Anchored Accumulation Calculus for Incompressible Navier–Stokes: Exact Viscous Propagation, Fourth-Order Forcing, and Numerical Verification

Abstract

We introduce and numerically validate an Anchored Accumulation Calculus (AAC) realization of the periodic, constant-density, incompressible Newtonian Navier–Stokes equations. The central construction is operator-exact in the linear viscous part: after Fourier transformation and Leray projection, each solenoidal mode obeys a scalar stiff decay equation forced by the projected nonlinearity. AAC applies the exact exponential propagator to that viscous operator and accumulates the nonlinear forcing through a cubic Lagrange history integrated exactly against the exponential kernel. The fourth-order member, AAC4, therefore does not approximate the stiff linear propagator by an explicit polynomial; instead it evaluates the corresponding Duhamel accumulation mode by mode. The paper first derives AAC from the variation-of-constants formula and proves four structural properties: exactness for the isolated linear viscous mode, preservation of the divergence-free Fourier subspace, fourth-order local consistency and conditional global fourthorder convergence for the smooth semidiscrete problem, and reduction to classical Adams– Bashforth 4 as the viscosity tends to zero. A separate proposition establishes the nonlinear energy orthogonality used by the benchmark, while a linear stability analysis shows why the method removes the explicit viscous stability barrier that limits classical RK4. The implementation uses a Fourier pseudospectral discretization on the periodic cube, a Leray projection, quadratic nonlinear evaluation through the vorticity identity, coordinatewise two-thirds dealiasing, a real-to-complex half-spectrum FFT path, and deterministic threedimensional vortex-braid initial data. The numerical verification suite includes exact ABC decay, nonlinear Taylor–Green matching, temporal convergence, the AAC4–AB4 limit, nonlinear energy orthogonality, variable-step consistency, three-dimensional vortex stretching, resolution refinement, operator audits, a long-time stress calculation, hard-stress timestep refinement, cross-resolution/cross-method agreement, and resolution scaling.

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