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 In this paper we study the method of Pseudo-transient as a tool for solving some ODEs, PDEs or predator-prey system.To compute the stationary solutions of PDEs or equilibria of ODEs numerically, we need to compute for a long time period to ensure that we have the approximation of the solutions at large time. In this paper, we would like to introduce Pseudo-transient method to speed up our computation if the procedure between the initial condition and behavior at large time is not important. We report several numerical results and compare the advantages and disadvantages for the Pseudo-transient method and the usual ODE solvers.
 Contents1 Introduction 12 Pseudo-transient continuation 13 Examples and Numerical experiments 33.1 An ODE system . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 A predator-prey model . . . . . . . . . . . . . . . . . . . . . . 63.3 The Burgers equation . . . . . . . . . . . . . . . . . . . . . . . 144 Conclusion 20References 21Appendix 22
 References[1] J. Burns, A. Balogh, D. S. Gilliam, and V. I. Shubov. Numerical stationarysolutions for a viscous Burgers’ equation. J. Math. Systems Estim.Control, 8(2):16 pp. (electronic), 1998.[2] G. J. Butler and Paul Waltman. Bifurcation from a limit cycle in a twopredator-one prey ecosystem modeled on a chemostat. J. Math. Biol.,12(3):295–310, 1981.[3] Chuang-Hsiung Chiu. Lyapunov functions for the global stability of competingpredators. J. Math. Anal. Appl., 230(1):232–241, 1999.[4] S. B. Hsu, S. P. Hubbell, and Paul Waltman. Competing predators. SIAMJ. Appl. Math., 35(4):617–625, 1978.[5] C. T. Kelley and David E. Keyes. Convergence analysis of pseudotransientcontinuation. SIAM J. Numer. Anal., 35(2):508–523, 1998.[6] C. T. Kelley, C. T. Miller, and M. D. Tocci. Termination of Newton/chord iterations and the method of lines. SIAM J. Sci. Comput., 19(1):280–290, 1998. Special issue on iterative methods (Copper Mountain,CO, 1996).[7] Torsten Lindström. Global stability of a model for competing predators:an extension of the Ardito & Ricciardi Lyapunov function. NonlinearAnal., 39(6, Ser. A: Theory Methods):793–805, 2000.[8] Wim A. Mulder and Bram van Leer. Experiments with implicit upwindmethods for the Euler equations. J. Comput. Phys., 59(2):232–246, 1985.
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