Problem Set 1 Issued: Wednesday, February 11, 2015 Due: Monday, February 23, 2015

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS SPRING 05 Problem Set Issed: Wednesday Febrary 05 De: Monday Febrary 05 Gradng Note: Please provde yor soltons ether as hand-wrtten/hard-copy soltons or by sbmttng va corse webste. MATLAB codes shold be sbmtted va corse webste. The blk of the grades wll be gven to detaled explanatons and to algorthms and nmercal schemes that captre the essence of the nmercal problems. We know that sccessfl codng of nmercal schemes can be tme consmng and prone to small errors. Sch small errors or omssons n a code wll not be heavly penalzed. Problem Do problems.7 and.8 of Chapra and Canale pages Problem Consder the steady heat transfer wthn a thn rectanglar flat plate whose sdes are kept at fxed temperatres. Three sdes of the plate are at 0 o C and the last one at T = 80 o C. The dmensons of the rectanglar plate are a = 5 m and b = 4 m. For the gven bondary condtons T(xy) can be expressed analytcally by Forer seres (Erwn Kreyszg Advanced Engneerng Mathematcs John Wley and Sons 99): x y sn[(n) ] snh[(n) ] 4T T(x y) a a n (n ) b snh[ ( n ) ] a For the followng nmercal comptatons dvde the length a nto 40 segments and the wdth b nto segments. To add term n the seres please se the lmted precson addton fncton radd.m (was presented n lectre and s avalable on corse webste). Page of 7

2 a) Determne and plot (e.g. -D srface plot) the temperatre dstrbton T(xy) wthn the plate sng sgnfcant dgts n the radd fncton n the followng two ways:. Carryng ot the smmaton from n = to n = 0.. Carryng ot the smmaton n the reverse order.e. from n = 0 to n =. b) Determne the dfference n the reslts of these two smmaton procedres and generate a srface plot for ths dfference. Whch method gves more accrate reslts? Why? MATLAB help: Look at meshgrd and srf commands n MATLAB and n the MATLAB revew and rectaton provded on Wednesday 4 Febrary 05. Problem Do problem 4. of Chapra and Canale page 08. Problem 4: Trncaton Error Rond-off Error Order of Accracy and Nosy Data. Ths problem stdes the errors occrrng drng the evalaton of the frst dervatve f '( x) of a gven fncton f (x) sng the forward and central dfference schemes (of corse n real fld problems f wold be nknown). For the example here f (x) = sn(x). Frst we evalate the effect of the dscretzaton step sze h on the estmate of f '( x). a) Set h =0 wth = [-0: 0.5: 0] and x = π 0.. Wrte a short program that evalates the total errors of both the forward and central dfference estmates of f '( x) for each of these vales of h. Plot the reslts.e. plot two total error crves as a fncton of h n a sngle fgre panel. For each case plot also the leadng term of the trncaton (dscretzaton) error. b) Brefly dscss and explan the reslts n terms of h. c) Brefly explan the behavor of the total error at the largest vales of h? (Hnt: whch of the forward or central dfference s most accrate at these vales of h?). Bons (do ths only f yo are nterested t s not needed for cll credt). We now evalate the effects dfferentatng nosy data.e. the effects that small random ncorrelated nose n f( x) have on the vales of dervatve estmates. Assme that f( x) s pertrbed wth nose that s randomly dstrbted zero-mean Gassan wth ampltde of %: n other words the pertrbed f (x) ( 0.0) sn(x) wth N(0). d) Set h = 0.0. For the vales of x = [0: h: π] evalate the pertrbed f( x) and the correspondng forward and central dfference estmates of f '( x). e) Plot the reslts as a fncton of x: e.g. three plots f( x) and then the two f '( x) each overlad on the npertrbed dervatvecos(x). Dscss yor reslts. f) Do the general propagaton error formla and/or the standard error estmate adeqately explan yor reslts n e)? Page of 7

3 g) Select a vale of x and evalate the total error n the forward and central dervatve estmates as h s decreased. Compare yor reslts to the theoretcal trncaton errors (wthot nose added). Problem 5 Do problem 8.4 of Chapra and Canale. Page of 7

4 Problem 6 (ths s a short problem) It s common n comptatons to determne random access memory (RAM) reqrements. a) Calclate the RAM (n megabytes) necessary to store a mltdmensonal array of sze 00 x 00 x 00 x 000 (for example a dscretzed temperatre feld n 4d: d n space and 000 tme steps). Ths array s doble precson and each vale reqres a 64-bt word. Recall that a 64-bt word = 8 bytes and megabyte = 0 6 bytes. b) If yor laptop has 6 Gb of RAM cold yo store the array n a) n yor RAM? For yor nformaton when yor memory reqrements exceed the sze of the avalable RAM the CPU often ses swap space. Swap space s sally a porton of the dsk space sed for sch temporary storage locaton wth the caveat that access to dsk s mch slower than RAM access. Problem 7: (Rond off Errors n Orthonormalzaton) In ths problem we stdy the effect of rond-off errors n a machne drng an orthonormalzaton process. Gven a matrx V( mn) m n wth lnearly ndependent colmns v v v n orthonormalzaton refers to the generaton of orthonormal vectors q q qn each of sze m sch that they span the same n dmensonal sbspace of the nner prodct of two vectors q and T q as q q q. q. j j j m as vectors v.. n. We defne Vectors q q... q n are sad to be orthonormal f they are all orthogonal to each other.e. q q 0 for all j and < j n and have nt magntde.e. q q q j.. n. The classcal Gram-Schmdt (CGS) process s a method for orthonormalzng a gven set of vectors. It works as follows: v q r v r q q r r v r q r q q... n v r q n n n q n r n nn T where r q. v j and r... n. The set of vectors q... n are j j orthonormal and span the same sbspace as v.. n. Page 4 of 7

