solving partial differential equations the plotting statements update the xtcurve on the screen in additionwe save fraction of the plots to files tmp_ pngtmp_ pngtmp_ pngand so on these plots can be combined to ordinary video files common tool is ffmpeg or its sister avconv these programs take the same type of command-line options to make flash video movie flvrun terminal terminalffmpeg - tmp_% png - -vcodec flv movie flv the - option specifies the naming of the plot files in printf syntaxand - specifies the number of frames per second in the movie on macrun ffmpeg instead of avconv with the same options other video formatssuch as mp webmand ogg can also be producedterminal terminalffmpeg - tmp_% png - -vcodec libx movie mp terminalffmpeg - tmp_% png - -vcodec libvpx movie webm terminalffmpeg - tmp_% png - -vcodec libtheora movie ogg the results of simulation start out as in figs and we see that the solution definitely looks wrong the temperature is expected to be smoothnot having such saw-tooth shape alsoafter some time (fig )the temperature starts to increase much more than expected we say that this solution is unstablemeaning that it does not display the same characteristics as the truephysical solution even though we tested the code carefully in the previous sectionit does not seem to work for physical applicationhow can that bethe problem is that is too largemaking the solution unstable it turns out that the forward euler time integration method puts restriction on the size of for the heat equation and the way we have discretized itthis restriction can be shown to be [ ( this is called stability criterion with the chosen parameters( tells us that the upper limit is : which is smaller than our choice above rerunning the case with equal to /indeed shows smooth evolution of xtfind the program rod_fe py and run it to see an animation of the xtfunction on the screen scaling and dimensionless quantities our setting of parameters required finding three physical properties of certain material the time interval for simulation and the time step depend crucially on the values for and lwhich can vary significantly from case to case oftenwe are more interested in how the shape of xtdevelopsthan in the actual uxand values for specific material we can then simplify the setting of physical parameters by scaling the problem