Numerical Methods in Geophysics: Implicit Methods
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1 Numerical Methods i Geophysics: What is a implicit scheme? Explicit vs. implicit scheme for Newtoia oolig rak-nicholso Scheme (mixed explicit-implicit Explicit vs. implicit for the diffusio equatio Relaxatio Methods Numerical Methods i Geophysics
2 What is a implicit method? Let us recall the ODE: d f (, t Before we used a forward differece scheme, what happes if we use a backward differece scheme? O( f (, t f (, t - Numerical Methods i Geophysics
3 What is a implicit method? or τ 0( τ ( Is this scheme coverget? Does it ted to the exact solutio as ->0? YES, it does (exercise Is this scheme stable, i.e. does decay mootoically? his requires 0 < < τ Numerical Methods i Geophysics
4 What is a implicit method? 0 < < τ his scheme is always stable! his is called ucoditioal stability... which does t mea it s accurate! Let s see how it compares to the explicit method... Numerical Methods i Geophysics
5 What is a implicit method? Explicit ustable - implicit stable - both iaccurate.5.4; tau0.7 red-aalytic blue-explicit gree-implicit 0.5 e mpe ra ture ime (s Numerical Methods i Geophysics
6 What is a implicit method? Explicit stable - implicit stable - both iaccurate ; ta u0.7 red-aalytic blue-explicit gree-implicit 0.5 emperature ime (s Numerical Methods i Geophysics
7 What is a implicit method? Explicit stable - implicit stable - both iaccurate ; ta u0.7 red-aalytic blue-explicit gree-implicit e mpera ture ime (s Numerical Methods i Geophysics
8 What is a implicit method? Explicit stable - implicit stable - both accurate ; ta u0.7 red-aalytic blue-explicit gree-implicit e mpera ture It does t look like we gaied much from ucoditioal stability! ime (s Numerical Methods i Geophysics
9 Mixed implicit-explicit schemes We start agai with... d f (, t Let us iterpolate the right-had side to / so that both sides are defied at the same locatio i time... f (, t (, t Let us examie the accuracy of such a scheme usig our usual tool, the aylor series. f Numerical Methods i Geophysics
10 Mixed implicit-explicit schemes... we leared that... ( t O d t d t d t... also the iterpolatio ca be writte as... ( ( 3 t O f d t df t f f f, ( t f d t df d, ( sice > Numerical Methods i Geophysics
11 Mixed implicit-explicit schemes... it turs out that... this mixed scheme is accurate to secod order! he previous schemes (explicit ad implicit were all first order schemes. Now our coolig experimet becomes: ( τ ( τ ( τ leadig to the extrapolatio scheme Numerical Methods i Geophysics
12 Mixed implicit-explicit schemes τ τ How stable is this scheme? he solutio decays if... < τ < τ Numerical Methods i Geophysics
13 Mixed implicit-explicit schemes < τ < τ his scheme is always stable for positive ad τ! If > τ, the solutio decreases mootoically! Let us ow look at the Matlab code ad the compare it to the other approaches. Numerical Methods i Geophysics
14 Mixed implicit-explicit schemes t0. tau.7;.; iput(' Give : '; troud(0/; t0; a(; i(; m(; for i:t, t(ii*; (i(i-/tau*(i; % explicit forward a(iexp(-*i/tau; % aalytic solutio i(i(i*(/tau^(-; % implicit m(i(-/(*tau/(/(*tau*m(i; % mixed ed plot(t,(:t,'b-',t,a(:t,'r-',t,i(:t,'g-',t,m(:t,'k-' xlabel('ime(s' ylabel('emperature' Numerical Methods i Geophysics
15 Mixed implicit-explicit schemes ; ta u0.7 red-aalytic blue-explicit gree-implicit black-mixed emperature ime (s Numerical Methods i Geophysics
16 Mixed implicit-explicit schemes ; ta u0.7 red-aalytic blue-explicit gree-implicit black-mixed 0.5 emperature ime (s Numerical Methods i Geophysics
17 Mixed implicit-explicit schemes ; ta u0.7 red-aalytic blue-explicit gree-implicit black-mixed emperature ime (s Numerical Methods i Geophysics
18 Mixed implicit-explicit schemes emperature ; tau0.7 e mpe ra ture ; ta u ime (s red-aalytic blue-explicit gree-implicit black-mixed ime (s he mixed scheme is a clear wier! Numerical Methods i Geophysics
