Chapter 8 Diusio Equatio This chapter describes dieret methods to discretize the diusio equatio 2 t α x 2 + 2 y 2 + 2 z 2 = 0 which represets a combied boudary ad iitial value problem, i.e., requires to prescribe boudary boudary t ad iitial coditios x,y,z,t = 0. 8.1 Explicit methods 8.1.1 Forward time cetered space scheme I oe dimesio ad usig iite dierece the FTCS scheme is The trucatio error is give by E = α x2 2 +1 = + s x 2 L xx s = α/ x 2 s 1 4 6 x 4 + O x 4 The ampliicatio actor is k x g = 1 4ssi 2 2 which yields s < 1/2 or stability. 145
CHAPTER 8. DIFFUSION EQUATION 146 t +1 Schematic: FTCS -1 i-2 i-1 i i+1 i+2 x Figure 8.1: Represetatio o the FTCS scheme. I two dimesios the same approach ca be used to yield k +1 = k + s x x 2 L xx k + s y y 2 L yy k with s x = α x / x 2, s y = α y / y 2 Stability requires s x + s y 1/2. A improved ie poit scheme ca be oud i α x = α y = αad x = y which or +1 k = αl xx k + αl yy k + α2 2 L xx L yy k yields stability or s 1/2. This scheme ca eicietly be implemeted with k = 1 + αl yy k k +1 = 1 + αl xx k 8.1.2 Richardso ad DuFort-Frakel schemes Oe could have the idea that is is more accurate to employ a cetered dierece or the temporal derivative which give the Richardso scheme +1 1 = αl xx = α 2 x 2 1 2 + +1 ad is secod order accurate or the time derivative. However the stability aalysis shows that this scheme is ucoditioally ustable. A small modiicatio o this scheme where the term 2 is
CHAPTER 8. DIFFUSION EQUATION 147 split ito two time levels accordig to 2 = 1 + +1 scheme: leads to the so-called Duort-Frakel +1 1 = αl xx = α 2 x 2 1 1 +1 + +1 Although it appears as i this were implicit it is straightorward to re-arrage terms to yield +1 = 2s 1 + 2s 1 + +1 + 1 2s 1 + 2s 1 This scheme is actually ucoditioally stable. However, carryig out the cosistecy test, i.e., expadig terms i the correspodig Taylor series yields [ t α 2 ] 2 x 2 + α 2 x t 2 + O 2, x 2 = 0 Symbolic represetatio o the DuFort-Frakel scheme. Thus it is ot suiciet to coduct the limit o, x 0 to achieve cosistecy as log as / x remais iite. Usig the relatio 2 α = s x demostrates that cosistecy ca be achieved i s is kept costat ad 0 because x 2 or costat s. The ampliicatio actor or the DuFort-Frakel scheme is 8.1.3 Three-level scheme The geeral method suggest a approach such as g = 2scosk 1 + 2s ± 1 1 4s 1 + 2s 2 + 4s 2 cosk a +1 + b + c 1 dl xx + el xx 1 = 0
CHAPTER 8. DIFFUSION EQUATION 148 Usig this method ad applyig cosistecy yields the equatio 1 1 + γ +1 1 γ [ = α 1 1 βl xx + βl xx 1 ] The error resultig rom this scheme is 1 E = αs x 2 2 + γ + β 1 4 12s x 4 + O x 4 Symbolic represetatio o the the-level scheme. such that this scheme becomes 4th order accurate or β = 0.5 γ + 1/12s. The equatio or the ampliicatio actor is 1 + γg 2 [1 + 2γ + 2s1 βcosk 1] + [γ 2βscosk 1] = 0 The discussio o the parameter space is somewhat complicated but the geeral result is that there is a stability limit with values o s icreasig rom about 0.35 to 5 i γis raised rom 0 to about 6. s 0.5 0.4 ustable stable 0.3 2.0 4.0 6.0 γ Figure 8.2: Illustratio o the stability space or the three level scheme. The trucatio error or the previous methods or dieret grid resolutio is show i the ollowig table. It illustrates that or speciic values o s the simple FTCS ad DuFort-Frakel methods ca achieve 4th order accuracy. 8.1.4 Hopscotch method This is a airly origial method which uses a two stage FTCS algorithm. Here we describe the two-dimesioal variat. I the irst stage which is carried out or + k + = eve the usual FTCS scheme is applied
