Study on the detection of groundwater boundary based on the Trefftz method

Detecting water head is crucial in groundwater utilization, and requires quick and accurate solutions. This study employs a method combining the collocation Trefftz method (CTM) and the fictitious time integration method (FTIM) for groundwater head detection and restoration. The Laplace equation is solved using CTM, and the Trefftz basis function is linearly combined to fit the exact solution while adding characteristics length for numerical stability. The FTIM solves the nonlinear algebraic equations system for water head detection. Numerical examples quantify the method's accuracy, and boundary restoration results are compared with the Picard successive approximation method, showcasing FTIM's advantages in convergence steps and precision. The solution comparison under irregular boundary conditions further verifies the proposed method's efficacy (MAE <= 10-15). The CTM-FTIM calculated water level boundary aligns with the actual boundary, and its noise immunity is verified using real observation well data (MAE <= 10-2, Itertimes <= 5000). The CTM-FTIM method eliminates meshing needs for quick solutions in irregular regions, accurately determining water levels in the study domain using few known boundary points, solving infinite domain groundwater head detection.