Stable, Compact, and Direct Ghost-Cell Reconstruction: A Non-Iterative Approach for Embedded-Boundary Methods
A direct, analytical, non-iterative ghost-cell reconstruction framework is developed for Cartesian-grid embedded-boundary methods. Analytical expressions impose Dirichlet and Neumann boundary conditions directly at the embedded boundary, eliminating intermediate image-point reconstruction, matrix inversion, and precomputation or storage of geometry-dependent reconstruction weights. For the Cartesian stencil considered, dependencies among neighboring ghost cells form a directed acyclic graph. A topological ordering partitions ghost cells into dependency levels, permitting level-by-level reconstruction without iterative updates. The formulation is combined with hybrid ghost cells (HGC), whose centers may lie on either side of the embedded boundary. This placement satisfies the linear reconstruction-stability criterion previously derived for scalar advection while retaining a compact nearest-neighbour Cartesian stencil. An extended-stencil classical ghost-cell formulation (CGC_ES) serves as a stability-preserving reference, distinguishing the effects of reconstruction stability and stencil compactness. The same analytical relations provide solution values and spatial gradients directly on the embedded boundary, enabling evaluation of pressure forces, wall stresses, drag, and lift without separate surface reconstruction or filtering. Simulations of flow past a circular cylinder and an airfoil show that the linear criterion remains a useful indicator of reconstruction stability for the nonlinear incompressible Navier-Stokes cases considered. Classical ghost-cell reconstruction develops spurious oscillations when the criterion is violated, whereas CGC_ES and HGC remain stable. HGC additionally satisfies the stability requirement with a compact nearest-neighbour stencil.