CHAPTER 8. DIFFUSION EQUATION 149 Table 8.1: RMS errors or dieret explicit schemes ad varyig parameters as a uctio o grid resolutio. Case s γ RMS RMS RMS approx. x = 0.2 x = 0.1 x = 0.05 cov. rate FTCS 1/6 0.007266 0.00049 0.000033 3.9 FTCS 0.3 0.6437 0.1634 0.0413 2.9 FTCS 0.41 1.2440 0.3023 0.0755 2.0 DuF-F 0.289 0.0498 0.00233 0.00012 4.3 DuF-F 0.3 0.0244 0.0136 0.00395 1.8 DuF-F 0.41 0.8481 0.2085 0.0525 2.0 3L-4th 0.3 0.0 0.0711 0.00416 0.00022 4.2 3L-4th 0.3 0.5 0.1372 0.00665 0.00029 4.5 3L-4th 0.3 1.0 0.2332 0.00916 0.00054 4.1 3L-4th 0.41 1.0 0.7347 0.0229 0.00140 4.0 +1 k = αl xx k + αl yy k + k + = eve I the secod stage the same scheme is applied but a ow o all grid poits with +k + = odd ad b the secod order derivative is usig the ewly computed time level + 1. This is possible ow i a explicit method because all grid poits adacet to the oe to be updated have bee updated i the irst stage. 1. Stage schematic: 2. stage schematic: k +1 = αl xx +1 k + αl yy k +1, + k + = odd The Figure above shows a schematic o the time update. The grid topology is show i the ollowig igure. Here the irst stage updates or istace all grid poits idicated i blue. The 2d stage the uses the spatial derivative rom those grid poits to update the orage poits. The resultig patter looks like a chess board. Note that this method is ot oly very eiciet ad simple but also ucoditioally stable i 2 dimesios?. The error associated with the scheme is O, x 2, y 2. 8.2 Implicit methods 8.2.1 Fully implicit scheme This method is equivalet to the FCTS method, however, with the 2d derivative operator evaluated at the ew time level.
CHAPTER 8. DIFFUSION EQUATION 150 y x Figure 8.3: Represetatio o the grid topology used or the Hopscotch method. Cosistecy yields E = 2 1 + 1 4 6s x 4 ad the ampliicatio actor is g = +1 + O x 4 = αl xx +1 k x 1 Schematic o the ully implicit 1 + 4ssi 2 scheme. 2 which demostrates that the scheme is ucoditioally stable. The solutio to this method ca be oud by solvig 1 + 2s s 0 0 0 s 1 + 2s s 0 0 0 s 1 + 2s s 0 0 0 s 1 + 2s s 0 0 0 s 1 + 2s +1 1 +1 2 +1 3 +1 4 +1 5 = d 1 d 2 d 3 d 4 d 5 o rak N correspodig to the umber o grid poits. The tridiagoal system is easily solved usig the Thomas algorithm. 8.2.2 Crak-Nicholso scheme This method uses a mixture o spatial derivative usig time levels ad + 1.
CHAPTER 8. DIFFUSION EQUATION 151 +1 = α 2 L xx + +1 which geerates a error o order E = O 2, x 2. Note that the scalig o 2 i the error is caused by the cetered time derivative. The equatio or the ampliicatio actor is or g 1 + 2sgsi 2 k 2 + 2ssi2 k 2 = 0 Schematic o the iite dierece Crak-Nicholso scheme g = 1 2ssi2 k 2 1 + 2ssi 2 k 2 which implies ucoditioal stability. Fiite elemet Crak-Nicholso Note that this is easily expaded to the iite elemet Crak-Nicholso scheme by applyig the correspodig mass operators to the time derivative term 1 M x +1 = α 1 2 L xx + 1 2 L xx +1 Schematic o the iite elemet Crak-Nicholso scheme The Crak-Nicholso scheme ca also be geeralized i substitutig the actors o 1/2 by a variable parameter i the ollowig maer +1 I this case the method is ucoditioally stable or [ = α 1 βl xx + βl xx +1 ] ad has the restrictio i β > 0.5. s 0 β 1 2 1 21 2β
