Seite - 166 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

166 5 SolvingPartialDifferentialEquations The initial conditionsare u0.0/D s.0/; (5.12) ui.0/DI.xi/; i D1;:: :;N : (5.13) Wecanapplyanymethodfor systemsofODEs tosolve (5.9)–(5.11). 5.1.2 ConstructionofaTestProblemwithKnownDiscreteSolution At this point, it is tempting to implement a real physical case and run it. However, partial differential equations constitute a non-trivial topicwheremathematical and programmingmistakes comeeasy. Abetter start is therefore to address a carefully designedtest examplewherewecancheck that themethodworks. Themostattrac- tive examples for testing implementations are thosewithout approximationerrors, becauseweknowexactlywhat numbers the programshould produce. It turns out that solutionsu.x;t/ that are linear in timeand in spacecanbeexactly reproduced bymost numericalmethods for partial differential equations. Acandidate solution mightbe u.x;t/D .3tC2/.x L/: Inserting thisu in thegoverningequationgives 3.x L/D0Cg.x;t/ ) g.x;t/D3.x L/: Whatabout theboundaryconditions?Werealize that@u=@xD 3tC2 forx DL, which breaks the assumption of @u=@x D 0 at x D L in the formulation of the numericalmethodabove.Moreover,u.0;t/D L.3tC2/, sowemust set s.t/D L.3tC2/ands0.t/D 3L. Finally, theinitialconditiondictatesI.x/D2.x L/, but recall that we must have u0 D s.0/, and ui D I.xi/, i D 1;:: :;N: it is important thatu0 starts out at the right value dictated by s.t/ in case I.0/ is not equal thisvalue. Firstweneedtogeneralizeourmethodtohandle@u=@xD ¤0atxDL.We thenhave uNC1.t/ uN 1.t/ 2 x D ) uNC1 DuN 1C2 x; which inserted in (5.7)gives duN.t/ dt Dˇ2uN 1.t/C2 x 2uN.t/ x2 CgN.t/: (5.14) 5.1.3 Implementation:ForwardEulerMethod In particular, we may use the Forward Euler method as implemented in the general function ode_FE in the module ode_system_FE from Sect. 4.2.6. The ode_FE function needs a specification of the right-hand side of theODE system.