Modeling Convection Diffusion with Exponential Upwinding

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1 Appled Mathematcs, 013, 4, Publshed Onlne August 013 ( Modelng Convecton Dffuson wth Exponental Upwndng Humberto C. Godnez *, Valpuram S. Manoranjan # Department of Mathematcs, Washngton State Unverst, Pullman, USA Emal: # mano@wsu.edu Receved Ma 13, 013; revsed June 13, 013; accepted June 0, 013 Coprght 013 Humberto C. Godnez, Valpuram S. Manoranjan. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n an medum, provded the orgnal work s properl cted. ABSTRACT Ths paper shows the usefulness of the exponental upwndng technque n convecton dffuson computatons. In partcular, t s demonstrated that, even when convecton s domnant, f exponental upwndng s emploed n conjuncton wth ether the Jacob or the Gauss-Sedel teraton process, one can obtan computed solutons that are accurate and free of unphscal oscllatons. Kewords: Convecton Domnated Dffuson; Exponental Upwndng; Iteraton Matrx; Convergence; Spectral Radus 1. Introducton Convecton dffuson equatons are commonl used to descrbe a wde varet of phscal phenomena. For example, when one studes how temperature s beng convected b a movng flud [1] or when modelng nsect dspersal n a wnd regon or n descrbng transport of contamnants n groundwater, convecton dffuson models are extremel useful. In man of these phscal processes there s a massve amount of data to be analzed and studed. Therefore, when studng these processes and the assocated convecton dffuson equatons computatonall, t s desrable to mplement methods that are amenable to parallel computng. If we develop a computatonal method dscretzng a convecton dffuson model ether n two or three dmensons, at the end, we are left to solve a lnear sstem of equatons. One could emplo the Jacob or the Gauss-Sedel teratve methods to solve such a lnear sstem of equatons, snce the could be easl adapted to parallel computng. However, there s a problem n emplong ether the Jacob or the Gauss-Sedel teratve method f the convecton terms are ver domnant n the convecton dffuson equaton. Because, n such cases, when carrng out the teratve computatons, unphscal oscllatons wll appear leadng to non convergence of the teratve process. Ths s due to the fact that the teraton matrces of the Jacob and the Gauss-Sedel methods, * Currentl at Los Alamos atonal Laborator, Los Alamos, M, USA. # Correspondng author. n such cases (.e., when convecton s ver domnant) would have spectral rad greater than one [,3]. Thus, volatng the condton for convergence of the teratve process. One wa to deal wth such unwanted oscllatons s b refnng the spatal mesh wdths. However, most of the tme, the refnements have to be so extreme, that t ma not be vable to carrout the computatons. So, one s forced to modf the computatonal method, for example, b usng an upwndng technque, or some other technque to suppress the undesrable oscllatons. There are a varet of upwndng technques to solve convecton dffuson problems [1]. In ths paper, we focus on an exponental upwndng technque and use t n conjuncton wth a fnte element method. Importantl, we stud the effect of exponental upwndng on the spectral rad of the resultng Jacob and Gauss-Sedel teraton matrces. The motvaton for applng an exponental upwndng technque s the followng. Let us consder the one-dmensonal convecton dffuson problem: d u d 0, 0 d u R x L x d (1) x wth u0 ul 0. () Introducng an ntegratng factor, (1) s equvalent to: d Rx du e 0. (3) dx dx Coprght 013 ScRes.

