysis cannot detect an arbitrary unstable boundary row; for example a row produces growth even beside an otherwise stable heat stencil.
After transforming time and tangential variables, the uniform Kreiss--Lopatinskii condition requires the boundary equations to determine every decaying normal mode with an inverse bounded uniformly over transformed frequencies. It rules out growing or weakly controlled boundary modes.
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