Journals: Mary C. Hill

Journal Articles Mary C. Hill

Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes
Mehl, S.W. and Hill, M.C.

2002, Advances in Water Resources, vol. 25, p. 497-511
ABSTRACT
A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback.
Results indicate: (1) The new method exhibits quadratic convergence for homogenous systems and convergence equivalent to uniform grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%.

Figure: Locally refined mesh with shared nodes along the interface.

Figure: Numerical test revealed quadratic convergence for homogeneous systems and less than quadratic for heterogeneous systems, indicating that the new method performs similarly to other methods.
Table 3 – Comparison of errors and CPU time for several grid refinement schemes. The system is depicted in Figure 12 with the transmissivity set that ranges from 1.200x10³ to 4.25x10⁰ m²/s
[Computation times using a Linux workstation, Pentium II – 333MHz, 64Mb Ram.]

Gridding Mean Head Error (%) Mean Cell-to-Cell Flux Error (%) Interior Mean Head Error (%) Interior Mean Cell-to-Cell Flux Error (%) CPU Time (s)
Fine grid (“Truth”) 0.000 0.000 0.000 0.000 716
Variably Spaced 0.014 0.078 0.023 0.034 57
MODTMR-Head¹ 0.282 4.901 0.393 2.140 3
MODTMR-Flux² 3.641 17.016 6.801 7.074 4
Iterative-Linear³ 0.061 1.758 0.099 0.142 28
Iterative-Darcy⁴ 0.056 1.269 0.089 0.140 28
¹Local model uses specified-head boundary conditions derived from the regional model.
²Local model uses specified-flux boundary conditions derived from the regional model.
³Iterative method developed for this work using linear interpolation.
⁴Iterative method developed for this work using Darcy weighted interpolation.

Table 4 – Comparison of errors and CPU time for several grid refinement schemes. The system is depicted in Figure 12 with the transmissivity set that ranges from 1.2x10⁵ to 4.0x10^-5 m²/s.
[Computation times using a Linux workstation, Pentium II – 333MHz, 64Mb Ram.]

Gridding Mean Head Error (%) Mean Cell-to-Cell Flux Error (%) Interior Mean Head Error (%) Interior Mean Cell-to-Cell Flux Error (%) CPU Time (s)
Fine grid (“Truth”) 0.000 0.000 0.000 0.000 1929
Variably Spaced 0.043 0.190 0.031 0.116 520
MODTMR-Head¹ 0.105 8.433 0.045 1.429 4
MODTMR-Flux² 2.090 25.381 2.211 15.687 4
Iterative-Linear³ 0.171 4.143 0.084 0.903 83
Iterative-Darcy⁴ 0.152 2.348 0.079 0.806 76
¹Local model uses specified-head boundary conditions derived from the regional model.
²Local model uses specified-flux boundary conditions derived from the regional model.
³Iterative method developed for this work using linear interpolation.
⁴Iterative method developed for this work using Darcy weighted interpolation.

mchill@usgs.gov
Last Modified: August 14, 2000