2 H. C. GODIEZ, V. S. MAORAJA 81 Spatal dscretzaton of (3) wll gve e Rh 1 1 e Rh e Rh 1 e Rh U U U (4) wth the followng boundar condtons: U0 0 U L h,andu uh. s the total number of x mesh ponts. Equaton (4) could be vewed as the exponentall upwnded dfference form of Equaton (1). If we solve (1)-() smpl b ntegratng twce, we have the soluton: u C1 C erx, C and 1 C are constants that can be determned b the condtons gven n (). ow, the soluton for the dfference Equaton (4) s: U D D 1 erh, D 1 and D are constants that can agan be determned b the boundar condtons n (). So, t s clear (snce h = x ) that the soluton for the dfferental Equaton (1) and the soluton for the dfference Equaton (4) are exactl the same. Ths means that solvng (3) numercall, usng the exponentall upwnded form (4), wll gve us the exact soluton of (1)-(). Although we know that ths s not the case when dealng wth convecton dffuson equatons n hgher dmensons, we wsh to examne the usefulness of ths tpe of dea (.e., exponental upwndng) and the effect that t wll have on the Jacob and the Gauss-Sedel teratve methods, partcularl, on the spectral rad of the respecttve teraton matrces.. Convecton Dffuson Stead State Model Let us consder the two-dmensonal stead state convecton dffuson model gven b u u Ru f x, x, (5) xx x ux, 0 on, f x, s a gven functon and R, a non-zero parameter. If we appl a fnte element method, we are left wth a sstem of lnear equatons GU f (7) j j G j aj, R j dxd, x x x x x f (6) f, fdxd, and are the basc lnear pecewse functons for 1,,. When R s relatvel small, the sstem of lnear equabe solved easl wth ether the tons gven b (7) can Jacob or the Gauss-Sedel teratve methods. However, as R ncreases (.e., as convecton becomes domnant), the spectral rad of both the Jacob and the Gauss-Sedel teraton matrces wll begn to grow and eventuall become greater than one, gvng us non-convergence (of the teratve processes) n the form of oscllatons [4]. For our computatonal modelng, we choose f x, such that u x, snxsn s the exact soluton of (5)-(6) x, 0,1 0,1 and 1 x h. Fgure 1 shows the osc llatons that 0 are generated when the Jacob teratve process s used wth R = 85. For bgger values of R the oscllatons wll overshadow the soluton completel gvng us extremel large oscllatons. For comparson, Fgure presents the exact soluton of the problem (5)-(6). One wa to avod ths problem s to solve (7) wth an Fgure 1. umercal soluton of (5)-(6) wth R = 85 usng 1000 teratons. Fgure. Exact Soluton of (5)-(6) wth R = 85. Coprght 013 ScRes.

3 8 H. C. GODIEZ, V. S. MAORAJA egenspectrum envelopng technque proposed n [4]. In that paper, the authors stud a two step-jacob and Gauss- Sedel teratve methods b constructng an ellpse, n the complex plane, that envelopes the egenvalues of the teraton matrces. In ths paper, we seek the same objectve the need to elmnate unphscal oscllatons when solvng convecton domnated dffuson problems. However, we wll tr to accomplsh ths b modfng the numercal scheme for the partal dfferental equaton, nstead of modfng the teratve method drectl. Let us rewrte Equaton (5) as x e Rx e Rx u u e Rx f x,. (8) x We are applng the same dea, dscussed n the ntro- ducton, to onl one of the spatal varables, n ths case x. The weak form of (8) s Au, v e Rx f, v e Rx u u Au and v H. H s the x x space of admssble test functons. ow, applng a Galerkn method we get Rx auv, e fv, (9), e Rx x x d d. a u v u v u v x Let,, U x be the Galerkn approxmaton for u(x, ) and U x U x, the are the basc 1 lnear pecewse functons for 1,, and v(x, ) = (x, ). Substtutng U x, n (9) we get the lnear sstem of equatons gven b GU f G j a j, and e Rx f, f. Wth ths new sstem of equatons, we analze the spectral rad of the Jacob and the Gauss-Sedel teraton matrces to determne whether the respectve teratve processes wll converge for large R values. Fgures 3 and 4 show the egenspectra of the respectve teraton matrces when R = 100, a large value. We see that n both cases the spectral rad of the teraton matrces are less than one. Ths means that the teratons wll lead to convergence. Our computatons show that ths s ndeed the case. Also, we do not get an unwanted oscllatons. Fgure 3. Egenspectrum of the Jacob teraton matrx wth R = 100. Coprght 013 ScRes.

