Numerical convergence does not mean mathematical convergence: Examples of simple saturated steady-state groundwater models with pumping wells

Chapuis, Robert P.; Duhaime, François; Weber, Simon; Marefat, Vahid; Zhang, Lu; Blessent, Daniela; Bouaanani, Najib; Pelletier, Dominique

doi:10.1016/j.compgeo.2023.105615

Computers and Geotechnics

2023

DOI: 10.1016/j.compgeo.2023.105615

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Numerical convergence does not mean mathematical convergence: Examples of simple saturated steady-state groundwater models with pumping wells

Robert P. Chapuis

François Duhaime

Simon Weber

et al.

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2024

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Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

Chapuis,

Taveau,

Duhaime

et al. 2024

Num Anal Meth Geomechanics

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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.

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Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

Chapuis,

Taveau,

Duhaime

et al. 2024

Num Anal Meth Geomechanics

View full text Add to dashboard Cite

show abstract

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Numerical convergence does not mean mathematical convergence: Examples of simple saturated steady-state groundwater models with pumping wells

Cited by 1 publication

References 41 publications

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

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