CHAPTER 8. DIFFUSION EQUATION 152 8.2.3 Geeralized three level schemes Aother geeralizatio is to cosider a weighted time dierecig over three time levels: where the particular choice 1 + γ +1 γ [ ] = α 1 βl xx + βl xx +1 γ = 1 2 β = 1 yields a error E = O 2, x 2 with ucoditioal stability. Fially we ca apply a geeralized mass operator M x = δ,1 2δ,δ which yields +1 1 + γm x γm x [ ] = α 1 βl xx + βl xx +1 Note that the Crak-Nicholso schemes are recovered usig γ = 0, β = 1/2. The FEM Crak-Nicholso scheme is recovered i this case with δ = 1/6. The error or this scheme is 1 E = αs x 2 2 + γ β + δ 1/12 4 s x 4 + O x 4 Schematic o the geeral three level schemes such that the scheme becomes ourth order accurate or The algebraic equatios or this scheme are β = 1 2 + γ + δ 1/12 s a 1 +1 +1 + c +1 +1 = 1 + 2γM x γm x 1 a i = c i = 1 + γδ sβ b = 1 + γ1 2δ + 2sβ L xx = 1 2 + +1 The speciic choice o δ = 1/12 yields ourth order accuracy or β = 1 2 + γ. + 1 βs L xx A summary o the methods implemeted i the program Diim. o implicit schemes is give i the ext table. The ollowig table is a summary o the RMS error obtaied or implicit methods with the program Diim.. All results use s = 1.0
CHAPTER 8. DIFFUSION EQUATION 153 Method - me M x β 1 - FDM-2d order 0, 1, 0 0.5 + γ 2 - FEM-2d order 1/6, 2/3, 1/6 0.5 + γ 3 - FDM-4th order 0,1,0 0.5 + γ 1 12s 4 - FEM-4th order 1/6,2/3,1/6 0.5 + γ + 1 12s 5 - Composite 1/12, 5/6, 1/12 0.5 + γ Table 8.2: Overview o implicit methods implemeted or the oe-dimesioal diusio equatio. Method γ RMS RMS RMS approx. x = 0.2 x = 0.1 x = 0.05 cov. rate 1 0.0 0.3895 0.1466 0.03993 1.9 2 0.0 0.8938 0.1787 0.04185 2.1 3 0.0 0.2393 0.01526 0.00105 3.9 4 0.0 0.2393 0.01522 0.00090 4.1 5 0.0 0.2393 0.01525 0.00103 3.9 1 1.0 2.090 0.03003 0.03245 3.0 2 1.0 1.760 0.2475 0.04668 2.4 3 1.0 2.367 0.1246 0.00813 3.9 4 1.0 1.395 0.0927 0.00591 4.0 5 1.0 1.867 0.1087 0.00710 3.9 Table 8.3: RMS errors or dieret explicit schemes ad varyig parameters as a uctio o grid resolutio. 8.2.4 Boudary coditios Thus ar we have mostly implied Dirichlet boudary coditios which are straightorward to implemet. Vo Neuma coditios provide more o a challege. The most straightorward implemetatio is a oe sided dierece here or the boudary at x mi 2 +1 1 +1 = c +1 x which yields a equatio or the boudary value +1 1. However, this gives oly irst order accuracy while the overall methods usually give at least 2d order accuracy. Better approach: Itroduce artiicial mathematical boudary with = 0 ad 2 +1 0 +1 = c +1 x where c +1 is the gradiet o at the boudary or time t +1. For the FTCS scheme this gives
CHAPTER 8. DIFFUSION EQUATION 154 Similar or the ully implicit method this yields +1 1 = 1 2s 1 + s 2 + 0 = 1 2s 1 + 2s 2 c+1 x 1 + 2s 1 +1 s 2 +1 + 0 +1 = 1 or 1 + 2s 1 +1 2s 2 +1 = 1 2sc+1 x
CHAPTER 8. DIFFUSION EQUATION 155 8.2.5 Summary o methods or the oe-dimesioal diusio equatio Scheme Algebraic equatio Trucatio error Ampl. actor g Stability FTCS +1 = αlxx E = α x2 2 s 1 6 4 x 4 1 4ssi 2 Θ 2 s 1/2 DuFort-Frakel +1 = 2s 1+2s 1 + +1 + 1 2s 1+2s 1 E = α x2 s 2 1 12 4 x 4 2scosΘ+1 4ssi 2 Θ 1/2 1+2s oe Crak-Nicholso +1 = α 2 L xx + +1 E x2 = α 12 4 x 4 1 2ssi 2 Θ/2 1+2ssi 2 Θ/2 oe 3level implicit 3 2 +1 1 2 = αlxx +1 E x2 = α 12 4 x 4 1± 4 i[ 3 16 3 +s1 cosθ]1/2 2[1+ 4 s1 cosθ] oe 3 FEM Crak-Nicholso Mx +1 = α 2 L xx + +1 E = α x2 12 4 x 4 2 3s+1+3s cosθ 2+3s+1 3s cosθ oe Remarks s = α x 2 Lxx = 1 1, 2,1 +1 x 2 = +1 Mx = 1/6,2/3,1/6 = 1
