Algorithm for the Cauchy Problem for Difference Equations with Variable Coefficients and Initial Data on a Parallelepiped

We consider the Cauchy problem for a scalar linear three-dimensional difference equation with variable coefficients and a fixed, finite rectangular stencil in a semi-infinite parallelepiped. Prescribed data occupy the complement of the unknown interior region, including the lower layers and lateral strips. We extend a layer-by-layer construction for constant-coefficient equations to coefficient functions evaluated at the lattice nodes of the corresponding function values. A symbolic block Toeplitz template is assembled once and evaluated column-wise at the coordinates of the unknowns on each layer. The resulting numerical matrices are generally not block Toeplitz. Precomputed coefficient arrays supply the values needed to assemble the right-hand sides. We verify that the assembled systems coincide with the original difference equation. If every layer matrix up to the prescribed target layer is nonsingular, the algorithm uniquely determines all unknown values on these layers, including the value at the target point. A worked example illustrates the coefficient arrays, matrix assembly, and successive solutions; exact rational verification confirms the displayed results. Storage requirements and dense-solver operation counts are also discussed.