Page - 175 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

5.1 FiniteDifferenceMethods 175 time, is thevisualizationon the screen, but for that purposeonecanvisualize only a subset of the time steps. However, there are occasions when you need to take larger time steps with the diffusion equation, especially if interest is in the long- termbehavioras t !1. Youmust then turn to implicit methods forODEs. These methodsrequire thesolutionsof linearsystems, if theunderlyingPDEis linear, and systemsofnonlinearalgebraic equations if theunderlyingPDEisnon-linear. The simplest implicitmethod is theBackwardEuler scheme,whichputs no re- strictions on t for stability, but obviously, a large t leads to inaccurate results. TheBackwardEuler schemefora scalarODEu0 Df.u;t/ reads unC1 un t Df.unC1;tnC1/: This equation is to be solved forunC1. Iff is linear inu, it is a linear equation, but iff is nonlinear inu, oneneedsapproximatemethods for nonlinear equations (Chap.6). Inourcase,wehaveasystemof linearODEs(5.9)–(5.11).TheBackwardEuler schemeapplied toeachequation leads to unC10 un0 t D s0.tnC1/; (5.16) unC1i uni t D ˇ x2 .unC1iC1 2unC1i CunC1i 1 /Cgi.tnC1/; (5.17) i D1;:: :;N 1; unC1N unN t D 2ˇ x2 .unC1N 1 unC1N /Cgi.tnC1/: (5.18) This is a systemof linear equations in the unknownsunC1i , i D 0;:: :;N , which is easy to realize bywriting out the equations for the caseN D 3, collecting all theunknowntermson the left-handsideandall theknowntermson the right-hand side: unC10 Dun0 C ts0.tnC1/; (5.19) unC11 t ˇ x2 .unC12 2unC11 CunC10 /Dun1 C tg1.tnC1/; (5.20) unC12 t 2ˇ x2 .unC11 unC12 /Dun2 C tg2.tnC1/: (5.21) Asystemof linearequations like this, isusuallywrittenonmatrixformAuDb, whereA is a coefficientmatrix,u D .unC10 ;: : :;nnC1N / is the vector of unknowns, andb is avectorofknownvalues. Thecoefficientmatrix for thecase (5.19)–(5.21) becomes AD 0 B @ 1 0 0 t ˇ x2 1C2 t ˇ x2 t ˇ x2 0 t 2ˇ x2 1C t 2ˇ x2 1 C A