Numerical Methods in Geophysics: High-Order Operators
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- Drusilla Sparks
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1 : High-Order Opertors Why do we eed higher order opertors? Tylor Opertors Oe-sided opertors High-order Tylor Extrpoltio Ruge-Kutt Method Truted Fourier Opertors - Derivtive d Iterpoltio Aury o high-order shemes derivtives
2 Why do we eed higher-order opertors? For relisti prolems the irst d seod order methods re ot urte eough. So r we oly used iormtio rom the erest eighourig grid poits. Could we improve the ury o the derivtive opertors y usig more iormtio (o oth sides? Legth o opertor Derivtive g g g g g Derivtive
3 Tylor Opertors... like so ote we look t Tylor series... Rememer how we derived the seod-order sheme ' dx ' dx ( ( ' dx the solutio to this equtio or d leds to system o equtio whih e st i mtrix orm Iterpoltio Derivtive / dx
4 Tylor Opertors... i mtrix orm... Iterpoltio Derivtive... so tht the solutio or the weights is... dx / dx /
5 Tylor Opertors... d the result... Iterpoltio Derivtive / / dx C we geerlise this ide to loger opertors? Let us strt y extedig the Tylor expsio eyod (x±dx:
6 Tylor Opertors * * * *d ( x ( x dx dx (dx ( dx ' ' (dx! ( dx! '' '' ( dx ( x dx ( dx ' ''! (dx! ( dx! ( dx! (dx ( x dx (dx ' ''! ''' (dx! ''' ''' '''... gi we re lookig or the oeiiets,,,d with whih the utio vlues t x±(dx hve to e multiplied i order to oti the iterpolted vlue or the irst (or seod derivtive!... Let us dd up ll these equtios like i the previous se...
7 Tylor Opertors d ( d ( ' d dx ( ' ' d dx 8 8 ( ' ' ' d dx... we ow sk or the oeiiets,,,d, so tht the let-hd-side yields either,,,...
8 Tylor Opertors... i you wt the iterpolted vlue... d d d 8 8 d... you eed to solve the mtrix system...
9 Tylor Opertors... Iterpoltio... 8/ / / 8/ / / d... with the result ter ivertig the mtrix o the lhs... / / / / d
10 Tylor Opertors... irst derivtive... / 8/ / / 8/ / / dx d... with the result... / 4/ 4/ / dx d
11 Tylor Opertors... third derivtive... / 8/ / / 8/ / / dx d... with the result... / / dx d... why did it ot work or the d derivtive?
12 Tylor Opertors * * 4 (dx (dx (dx ( x dx (dx ' '' '''!! 4! 4 ( dx ( dx ( dx ( x dx ( dx ' '' ''' ''''!! 4! '''' * ( x *d *e ( dx ( dx ( x dx ( dx ' '' '''!! ( dx 4! (dx (dx ( x dx (dx ' '' '''!! 4 '''' (dx 4! 4 ''''... ote tht we hd to dd the 4th derivtives whih will give us the required ostrits o the oeiiets,,,d,e
13 Tylor Opertors e d ( e d ( ' e d dx ( ' ' e d dx 8 8 ( ' ' ' e d dx 4 4 ( ' ' ' ' 4 e d dx... so illy we ed up with the system...
14 Tylor Opertors... i you wt the seod derivtive... e d e d e d 8 8 e d 4 4 e d... you eed to solve the mtrix system ould we id iterpoltio weights like this?
15 Tylor Opertors... seod derivtive... / / 4 / 4 / / 8/ / / 8/ / / dx e d... with the result... / 4/ 5/ 4/ / dx e d
16 Tylor Opertors... Forerg (99 gives losed-orm expressio or the irst derivtive weights... w ( j( p/ j ( p/! j!( p/ p, j j! i j,,..., p/ i j... where p(eve is the order o ury, j is the x-positio o the weight. Forerg, B., A prtil guide to pseudospetrl methods, Cmridge Uiversity Press.
17 Oe-sided opertors Beore we look t how the opertors look like s they grow loger we ivestigte whether we pproximte derivtive er physil oudry... derivtive domi oudry Let us ollow the sme route s eore d use Tylor series. Let s strt with irst order sheme.
18 Oe-sided opertors... so we hve to look or iormtio o oe side oly ' dx ' ( dx ( ( ' dx the solutio to this equtio or d leds to system o equtios whih e st i mtrix orm Derivtive / dx... d the solutio is...
19 Oe-sided opertors dx This is our well kow deiitio o the etered derivtive, ut it will e deied ot right t the oudry ut dx/ wy rom it! Derivtive Let us exted this to the right d id higher-order opertors
20 Oe-sided opertors * * ( dx ' ( dx! '' ( dx! ''' * (dx ' (dx! '' (dx! ''' *d (dx ' (dx! '' (dx! '''... gi we multiply y our oeiiets d dd everythig up...
