Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

Abstract

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

“…This simplification enables a clearer determination of the properties of the numerical scheme, such as accuracy and stability. Recently, the ST coupled approach has been widely combined with meshless methods such as the ST localized RBFCM [38], ST kernel-based method [39], ST localized method of fundamental solutions [40,41], ST Trefftz Method [42,43], ST backward substitution method [44], and ST-GFDM [45][46][47][48][49][50][51][52][53][54]. Based on the flexibility of the ST-GFDM, researchers have applied this meshless method for engineering problems in the past few years.…”

Study on Seepage Mechanism and Stability of Unsaturated Slope Based on Trefftz Method

“…Nevertheless, for complex study areas, such methods encounter the problem of difficult grid creation, and the problem boundary needs to be simplified to be analyzable [7]. In recent years, many scholars have studied meshless methods that are exempted from the establishment of a mesh [8], the collocation Trefftz method does not require the creation of a grid and integration of the boundary [9]. The basis function satisfies the control equation, and the points are only distributed on the boundary.…”

A Novel Space-Time Marching Method for Solving Linear and Nonlinear Transient Problems

“…The transient problems related to convection, diffusion, reaction, and sources/sinks in porous media [1][2][3] are often found in a wide range of natural phenomena and must be solved by partial differential equations (PDEs) [4]. Accordingly, numerical methods have been proposed to simulate the transient problem, including the finite element method [5], the finite difference method (FDM) [6], the generalized finite difference method [7,8], the Trefftz method [9], and the radial basis function collocation method (RBFCM) [10]. Currently, the RBFCM is one of the most widely used numerical methods to simulate engineering problems due to its simplicity and effectiveness [11].…”

“…In transient applications, the implicit Crank-Nicolson method is the most common method and has been improved over time. Additionally, the ST [9,21], and the ST marching [22] methods have also been proposed. However, these two methods may exhibit error accumulation for simulations over a long period of time.…”