A Family of Block-Centered Schemes for Contaminant Transport Equations with Adsorption via Integral Method with Variational Limit

TL;DR

This paper introduces a flexible family of high-order conservative numerical schemes for contaminant transport with adsorption, utilizing an integral method with variational limits on block-centered grids, and includes boundary treatments for improved accuracy.

Contribution

It develops a unified framework for high-order schemes that can reproduce and extend classical compact schemes, with stability, convergence analysis, and boundary treatments.

Findings

Schemes achieve high-order accuracy consistent with theoretical predictions.

Parameter variations significantly affect error and scheme accuracy.

Numerical experiments confirm effectiveness across different adsorption models.

Abstract

This paper develops a class of high-order conservative schemes for contaminant transport with equilibrium adsorption, based on the Integral Method with Variational Limit on block-centered grids. By incorporating four parameters, the scheme can reproduce classical fourth-order compact schemes and further extend to sixth- and eighth-order accurate formulations, all within a unified framework. Under periodic boundary conditions, we analyze the stability, convergence, and mass conservation of the parameterized numerical scheme. Numerical experiments are then conducted to examine the impact of parameter variations on errors, explore the relationship between parameters and the fourth-, sixth-, and eighth-order schemes, and verify that the schemes' high-order accuracy aligns with theoretical predictions. To enhance the applicability of the proposed method, we further develop two fourth-order compact boundary treatments that ensure uniform accuracy between boundary and interior regions. Numerical results confirm the effectiveness of the proposed schemes across various adsorption models.