CHAPTER 8. DIFFUSION EQUATION 156 8.3 Splittig schemes 8.3.1 ADI method I two-dimesios implicit methods are usually computatioally very expesive. Cosider the two-dimesioal versio o the ully implicit method. 1 + 2s x + 2s y k +1 s x 1,k +1 + +1,k +1 s y,k 1 +1 +,k+1 +1 = k The problem with this equatio is the iversio o the matrix deied by the rhs o the equatio. I the oe-dimesioal case the resultig matrix was a tridiagoal baded matrix. I the twodimesioal case the matrix is ot aymore baded but has elemets that are ar oset o the diagoal eve though the matrix is sparse. While there are some techiques to deal with spares matrices Gauss elimiatio is still rather ad ote prohibitively expesive. The alterative to a ully implicit solutio is the ADI method which is illustrated usig the ollowig basic equatios. k k /2 α xl xx k α yl yy k = 0 k +1 k α x L xx k /2 α yl yy k +! = 0 Here the i the irst equatio is iterpreted as a auxiliary itermediate time level + 1/2. The correspodig algebraic equatios are cast i the orm 1 2 s x 1,k + 1 + s x k 1 2 s x +1,k = 1 2 s y,k 1 + 1 s y k + 1 2 s y,k+1 1 2 s y +1,k 1 + 1 + s y +1 k 1 2 s y +1,k+1 = 1 2 s x 1,k + 1 s x k + 1 2 s x +1,k Note that these equatios are almost idetical to the oes used or the correspodig elliptic equatio solver usig the ADI scheme 7.4 ad 7.5. These equatios are used i two stages to evolve the system rom time level to time level + 1. Note that - as i the prior itroductio o the ADI scheme - each step requires oly the solutio o a implicit equatio i oe dimesio. I the irst step equatio the system is solved or k cosidered ixed. Sice the solutio is sought oly or the x grid, i.e., oe-dimesioal, the resultig matrices are baded tridiagoal ad easy to solve with the Thomas algorithm. Similarly the secod step is coducted oly or the y grid with x or ixed such that the secod step also ivolves oly oe-dimesio ad thus the solutio o a baded tridiagoal matrix. The vo Neuma stability aalysis is used to determie a ampliicatio actor or each hal step. The product o the resultig ampliicatio actors yields
CHAPTER 8. DIFFUSION EQUATION 157 k+1 k k-1-1 +1-1 +1 Figure 8.4: Schematic o the two stages o the simple iite dierece ADI method. [ ][ ] g = g g 1 2s y si 2 k y/2 1 2s x si 2 k x/2 = 1 + 2s x si 2 k x/2 1 + 2s y si 2 k y/2 which implies g 1 ad thereore ucoditioal stability. The scheme has a error o O 2, x 2, y 2. Note that boudary coditios eed to to be cosidered careully to isure that the global error ideed remais secod order. Usig Dirichlet coditios the evaluatio o the boudary at x max = 1 or the itermediate step usig x,k = b+1/2 k yields a error o O. The correct approach or this boudary should be x,k = 1 2 8.3.2 Geeralized two level scheme b k + b +1 1 k 4 L yy b k + b +1 k As i the case o the oe-dimesioal schemes splittig schemes ca easily be geeralized by itroducig weights or the spatial ad temporal derivatives at dieret time levels. For two time levels this geeralizatio is +1 k = 1 β[α x L xx + α y L yy ] k + β [α xl xx + α y L yy ] +1 k 8.1 agai with k +1 = k +1 k +1. Oe ca rewrite this as a equatio or k terms to the let side i the ollowig maer by movig the β [1 β α x L xx + α y L yy ] +1 k = α x L xx + α y L yy k
CHAPTER 8. DIFFUSION EQUATION 158 Up to O 2 this ca be rewritte as 1 βα x L xx 1 βα y L yy +1 k = α x L xx + α y L yy k where we have added a term β 2 2 α x α y L xx L yy k +1 which however is O 2. With the deiitio k = 1 βα yl yy k +1 oe obtais the ollowig two stage scheme: 1 1 βα x L xx k = α xl xx + α y L yy k 2 1 βα y L yy k +1 = k Each o these steps oly ivolves the iversio o a tridiagoally baded matrix. Similar to the oe-dimesioal equivalet the resultig scheme is ucoditioally stable or β 0.5 also i 3D ad the resultig error is o the order O 2, x 2, y 2 or β = 0.5. Note that a extesio to three time levels is straightorward as i prior examples o oe-dimesioal schemes 1 + γ k +1 γ k [ ] = 1 β α x L xx k + α yl yy k [ ] +β α x L xx k +1 + α y L yy k +1 yields with some mior algebra 1 2 1 β 1 + γ α xl xx k = 1 + γ α xl xx + α y L yy k + γ 1 + γ k 1 β 1 + γ α yl yy k +1 = k Agai this scheme requires oly a oe-dimesioal implicit solutio at each stage. 8.3.3 Fiite elemet methods Similar to oe-dimesioal implicit schemes it is straightorward to exted two or three-dimesioal implicit schemes to iite elemets. I geeral the iclusio o iite elemets ca be doe usig a mass operator such that the diusio equatio i operator orm becomes M x M y t = α x M y L xx k + α y M x L yy k k
