A Spacetime Meshless Method for Modeling Subsurface Flow with a Transient Moving Boundary

Abstract

In this paper, a spacetime meshless method utilizing Trefftz functions for modeling subsurface flow problems with a transient moving boundary is proposed. The subsurface flow problem with a transient moving boundary is governed by the two-dimensional diffusion equation, where the position of the moving boundary is previously unknown. We solve the subsurface flow problems based on the Trefftz method, in which the Trefftz basis functions are obtained from the general solutions using the separation of variables. The solutions of the governing equation are then approximated numerically by the superposition theorem using the basis functions, which match the data at the spacetime boundary collocation points. Because the proposed basis functions fully satisfy the diffusion equation, arbitrary nodes are collocated only on the spacetime boundaries for the discretization of the domain. The iterative scheme has to be used for solving the moving boundaries because the transient moving boundary problems exhibit nonlinear characteristics. Numerical examples, including harmonic and non-harmonic boundary conditions, are carried out to validate the method. Results illustrate that our method may acquire field solutions with high accuracy. It is also found that the method is advantageous for solving inverse problems as well. Finally, comparing with those obtained from the method of fundamental solutions, we may obtain the accurate location of the nonlinear moving boundary for transient problems using the spacetime meshless method with the iterative scheme.

Modeling of Flow and Transport in Saturated and Unsaturated Porous Media

“…In Ku et al [23], the authors extended the previous method to the solution of subsurface flow problems with a transient moving boundary. The proposed solution is based on the spacetime collocation method with complete Trefftz basis functions.…”

Modeling Tide–Induced Groundwater Response in a Coastal Confined Aquifer Using the Spacetime Collocation Approach

“…As for the boundary data, it is assigned on both right and left of the spacetime region, as illustrated in Figure 2. To evaluate the accuracy of the proposed approach, the maximum absolute error (MAE) is evaluated by the following equation [32][33][34].…”

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

“…It mathematically models how a substance's concentration changes in space and time due to the process of diffusion [4,5]. The diffusion equation finds extensive applications across various scientific and engineering disciplines including groundwater, environmental science, geophysics, transport phenomena, heat conduction, and engineering [6,7]. Solving the diffusion equation involves finding the distribution of a quantity over space and time for given initial and boundary conditions.…”

“…Numerical techniques like finite difference, finite element, spectral, or meshfree methods are commonly used [8,9]. Among these methodologies, the meshfree methods have attracted the attention of researchers from various scientific fields due to their ability to handle diffusion equations with complex and irregular geometries [6,8,10]. The radial basis function (RBF) collocation method is a meshfree approach used to analyze governing equations, where the unknowns are represented by function approximation.…”