21 Oe-sided opertors d ( d dx ' ( d 9 dx ' ' ( d 4 7 dx ' ' ' ( d... to oti the derivtives we hve to solve the system...
22 Oe-sided opertors... i you wt the irst derivtive... d d 9 d 4 7 d... you eed to solve the mtrix system...
23 Oe-sided opertors... Iterpoltio... / 7 / 4 / / 9 / / dx d... with the result ter ivertig the mtrix o the lhs... / / / dx d
24 Tylor Opertors.5 Order o ury:.5 Order o ury: 4.5 -poit.5 4-poit Order o ury: 8.5 Order o ury:.5 8-poit.5 -poit
25 Tylor Opertors - oe sided 4 -poit 5-poit poit 8-poit Note the explodig oeiiets with iresig opertor legth
26 Tylor Opertors - Summry Fiite-dieree opertors with high-order ury e derived usig Tylor series. For two-sided opertors the oeiiets rpidly derese. For oe-sided opertors the oeiiets get lrger with iresig opertor legth. Now tht we improved the ury o the spe derivtives, how we improve the ury o the time extrpoltio? Let s look t the Tylor sheme...
27 High-order Tylor extrpoltio Let us look t the ousti wve equtio p ( p x z S.. we ow kow how to urtely lulte the r.h.s. o this equtio... our stdrd FD sheme or the time extrpoltio yields p( t dt p( t p( t dt dt p... extedig this to higher orders leds to the sheme... p( t dt p( t p( t dt N dt p (!... this hs iterestig osequees s we oly eed the eve orders o the time derivtive, whih we esily lulte...
28 High-order Tylor extrpoltio... sie... S p p z x t (... we hve lso... S p p t t z x t 4 (... or... S p p t t z x t 4 4 (... so we loop through our lgorithm s log s we wt (N times to hieve higher-order ury i the time-extrpoltio sheme... N p dt dt t p t p dt t p! ( ( ( (.. however we hve to e reul how the sptil d temporl opertors ehve d whether the ury o the solutio to the pde tully improves!
29 High-order extrpoltio... ote we hve to extrpolte irst order system... t T ( T, t... or... t u ρ( x x τ... d we iitilly used simple sheme like... T T dt ( T, t j j j j... this sheme is lso kow s the Euler sheme d is o little prtil use...
30 High-order extrpoltio... how out preditig vlue d the vergig... * T j Tj dt ( T j, t j this is our irst guess (equivlet to the Euler sheme d ow we use this vlue to improve our solutio... T [ (, (, ] * T t T t j Tj dt j j j j
31 High-order extrpoltio... ledig to geerl lgorithm like... For,,,,...N- Ed x y k k x dx dx y dx ( x ( x, y, ( k y k k preditor orretor lled preditor-orretor or modiied Euler or... sheme how does this pply to our oolig prolem?
32 High-order extrpoltio (, (, ( k k y y k y x dx k y x dx k dx x x For,,,,...N- Ed T /τ dt dt T dt T T τ... this ws the simplest sheme... with the modiied Euler sheme we get ( ( k k T T k T dt k T dt k dt t t τ τ For,,,,...N- Ed
33 High-order extrpoltio... the ext more urte sheme is the ourth order Ruge-Kutt method, extesio o the preditor-orretor sheme... (, ( /, ( /, (, ( 4 4 / / k k k k y y k y x dx k k y x dx k k y x dx k y x dx k dx x x For,,,,...N- Ed
34 High-order extrpoltio... Mtl smple ode... or i:t, t(ii*dt; T(iT(i-dt/tu*T(i; % Euler T(iexp(-dt*i/tu; % Alytil solutio Ti(iT(i*(dt/tu^(-; % impliit Tm(i(-dt/(*tu/(dt/(*tu*Tm(i; % mixed impliit-expliit k-dt/tu*te(i; k-dt/tu*(te(ik; Te(iTe(i/*(kk; % preditor-orretor k-dt/tu*tr(i; k-dt/tu*(tr(ik/; k-dt/tu*(tr(ik/; k4-dt/tu*(tr(ik; Tr(iTr(i/*(k*k*kk4; % Ruge-Kutt ed... with the results...
35 High-order extrpoltio Compriso o low order impliit, mixed impliit-expliit (Crk-Niholso, modiied Euler (preditor-orretor, Ruge-Kutt (ourth order or Newtoi Coolig dt; t u.7 Temperture.5 lue - Euler red - Crk-Niholso lk - lyti solutio mget - Ruge-Kutt gree - impliit Ruge-Kutt is the wier! Time (s
36 Temperture High-order extrpoltio Compriso o low order impliit, mixed impliit-expliit (Crk-Niholso, modiied Euler (preditor-orretor, Ruge-Kutt (ourth order or Newtoi Coolig dt.5; tu.7 lue - Euler red - Crk-Niholso lk - lyti solutio mget - Ruge-Kutt gree - impliit..5.5 Time (s
37 Fourier Coeiiets We will ow pproh the prolem o idig high-order spe opertors rom ompletely dieret viewpoit: Fourier Itegrls. Let us rell ( x F( k π F( k e ( x e ikx dx ikx dk... where (x is ritrry utio d F(k is its Fourier spetrum. Note tht there re severl dieret deiitios, whih distiguish themselves through ormlistio ostts d the sig ovetio i the expoet.
