An upwind discontinuous finite volume element method on triangular meshes for the two-dimensional unsaturated soil water movement problem

Fang Wang; Fan Chen; Changqing Lv; Xiaoyan Qin; Fang Wang; Fan Chen; Changqing Lv; Xiaoyan Qin

doi:10.3934/math.2026975

AIMS Mathematics

2026, Volume 11, Issue 8: 24152-24178. doi: 10.3934/math.2026975

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Research article

An upwind discontinuous finite volume element method on triangular meshes for the two-dimensional unsaturated soil water movement problem

Department of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China

Received: 31 May 2026 Revised: 29 July 2026 Accepted: 30 July 2026 Published: 07 August 2026
MSC : 65M12, 65M15, 65M60

The mathematical model for unsaturated soil water movement is essentially the Richards equation, which is a time-dependent nonlinear partial differential equation. This paper proposes an upwind discontinuous finite volume element method (DFVEM) on triangular meshes for solving two-dimensional unsaturated soil water flow problems. By employing an upwind strategy for the convection term, the method effectively avoids numerical dispersion and non-physical oscillations while preserving the local mass conservation and flexibility of the standard DFVEM. The error estimates are derived in the $ L^2 $ norm and $ |\!|\!|\cdot|\!|\!|_{1, h} $ norm. Numerical experiments are conducted to verify the theoretical analysis and demonstrate the efficiency of the proposed method.
- unsaturated soil water movement,
- upwind discontinuous finite volume element method,
- error estimate
Citation: Fang Wang, Fan Chen, Changqing Lv, Xiaoyan Qin. An upwind discontinuous finite volume element method on triangular meshes for the two-dimensional unsaturated soil water movement problem[J]. AIMS Mathematics, 2026, 11(8): 24152-24178. doi: 10.3934/math.2026975

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Abstract

The mathematical model for unsaturated soil water movement is essentially the Richards equation, which is a time-dependent nonlinear partial differential equation. This paper proposes an upwind discontinuous finite volume element method (DFVEM) on triangular meshes for solving two-dimensional unsaturated soil water flow problems. By employing an upwind strategy for the convection term, the method effectively avoids numerical dispersion and non-physical oscillations while preserving the local mass conservation and flexibility of the standard DFVEM. The error estimates are derived in the $ L^2 $ norm and $ |\!|\!|\cdot|\!|\!|_{1, h} $ norm. Numerical experiments are conducted to verify the theoretical analysis and demonstrate the efficiency of the proposed method.

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