Page - 290 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

290 9 SolvingPartialDifferentialEquations 9.2 FiniteDifferenceMethods We shall now construct a numerical method for the diffusion equation. We know how to solve ODEs, so in a way we are able to deal with the time derivative. Very often in mathematics, 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 to reduce the PDE to ODEs. One important technique for achieving this, is based on finite difference discretizationof spatialderivatives. 9.2.1 ReductionofaPDEtoaSystemofODEs Introducea spatial mesh inΩ withmesh points x0 =0<x1 <x2 < · · ·<xN =L. Thespacebetweentwomeshpointsxi andxi+1, i.e. the interval [xi,xi+1], is called acell.Weshallhere, forsimplicity,assumethateachcellhas thesamelengthΔx= xi+1−xi, i=0,.. .,N−1.