38 Fourier Coeiiets... how we express the derivtive o utio usig these expressios? x ( x x F( k e ikx dk ikf( k e ikx dk... euse F(k lerly does ot deped o x. Let us deie... P( k ik... ote tht we use pitl letters to deote ields i the wveumer domi...
39 Fourier Coeiiets... so tht... x ( x P( k F( k... ow ell rigs... d we rememer the Covolutio Theorem whih sys multiplitio i the wveumer domi is ovolutio i the spe domi whih e expressed s e ikx dk x ( x p( x x' ( x'... ote the smll letters s we re ow i the spe domi! dx I the disrete d d-limited world this itegrl turs ito ovolutio sum. This is the most geerl wy o desriig dieretil opertor. It omprises ll the ses rom two-poit, lol opertors up to the ext spetrl opertor.
40 Fourier Coeiiets.. we ow wt to express the opertor p(x i the spe domi... we irst hve to get rid o the iiities s we re i disrete domi where we hve mximum rqeuey (wveumer, the Nyquist requey. This d-limittio e expressed usig Heviside utios. H(x i x> i x< i our exmple the limittio i k e expressed s ( H ( k k H ( k k P( k ik Ny Ny we ow hve to trsorm this k ito the spe domi p( x ikx P( k e dk
41 Fourier Coeiiets... to oti... p( x x Ny Ny Ny π x [ si( k x k xos( k ]... i stggered sheme we eed to disretise spe like... x / k ( / dx Ny π / dx... ledig to... p( x ( π (( / dx
42 Fourier Coeiiets p( x ( π (( / dx... these re our dieretil weights
43 to shorte the opertor we tper with Gussi utio
44 Fourier Coeiiets - Iterpoltio... the sme pproh e pplied to the prolem o iterpoltio... ( x dx/ F( k e ik( x dx/ dk F( k e ikdx/ e ikx dk F( k I( k e ikx dk I ( k e ikdx / i.e. I(k is ow our iterpoltio opertor expressed i the wveumer domi. Agi we re lookig or the equivlet represettio i the spe domi, whih we get y iverse Fourier Trsorm
45 Fourier Coeiiets - Iterpoltio... i the d-limited world our opertor is... I ( k ikdx e / ( H ( k k H ( k k... whih i the spe domi yields... Ny Ny i( x si( k Ny ( x dx/ π ( x dx/ disretisig with x / k ( / dx Ny π / dx we oti i( x ( πdx( /
46 Fourier Coeiiets - Iterpoltio
47 Fourier Coeiiets - Iterpoltio the il 8-poit opertor
48 - Aury... s metioed erlier the derivtive opertor i the wveumer domi is x ( x x F( k e ikx dk ikf( k e ikx dk P( k ik whih i the spe domi led to the ovolutiol opertor p(x lookig like ext shorteed p(x.4... ~ p ( x
49 - Aury... this suggests tht we ow Fourier trsorm our shorteed opertor d ompre it with the ext oe i the k-domi ~ FFT( ~ p ( k ik whih lso mes tht we ow hve umeril wveumer s utio o the ext wveumer we propgte i our system..5 Ext derivtive opertor Im ( P(k % Nyquist
50 - Aury... we ow ompre some o our high-order Tylor opertors with the ext opertor i the wveumer domi... Poits poits 4 k-domi Mx rel.err < 4/5 Nyquist : % Nyquist Cum error < 4/5 Nyquist : % Nyquis t
51 - Aury... we ow ompre some o our high-order Tylor opertors with the ext opertor i the wveumer domi... 4 Poits poits 4 k-domi Mx rel.err < 4/5 Nyquist :. 5 Cum error < 4/5 Nyquist : % Nyquist -. 5 % Nyquis t
52 - Aury... we ow ompre some o our high-order Tylor opertors with the ext opertor i the wveumer domi... 8 Poits poits 4 k-domi Mx rel.err < 4/5 Nyquist : Cum error < 4/5 Nyquist : % Nyquist % Nyquis t
53 - Aury... we ow ompre some o our high-order Tylor opertors with the ext opertor i the wveumer domi... poits Poits 4 k-domi - - Mx rel.err < 4/5 Nyquist : Cum error < 4/5 Nyquist : % Nyquist % Nyquis t
54 Fourier Coeiiets Summry Fiite-dieree opertors e regrded s (truted spetrl (glol opertors i geerl wy. I pseudo-spetrl methods, the spe derivtives re lulted i the wveumer domi. The legth o the opertor determies its ury. FD improvig ury Spetrl iresig legth o opertor x
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