Using H-Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady-State Conditions
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John Wiley and Sons Ltd
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Unsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H-convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady-state conditions. The quantitative H-convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H-convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers. © 2024 John Wiley & Sons Ltd.
Palabras clave
Groundwater, Mathematical convergence, Numerical analysis, Numerical convergence, Numerical errors, Unsaturated, Convergence of numerical methods, Groundwater geochemistry, Nonlinear equations, Seepage, Sensitivity analysis, Convergence domains, Convergence test, Element sizes, H convergences, Mathematical convergence, Numerical convergence, Numerical errors, Steady-state condition, Unsaturated, Unsaturated seepage, Mesh generation