R-layer Mode Theory XXVIII: Computational Development of R-waves — Numerical Framework for Multi-Resolution Dynamics

R-layer Mode Theory Volume XXVIII develops the computational framework required to simulate R-wave dynamics across the deep, mid, and global resolutions of the R-layer hierarchy. Building on the analytical structure established in Volume XXVII, this volume introduces a mathematically rigorous discretization of the R-wave operator, hybrid spectral–finite difference schemes, and multi-resolution algorithms that preserve the hierarchical geometry of R-waves. As stated in the introduction: “R-waves exhibit a hierarchical internal structure: deep components require high spectral resolution, mid-scale components require stabilization, and global components require long-range accuracy.” The volume constructs: discrete resolution operators for deep (spectral), mid (finite difference), and global (finite difference + curvature) components discrete cross-scale coupling operator, generally non-normal discrete R-wave operator Lh=c1Ddeep,h+c2Dmid,h+c3Dglobal,h+Ch consistency and convergence analysis for hybrid discretizations A full numerical stability theory is developed using: spectral radius estimates von Neumann analysis Strang-type operator splitting stability generalized multi-resolution CFL conditions The volume then introduces a multi-resolution algorithm: resolution decomposition via projection operators spectral update for deep components finite-difference update for mid/global components coupling update via operator splitting filtering and recomposition to prevent aliasing Computational experiments demonstrate: deep-scale oscillatory decay mid-scale coherent propagation global-scale curvature response spontaneous formation of hybrid modes and cross-scale transitions Appendices provide discrete operator theory, derivation of the multi-resolution CFL condition, and implementation details including FFT, finite differences, and time-stepping schemes. Volume XXVIII establishes the numerical foundation for large-scale simulations and applied developments in subsequent volumes.