A Fixed-Grid Affine-Constrained Multiwavelet Coefficient Method for Buckley--Leverett Shock Capturing

We present a fixed-grid conservative affine-constrained modal/multiwavelet coefficient method for one-dimensional Buckley–Leverett saturation transport. The saturation is evolved directly in a local orthonormal coefficient basis with a mean/detail structure: the first mode carries the conservative cell average, whereas higher modes carry zero-mean local details. The hyperbolic inflow condition is imposed as a linear trace constraint on the coefficient vector and enforced by affine lifting. For $(p>1)$, the boundary reprojection is applied in the detail subspace of the inflow cell, so that the prescribed trace is restored without modifying the conservative cell-average update. The transport operator is discretized in conservative weak form with monotone numerical fluxes, and shock-induced oscillations are controlled by a troubled-cell limiter acting on modal details. The method is validated on a Berea-core waterflood benchmark against an independent $\mathtt{\text{pywaterflood}}$ reference solution using the same Corey fractional-flow closure, physical parameters, and pore-volume-injected scaling. The affine-constrained coefficient solver reproduces the reference breakthrough curve and saturation profiles, preserves the imposed inflow trace to roundoff accuracy, controls saturation bounds through mean-preserving detail rescaling, and gives small accumulated global mass-balance defects. Mesh-refinement, flux-comparison, and modal-order studies show that $(p=2)$, corresponding to a piecewise-linear local representation, provides the most favorable accuracy–cost compromise among the tested orders for this shock-dominated benchmark.