Page - 182 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

182 5 SolvingPartialDifferentialEquations Remarks In 2D and 3Dproblems, where theCPU time to compute a solution of PDEcanbehoursanddays, it isvery important toutilize symmetryaswedoabove to reduce the sizeof theproblem. Also note the remarks inExercise 5.6 about the constant areaunder theu.x;t/ curve: here, the area is 0.5 andu ! 0:5 as t ! 0:5 (if themesh is sufficiently fine -onewill get convergence to smaller values for small if themesh is notfine enoughtoproperly resolvea thin-shaped initial condition). Exercise5.9:Computesolutionsas t !1 Manydiffusionproblems reach a stationary time-independent solution as t ! 1. ThemodelproblemfromSect. 5.1.4 isoneexamplewhereu.x;t/D s.t/D const for t !1.Whenudoesnotdependon time, thediffusionequation reduces to ˇu00.x/Df.x/; inonedimension, and ˇr2uDf.x/; in2Dand3D.This is the famousPoisson equation,or iff D 0, it is knownas the Laplaceequation. In this limit t !1, there isnoneed foran initial condition,but theboundaryconditionsare the sameas for thediffusionequation. Wenowconsideraone-dimensionalproblem u00.x/D0; x2 .0;L/; u.0/DC; u0.L/D0; (5.38) which is known as a two-point boundary value problem. This is nothing but the stationary limit of the diffusion problem in Sect. 5.1.4. How canwe solve such a stationaryproblem(5.38)? The simplest strategy,whenwealreadyhavea solver for the corresponding time-dependent problem, is to use that solver and simulate until t !1,whichinpracticemeansthatu.x;t/nolongerchangesintime(within some tolerance). A nice feature of implicitmethods like theBackwardEuler scheme is that one can takeonevery long time step to “infinity”andproduce the solutionof (5.38). a) Let(5.38)bevalidatmeshpointsxi inspace,discretizeu00byafinitedifference, andsetupasystemofequationsfor thepointvaluesui,i D0;:: :;N ,whereui is theapproximationatmeshpointxi. b) Showthat if t !1 in (5.16) - (5.18), it leads to thesameequationsas ina). c) Demonstrate, by runningaprogram, that youcan takeone large time stepwith theBackwardEuler schemeandcompute the solutionof (5.38). Thesolution is veryboringsince it is constant:u.x/DC. Filename:rod_stationary.py. Remarks If the interest is in the stationary limit of a diffusion equation, one can either solve theassociatedLaplaceorPoissonequationdirectly,oruseaBackward Euler schemefor the time-dependentdiffusionequationwithavery long timestep. Using a ForwardEuler schemewith small time steps is typically inappropriate in