19 he Diffusio equatio t k x k diffusivity he diffusio equatio has may applicatios i geophysics, e.g. temperature diffusio i the Earth, mixig problems, etc. A cetered time - cetered space scheme leads to a ucoditioally ustable scheme! Let s try a forward time-cetered space scheme... Numerical Methods i Geophysics
20 he Diffusio equatio s( s k where dx how stable is this scheme? We use the followig Asatz e iω X e ikdx after goig ito the equatio with i(ω kmdx m e X m Numerical Methods i Geophysics
21 he Diffusio equatio... which leads to... ( sx s sx 0 ad 4s so the stability criterio is s dx /(k his stability scheme is impractical as the required time step must be very small to achieve stability. Numerical Methods i Geophysics
22 he Diffusio equatio (matrix form ay FD algorithm ca also be writte i matrix form, e.g. ( s is equivalet to J J s Numerical Methods i Geophysics
23 he Diffusio equatio (matrix form... this ca be writte usig operators... c c Lc where L is the tridiagoal scaled Laplacia operator, if the boudary values are zero (blak parts of matrix cotai zeros Numerical Methods i Geophysics
24 he Diffusio equatio... let s try a implicit scheme usig the iterpolatio... ( (, ( t t t x ad ( dx k... so agai we have defied both sides at the same locatio... half a time step i the future... Numerical Methods i Geophysics
25 he Diffusio equatio... after rearragig... s s s s s s / ( ( / ( / ( ( / (... agai this is a implicit scheme, we rewrite this i matrix form... J J s s s s s s / ( ( / ( / ( ( / (... or usig operators... V c c U Numerical Methods i Geophysics
26 he Diffusio equatio... ad the solutio to this system of equatios is... c U V c... we have to perform a tridiagoal matrix iversio to solve this system. Stability aalysis usig the Z-trasform yields (( s s / ( X X (( s s / ( X X ( β /( β β s( cosθ... where we used... cosθ cos k x ( X X / Numerical Methods i Geophysics
27 he Diffusio equatio ( β /( β β s( cosθ... describes the time-depedet behaviour of the umerical solutio, as before we fid... ( β/( β... which meas the solutio is ucoditioally stable this scheme implies that the FD solutio at each grid poit is affected by all other poits. Physically this could be iterpreted as a ifiite iteractio speed i the discrete world of the implicit scheme! Numerical Methods i Geophysics
28 he Relaxatio Method Let us cosider a space-depedet problem, the Poisso s equatio : ( Φ x z F dx applyig a cetered FD approximatio yields... ( Φ Φ Φ Φ Φ F Φ i, i, i, i, i, i, Φ Φ Φ rearragig... Φ i, i, i, i, i, 4 Φ Fdx 4... so the value at (i, is the average of the surroudig values plus a (scaled source term... Numerical Methods i Geophysics
29 he Relaxatio Method Φ... the solutio ca be foud by a iterative procedure... Φ Φ Φ 4 m m m m m i, i, i, i, i, Φ Fdx 4... where m is the iteratio idex. Oe ca start from a iitial guess (e.g. zero ad chage the solutio util it does t chage aymore withi some tolerace e.g. m Φ m Φ < ε i i If there is a statioary state to a diffusio problem, it could be calculated with the time-depedet problem, or with the relaxatio method, assumig d/0. What is more efficiet? (Exercise Numerical Methods i Geophysics
30 - Summary ertai FD approximatios to time-depedet partial differetial equatios lead to implicit solutios. hat meas to propagate (extrapolate the umerical solutio i time, a liear system of equatios has to be solved. he solutio to this system usually requires the use of matrix iversio techiques. he advatage of some implicit schemes is that they are ucoditioally stable, which however does ot mea they are very accurate. It is possible to formulate mixed explicit-implicit schemes (e.g. rak-nickolso or trapezoidal schemes, which are more accurate tha the equivalet explicit or implicit schemes. Numerical Methods i Geophysics
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