Page - 297 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

9.2 FiniteDifferenceMethods 297 Fig. 9.1 Unstable simulation of the temperature in a rod The -i option specifies the naming of the plot files, and -r specifies the number of frames per second in the movie. On Mac, run ffmpeg instead of avconvwith the same options. Other video formats, such as MP4, WebM, and Ogg can also be produced: Terminal Terminal> ffmpeg -i tmp_%04d.png -r 4 -vcodec libx264 movie.mp4 Terminal> ffmpeg -i tmp_%04d.png -r 4 -vcodec libvpx movie.webm Terminal> ffmpeg -i tmp_%04d.png -r 4 -vcodec libtheora movie.ogg An Unstable Solution The results of a simulation start out as seen from the two snapshots in Fig.9.1. We see that the solution definitely looks wrong. The temperature is expected to be smooth, not having such a saw-tooth shape. We say that this solution is unstable, meaning that it does not display the same characteristics as the true, physical solution. Even though we tested the code carefullyin theprevioussection, itdoesnotseemtoworkforaphysicalapplication! Why Is the Solution Unstable? The problem is thatΔt is too large, making the solutionunstable. It turnsout that theForwardEuler time integrationmethodputsa restrictionon thesize ofΔt. For theheat equationand the way we havediscretized it, this restrictioncanbeshowntobe [14] Δt≤ Δx 2 2β . (9.15) Thisiscalledastabilitycriterion.Withthechosenparameters,(9.15)tellsusthat the upper limit isΔt =0.9527439,which is smaller thanour choiceabove.Rerunning