A conservative finite-volume Buckley–Leverett solver with bounded-interval multiwavelet state analysis

We develop a conservative finite-volume Buckley–Leverett solver equipped with a bounded-interval multiwavelet state-analysis layer. Because non-capillary Buckley–Leverett transport is a nonlinear hyperbolic conservation law with entropy-admissible shocks, the saturation equation is advanced by a conservative finite-volume method with monotone numerical fluxes. The accepted finite-volume state is then embedded in a bounded-domain multiwavelet hierarchy, reconstructed back to cell averages, and used for multiresolution diagnostics. The formal transport accuracy is therefore governed by the underlying finite-volume discretization, while the multiwavelet layer is used for representation, compression, and front-localization diagnostics. Its purpose is instead to quantify whether the deterministic physical-space saturation state can be represented faithfully, compressed in a controlled manner, and used to identify dynamically active front regions. Validation against reference Buckley–Leverett profiles for a Berea benchmark shows accurate saturation histories, spatial profiles, front-position diagnostics, and mass balance. The multiwavelet reconstruction tracks the internal finite-volume state with essentially exact fidelity. Additional thresholding tests show that a substantial fraction of detail coefficients can be discarded while maintaining small reconstruction errors and negligible global mass defect, and fine-level detail activity localizes the moving displacement front. The resulting formulation provides a conservative and reproducible first stage toward future transport-active adaptive multiwavelet solvers for porous-media flow.