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

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.