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.

This paper presents a space–time meshfree method for solving transient inverse problems in subsurface flow. Based on the transient groundwater equation, we derived the Trefftz basis functions utilizing the method of separation of variables. Due to the basis functions completely satisfying the equation to be solved, collocation points are placed on the space–time boundaries. Numerical solutions are approximated based on the superposition theorem. Accordingly, the initial and boundary conditions are both regarded as space–time boundary conditions. Forward and inverse examples are conducted to validate the proposed approach. Emphasis is placed on the two-dimensional boundary detection problem in which the nonlinearity is solved using the fictitious time integration method. Results demonstrate that approximations with high accuracy are acquired in which the boundary data on the absent boundary may be efficiently recovered even when inaccessible partial data are provided.

In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

In this study, a novel space-time (ST) marching method is presented to solve linear and nonlinear transient flow problems in porous media. The method divides the ST domain into subdomains along the time axis. The solutions are approximated using ST polyharmonic radial polynomial basis functions (RPBFs) in the ST computational domain. In order to proceed along the time axis, we use the numerical solution at the current timespan of the two ST subdomains in the computational domain as the initial conditions of the next stage. The fictitious time integration method (FTIM) is subsequently employed to solve the nonlinear equations. The novelty of the proposed method is attributed to the division of the ST domain along the time axis into subdomains such that the dense and ill-conditioned matrices caused by the excessive number of boundary and interior points and the large ST radial distances can be avoided. The results demonstrate that the proposed method achieves a high accuracy in solving linear and nonlinear transient problems. Compared to the conventional time marching and ST methods, the proposed meshless approach provides more accurate solutions and reduces error accumulation.