Page - 163 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

5.1 FiniteDifferenceMethods 163 What about the source termg in our examplewith temperature distribution in a rod? g.x;t/models heat generation inside the rod. Onecould thinkof chemical reactions at amicroscopic level in somematerials as a reason to includeg. How- ever, inmostapplicationswith temperatureevolution,g is zeroandheatgeneration usually takesplaceat theboundary(as inourexamplewithu.0;t/D s.t/). Before continuing,wemayconsider an exampleof how the temperature distri- butionevolves in the rod. At time t D 0,weassume that the temperature is10ıC. Thenwe suddenly apply a device atx D 0 that keeps the temperature at50ı Cat this end. What happens inside the rod? Intuitively, you think that the heat genera- tionat theendwillwarmup thematerial in thevicinityofxD0, andas timegoes by,moreandmoreof therodwill beheated,before theentire rodhasa temperature of50ıC(recall thatnoheat escapes fromthesurfaceof the rod). Mathematically, (with the temperature inKelvin) this examplehasI.x/D 283 K, except at the end point: I.0/ D 323K, s.t/ D 323K, andg D 0. Thefigure belowshowssnapshotsfromfourdifferenttimesintheevolutionofthetemperature. 5.1 FiniteDifferenceMethods Weshall nowconstruct a numericalmethod for the diffusion equation. We know how to solve ordinary differential equations, so in awaywe are able to dealwith the time derivative. Very often inmathematics, a new problem can be solved by reducing it to a series of problems we know how to solve. In the present case, it means that we must do something with the spatial derivative @2=@x2 in order