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

TL;DR

This paper introduces a novel fixed-grid multiwavelet coefficient method for accurately capturing shocks in Buckley--Leverett saturation transport, ensuring conservation, boundary condition enforcement, and minimal mass errors.

Contribution

The paper develops a conservative affine-constrained multiwavelet method that directly evolves saturation in a local coefficient basis with boundary trace enforcement and shock control.

Findings

The method accurately reproduces reference saturation profiles and breakthrough curves.

It maintains boundary conditions and controls saturation bounds effectively.

Piecewise-linear representation offers optimal accuracy-cost balance for shock-dominated problems.

Abstract

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 \texttt{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.