Page - 170 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

170 5 SolvingPartialDifferentialEquations The plotting statements update the u.x;t/ curve on the screen. In addi- tion, we save a fraction of the plots to files tmp_0000.png, tmp_0001.png, tmp_0002.png, and so on. These plots can be combined to ordinary video files. Acommontool isffmpegor its sisteravconv. These programs take the same typeof command-lineoptions. TomakeaFlash videomovie.flv, run Terminal Terminal> ffmpeg -i tmp_%04d.png -r 4 -vcodec flv movie.flv The-ioptionspecifies thenamingof theplotfiles inprintf syntax,and-rspecifies the number of frames per second in themovie. OnMac, run ffmpeg instead of avconvwith thesameoptions.Othervideoformats, suchasMP4,WebM,andOgg canalsobeproduced: 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 Theresultsofasimulation start outas inFigs. 5.1and5.2.Wesee that thesolu- tiondefinitely lookswrong. The temperature is expected tobe smooth, not having such a saw-tooth shape. Also, after some time (Fig. 5.2), the temperature starts to increasemuchmore than expected. We say that this solution isunstable,meaning that it doesnotdisplay the samecharacteristics as the true, physical solution. Even thoughwetested thecodecarefully intheprevioussection, itdoesnotseemtowork foraphysicalapplication!Howcan thatbe? The problem is that t is too large,making the solution unstable. It turns out that theForwardEuler time integrationmethodputsa restrictionon the sizeof t. For the heat equation and the waywe have discretized it, this restriction can be showntobe [15] t x 2 2ˇ : (5.15) This is called a stability criterion. With the chosenparameters, (5.15) tells us that the upper limit is t D 0:0003125, which is smaller than our choice above. Re- runningthecasewitha t equal to x2=.2ˇ/, indeedshowsasmoothevolutionof u.x;t/. Find theprogramrod_FE.pyand run it to see ananimationof theu.x;t/ functionon thescreen. Scalinganddimensionlessquantities Our settingof parameters requiredfinding threephysical propertiesof a certain material. The time interval for simulationand the timestepdependcruciallyon thevaluesforˇandL,whichcanvarysignificantlyfromcase tocase.Often,we aremore interested inhowtheshapeofu.x;t/develops, than in theactualu,x, and t values for a specificmaterial.Wecan thensimplify the settingofphysical parametersby scaling theproblem.