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# A Fictitious Time Integration Method for Two-Dimensional Quasilinear Elliptic Boundary Value Problems

Chein-Shan Liu1
Department of Mechanical and Mechatronic Engineering, Department of Harbor and River Engineering, Taiwan Ocean University, Keelung, Taiwan. E-mail: csliu@mail.ntou.edu.tw

Computer Modeling in Engineering & Sciences 2008, 33(2), 179-198. https://doi.org/10.3970/cmes.2008.033.179

### Abstract

Dirichlet boundary value problem of quasilinear elliptic equation is numerically solved by using a new concept of fictitious time integration method (FTIM). We introduce a fictitious time coordinate t by transforming the dependent variable u(x,y) into a new one by (1+t)u(x,y) =: v(x,y,t), such that the original equation is naturally and mathematically equivalently written as a quasilinear parabolic equation, including a viscous damping coefficient to enhance stability in the numerical integration of spatially semi-discretized equation as an ordinary differential equations set on grid points. Six examples of Laplace, Poisson, reaction diffusion, Helmholtz, the minimal surface, as well as the explosion equations are tested. It is interesting that the FTIM can easily treat the nonlinear boundary value problems without any iteration and has high efficiency and high accuracy. Due to the dissipation nature of the resulting parabolic equation, the FTIM is insensitive to the guess of initial conditions and approaches the true solution very fast.

### Keywords

Quasilinear elliptic equation, Laplace equation, Poisson equation, Helmholtz equation, Fictitious Time Integration Method (FTIM).

Liu, C. (2008). A Fictitious Time Integration Method for Two-Dimensional Quasilinear Elliptic Boundary Value Problems. CMES-Computer Modeling in Engineering & Sciences, 33(2), 179–198. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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