5 The Modfed Gram-Schmdt (MGS) process s another method for orthonormalzaton of vectors. It s very smlar to the classc Gram-Schmdt process. In ths method the comptaton of q s same as n the classcal Gram-Schmdt above. For k... n q s compted as follows: r v q v r q k k k k k r q r q k k k k k... k k k k r q r q k k k k k k k k k r q k k k k k k rk k k qk. r k k r q k k k k k k Smlar to the classcal Gram-Schmdt orthonormal q s are generated seqentally startng from () () to n. As an example to calclate q one wold frst calclate and sng: Answer the followng qestons: kk r v q v r q r q r q r q. r a. Wrte MATLAB fnctons cgs.m and mgs.m that perform the classcal and the modfed Gram-Schmdt orthonormalzaton respectvely. In partclar yor fnctons shold take a sngle matrx V( mn) m n as the npt and retrn the matrces Q (matrx wth colmns q... n) and R (pper tranglar matrx wth elements r j) as the two otpts. b. Generate the matrx V( m m) followng these steps n MATLAB: rand('state'00) A qr(rand(m)); B qr(rand(m)); S = dag(.^-(:m)); ' V = A*S*B ; c. For each of the followng vales of m: generate the V matrx sng part (b) and compte the correspondng Q R matrces as: [QcgsRcgs] = cgs(v); [QmgsRmgs] = mgs(v); Page 5 of 7

6 As error ndcators we wll se the devaton of Q Q from the dentty I( m m). For each of the above cases compte the errors: ecgs = norm(qcgs'*qcgs - eye(m)); emgs = norm(qmgs'*qmgs - eye(m)); Plot these errors as a fncton of m. What do yo observe? Explan brefly. Is one method speror to the other? d. For m = 00 plot the dagonal entres of Rcgs and Rmgs on a semlog scale.e. for example: semlogy(:mdag(rcgs) o ); What do yo observe? BONUS: Try to explan the behavor yo observe. Problem 8: Root fndng Fnte Dfference and Tme ntegraton. A sgnfcant model of poplaton growth was derved n 88 by Belgan Perre-Francos Verhlst. Hs model elmnated the nrealstc effects of nlmted exponental growth. He sggested that when a poplaton ncreases there s a tendency for ndvdals to compete wth each other and so redce growth. Hs model has come to be known as the logstc growth eqaton. Consder a trtle s poplaton modeled by sch a logstc growth eqaton: x x f ( x t) rx t K where: x s the nmber of trtles n the poplaton K = 700 s the carryng capacty r = 0. s the growth rate a) Usng Taylor seres apply the forward backward and central dfferences to the frst order (tme) dervatve n the above eqaton and derve the correspondng dscrete tmentegraton schemes. Re-arrange the terms sch that the nknown vales (at ftre tme steps) are on the left-hand sde and known vales are on the rght-hand sde. b) The backwards dfference formla reqres the evalaton of the fncton f(xt) at a tme n the ftre where the vale of x s nknown. Ths reslts n a root-fndng problem. Wrte a fncton NewtonRaphson that has a header of the type: Fncton x = NewtonRaphson(f df x0 TOL maxt) Where: f s the fncton handle of the fncton whose root yo reqre df s the fncton handle of the dervatve of f x0 s an ntal gess vale and x the fnal vale TOL s the error tolerance Maxt s the maxmm nmber of teratons A skeleton of ths fncton can be downloaded from the corse webste ste. Page 6 of 7

7 c) Compte the nmber of trtles from tme 0 to tme 0 years sng the forward central and backwards dfference schemes. Use and 40 tme ntervals (that s t 0 / 0 for example). For each of these for tme step cases plot the Poplaton vs. Tme vales of the backward central and forward dfference schemes and of the analytcal solton (all for estmates on the same graph for each tme step case). Dscss the reslts as the nmber of tme ntervals ncrease. Intally we have two trtles and the exact analytcal solton s gven by: rt e xt () rt ( e ) / K We manly provde yo ths so yo can ntalze the tme ntegraton wth the central dfference (yo need two tme-steps to start the scheme). Yo can also se the forward dfference solton to ntalze (whch wold be common practce) ether way s fne. d) Plot the L norm of the error verss the nmber of tme steps sed for the three schemes on the same graph. Use a loglog() plot. For the L norm se: norm(x-xexact)/sqrt(# ntervals+). Here x s a vector contanng the solton at each tmestep. That s x has a length= (#ntervals+). So calclatng the error for forward dfference sng 0 ntervals yo wll get a sngle nmber. e) Determne the order of convergence for each scheme. Do they converge at the expected order? f) For whch scheme and for whch nmber of dvsons do yo obtan the fewest trtles at 00 years? (Do yo thnk yor colleages wold be happy f yo sed ths nmercal solton n yor poplaton growth research?) Page 7 of 7

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