CHAPTER 8. DIFFUSION EQUATION 159 k+1 k k-1-1 +1-1 +1 Figure 8.5: Schematic o the two stages o the two level iite dierece ADI method. where this operator or liear iite elemets is give by M x = 1/6,2/3,1/6 ad as usual the secod derivative operator is L xx = 1 x 2 1, 2,1. Aother way to express the operatio is M x k = +1 k+1 m i ik, M y k = m m mk i= 1 m=k 1 Applyig the mass operator i combiatios with the secod derivative operator yields or istace M y L xx k = 1 6 L xx,k 1 + 2 3 L xx,k + 1 6 L xx,k+1 Note that oe ca urther geeralize the approach by usig istead o the em mass operator a operator deied as M x = δ,1 2δ,δ. Usig the mass operator or the two level scheme 8.1 yields k +1 M x M y [ = α x M y L xx + α y M x L yy 1 β k + β k +1 Similar to the iite dierece two level scheme oe ca re-arrage terms ad add a term o order O 2 to yield ] M x βα x L xx M y βα y L yy +1 k = M x + 1 βα x L xx M y + 1 βα y L yy k 8.2 Note that the terms proportioal to 2 have bee added compared to the origial equatio which limits the accuracy to O 2 which, however was ayhow the accuracy o the origial equatio. Similar to the iite dierece approach oe ca ow solve this equatio i two stages
CHAPTER 8. DIFFUSION EQUATION 160 1 M x βα x L xx k = M y + 1 βα y L yy k 8.3 2 M y βα y L yy +1 k = M x + 1 βα x L xx k 8.4 Here oe ca easily that this set o equatios is idetical to 8.2 by multiplyig the secod stage equatio with M x βα x L xx ad isert k rom the irst stage. Note that the choice o β = 0.5 yields the ADI iite elemet method. The two level iite elemet scheme with M x = δ,1 2δ,δ is ucoditioally stable or or x = y ad α x = α y. β 0.5 + δ 0.25 s Fially ote that the iite elemet method ca also easily be applied to a liear versio computig k +1 istead o the equatios 8.3 ad 8.4. Equatio 8.2 is easily re-writte as M x βα x L xx M y βα y L yy k +1 = α x M y L xx + α y M x L yy k R k which with the deiitio o k = M y βα y L yy k +1 leds itsel to the ollowig two stage splittig scheme 1 M x βα x L xx k = R k 2 M y βα y L yy k +1 = k k+1 k k-1-1 +1-1 +1 Figure 8.6: Schematic o the two stages o the two level iite elemet ADI method. The scheme is basically idetical with the oliear ormulatio. The advatage o the liear ormulatio is usually a better accuracy. The ollowig igure shows a schematic o the iite
CHAPTER 8. DIFFUSION EQUATION 161 elemet scheme. Dieret rom the iite dierece method the irst stage ivolves all odes i the viciity o, k. The overall umerical eort o the various splittig schemes i two dimesios is very comparable. Although the iite elemet method ivolve some more odes or the 1st stage step the dierece with iite dierece methods is ot sigiicat. Thus the evetual choice is more determied by properties o thee resultig schemes. Fially it is worth poitig out that oe area o icreased complexity is that o boudary coditios. These eed to be ormulated i agreemet with the discretizatio ad the solutio method. Thus choice o particular boudary coditios alters the algebraic equatios at the boudaries ad must be take ito accout i the solutio o the tridiagoal matrices or splittig schemes.