Question

QUESTION 4 [8 MARKS TOTAL] Q4 Hand calculations (see Workshops 21 and 22 for relevant content) Consider the 1-D diffusion equation: Eqn 3 да d²q at Ox? Write the centered in space, Crank-Nicolson (CN) discretization of this equation. Consider just ONE Fourier mode of the representation of the error, and write it in the simplified form (defined at time level n, and node j) as Eqn (4) Using this Fourier mode, undertake a von Neumann stability analysis for the CN discretization and derive (and clearly state) the amplification factor as a function of the timestep St. You can assume that the error satisfies the same difference equation as the solution for Q, (because it does). &;" = Aºe ik jox SHOW ALL YOUR WORKING MEC3456 Assignment Page 5 of 9 Compare the amplification factor for CN to the amplification factor for the Forward in Time, Centered in Space (FTCS) discretization derived in lectures. Is the CN method stable, conditionally stable or unstable? Comment in your PDF.

Answer 1

22 at k 2²9 9x² WFT AFI KFI + n 91-9 2q K 2 Light f At 1822 (5:23) n for time for space kjst An e m htt MFI nti K wt + *2E Ejt ELTE St * { 4 - 2 2 + Ej 1 ]* (Sn) Erros Equation Apply Von - Neuman stability criteria in equal iktisu kja A *Lante & Set +A Крti 25 npikliaison nt net ikisa A e kj sx 2 A. e St e A" eiklj-iisa LA" eik (j) Su + TA 1 httijo Consider Kosné o Anteiro - Anelja itj-1)0 K Ant é -2 Atle St 2 $ x ² + Anti pisto +Ahe hoit-10 -2Ah e is o + o Anti _A" inte C A ntleto i St = 24" & the eittho ete - 2A*** A 268014 An -ġo

G = Amplification factor = A An 10 . - 18 - وأعلم + (-ام ها کام + ce 2ی (uى أ- ي) Kst CE telo 19-1Ltl] كى Take ے A kst dzj" ( - ے فی +1 کی Glite secesa هه هه (1+ 7- هه و + أورمي - - ) تر و + (1- ي مهم 8 م -كم و محلهم د> 'رگ G Since و بی - 4 -1 الى مكة +ا < 1 This scheme is uncondition الله stable - Scheme

Write the expression for Amplification factor for FTCS scheme G |_ t sine (32 r = k St TS+72 IGIS 1 s1 ke tsi-ar sin 20% Arsin2% sa Ain 2 r st Thus, this scheme 13 B conditionally stable Conclesions CN scheme Unconditionally conditionally stable FTCS scheme stable