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Abstract

In this paper, we propose a constraint-type fictitious time integration method (FTIM) for solving multi-dimensional non-linear elliptic-type partial differential equations. Based on the variable transformation of FTIM, the original governing equation is transformed into a new parabolic equation of an evolution type by introducing a space-time variable, and a new time integration direction is obtained. However, the space-time variable depends on the governing equation, boundary condition and fictitious time variable, especially due to the nonlinear effect. Previous studies have not discussed the definition of these nonlinear parameter problems, which may result in severe numerical instability and inaccuracy. To completely overcome this nonlinear parameter problem, a space-time variable with a minimum fictitious time size is introduced into the algorithm. By imposing a constraint condition that involves the system energy in the space domain and the minimum fictitious time step, the proposed scheme can absolutely satisfy the stringent convergence criterion and can quickly approach the true solution, even under a very small time step. More importantly, the convergence speed depends only on a space-time variable. The accuracy and efficiency of the scheme are evaluated by comparing the estimation results with those of previous studies. The obtained results demonstrate that the proposed method efficiently finds the true solution and can significantly improve both the accuracy and convergence.

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