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We implement the numerical unified transform method to solve the nonlinear Schrödinger equation on the half-line. For the so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the numerical inverse scattering transform solves whole-line problems. In particular, the method computes the solution at any x and t without spatial discretization or time stepping. Contour deformations based on the method of nonlinear steepest descent are used so that the method’s computational cost does not increase for large x , t and the method is more accurate as x , t increase. Our ideas also apply to some cases where the boundary conditions are not linearizable.
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PyTheis—A Python Tool for Analyzing Pump Test Data
The Theis equation is an important mathematical model used for analyzing drawdown data obtained from pumping tests to estimate aquifer parameters. Since the Theis model is a nonlinear equation, a complex graphical procedure is employed for fitting this equation to pump test data. This graphical method was originally proposed by Theis in the late 1930s, and since then, all the groundwater textbooks have included this fitting method. Over the past 90 years, every groundwater hydrologist has been trained to use this tedious procedure for estimating the values of aquifer transmissivity (T) and storage coefficient (S). Unfortunately, this mechanical procedure does not provide any intuition for understanding the inherent limitations in this manual fitting procedure. Furthermore, it does not provide an estimate for the parameter error. In this study, we employ the public domain coding platform Python to develop a script, namely, PyTheis, which can be used to simultaneously evaluate T and S values, and the error associated with these two parameters. We solve nine test problems to demonstrate the robustness of the Python script. The test problems include several published case studies that use real field data. Our tests show that the proposed Python script can efficiently solve a variety of pump test problems. The code can also be easily adapted to solve other hydrological problems that require nonlinear curve fitting routines.
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- Award ID(s):
- 2019561
- PAR ID:
- 10336900
- Date Published:
- Journal Name:
- Water
- Volume:
- 13
- Issue:
- 16
- ISSN:
- 2073-4441
- Page Range / eLocation ID:
- 2180
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/w13162180
