Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method
Abstract
:1. Introduction
2. The Methodology
2.1. The Governing Equation
2.2. The Space–Time Collocation Trefftz Method
2.2.1. Space–Time Trefftz Functions
2.2.2. The Space–Time Collocation Scheme
2.2.3. The Iterative Scheme for Modeling Boundary Detection Problem
3. Accuracy and Convergence Analysis
4. Numerical Examples
4.1. An Inverse Transient Groundwater Problem—A Cassini-like Boundary
4.2. A Boundary Detection Problem—An Amoeba-like Boundary
4.3. A Boundary Detection Problem—A Flower-like Boundary
5. Conclusions
- (1)
- For modeling of the two-dimensional transient groundwater inverse problems, an innovative concept, that the collocation points are placed on the space–time boundaries, is presented. Both the initial and boundary data are regarded as boundary conditions on the space–time boundary. Using the space–time collocation, the initial value problem (IVP) for solving the transient groundwater equation is considered to be a problem of the inverse boundary value where the time marching for the IVP can be avoided.
- (2)
- Previous studies depict that the Trefftz method is limited to linear and stationary problems. This study demonstrates that we solved transient groundwater inverse problems using the space–time meshfree method utilizing Trefftz functions with very high accuracy. The iterative scheme of the FTIM is utilized to deal with the system of nonlinear algebraic equations, which is resulted from spatial discretization of the space–time collocation Trefftz method. The advantages of the proposed meshfree method for solving inverse problems are also presented. Accurate approximations are yielded in which inaccessible boundary may be recovered, even 7/8 portion of the overall space–time boundaries are inaccessible.
- (3)
- The approximations of the transient groundwater boundary detection problem in two dimensions were developed. The geometric shape of inaccessible parts of the domain remains unknown and is obtained by solving the nonlinear equation formulated from the Trefftz functions and the coefficients. The fictitious time integration method was successfully applied for the solution of the nonlinearity. Highly accurate results for the two-dimensional transient groundwater boundary detection problem were obtained using the proposed space–time meshfree method.
Author Contributions
Funding
Conflicts of Interest
References
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Ku, C.-Y.; Hong, L.-D.; Liu, C.-Y. Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method. Water 2020, 12, 3580. https://doi.org/10.3390/w12123580
Ku C-Y, Hong L-D, Liu C-Y. Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method. Water. 2020; 12(12):3580. https://doi.org/10.3390/w12123580
Chicago/Turabian StyleKu, Cheng-Yu, Li-Dan Hong, and Chih-Yu Liu. 2020. "Solving Transient Groundwater Inverse Problems Using Space–Time Collocation Trefftz Method" Water 12, no. 12: 3580. https://doi.org/10.3390/w12123580