4 H. C. GODIEZ, V. S. MAORAJA 83 Fgure 4. Egenspectrum of the Gauss-Sedel teraton matrx wth R = 100. In Table 1 we show the percentage error and the number of teratons for convergence (to the soluton) for dfferent values of R. In all the cases, the number of teratons the Gauss-Sedel method took was less than half the number of teratons the Jacob method took for convergence. 3. Convecton Dffuson Transport Equaton In our prevous model, the convecton was onl present n one spatal varable. In order to further nvestgate the effectveness of ths upwndng technque, we would lke to explore a problem wth convecton on both spatal varables (x and ) and see what effects t has on the egenspectra of the Jacob and the Gauss-Sedel teratve matrces. For ths purpose we analze the followng convecton dffuson transport equaton ut uxx u ux u 0 t T, 0, 0 (10) subject to the followng boundar and ntal condtons: ux,,0 fx,, x, (11) u xt,, g xt,,, x,. Table 1. umber of teratons and assocated error for teratve methods. R Jacob method Gauss-Sedel method Percentage error o. of teratons Percentage error o. of teratons Applng a Galerkn sem-dscrete fnte element method [5], n whch we dscretze onl the spatal varables, we are left wth the followng sstem of ordnar dfferental equatons: BU GU b (1) B G a, j j j j, jdxd, j j j j dd, x x x x and b Wt, aw,. Coprght 013 ScRes.

5 84 H. C. GODIEZ, V. S. MAORAJA W x,, t s a functon that satsfes the boundar condton, and are the basc lnear pecewse functons for 1,,. In order to dscretze n tme, we wll use the Crank- colson method [6,7] and so, (1) becomes: or n1 n n1 n U U U U B G b t t n1 t n U U t B G B G b. (13) Therefore, at each tme step, we need to solve the lnear sstem (13). The stablt of ths method wll depend on the amplfcaton matrx [8] 1 t t B G B G. When and are small compared wth and respectvel, we wll have convergence; however, when the are large (.e., when convecton s domnant), the numercal soluton wll have oscllatons and t wll not converge to the correct soluton. In the latter case, f we tr to solve the sstem of lnear equatons wth the Jacob or the Gauss-Sedel teraton method, the spectral rad of the teraton matrces wll be greater than one. Fgure 5 shows the numercal soluton for 11, and one can clearl see the oscllatons that result from Jacob teratons. In [8] the authors address ths problem b proposng an altern atng drecton mplct method wth exponental upwnd ng. In that paper the dea of applng exponental upwndn g to each spatal varable come s naturall, snce one s solvng the sstem one varable at a tme. So, t s ntutve that the dea ds cussed n the ntroduct on wll lead to good results. Fgure 5. umercal soluton of (10)-(11) usng 1000 teratons wth α = 11 and γ = 11. In ths paper, to avod the oscllatons presented n the computed soluton, we wll rewrte Equaton (10) as: x x x e ut e e u x e e u. x (14) We appl exponental upwndng to both spatal var- weak form of (14) s: ables. The x e ut, v Au, v (15) x x u u Au e e e e, x x uv, uvx d d and v H. H s the space of adms- sble test functons. So the Galerkn form of (15) s x e ut, v au, v (16) x x au, v e uxvx e uvd xd. Let U x,, t be the Galerkn approxmaton for u(x,, t). Snce we have an nhomogeneous Drchlet boundar condton, we choose,,,,, U x t W x t U t x (17) 1 s are the pecewse blnear bass functons for 1,,, and v x,. W x,, t satsfes the boundar condton and we wll nterpolate the boundar condton to nclude t n our Galerkn approxmaton. So we have W x,, t U, 1, 1 x, 1 U, 1 g x, j, t and k, 1 are the pece- ow our wse blnear basc functons on the boundar. Galerkn approxmaton takes the form,,,, 1, 1,. U x t U t x U x 1 1 Substtutng (17) nto (16) we get the followng sstem of ordnar dfferental equatons n terms of t d U t j, U jaj, Wt, aw, j1 dt 0 for 1,, Coprght 013 ScRes.

6 H. C. GODIEZ, V. S. MAORAJA 85 and the ntal condton becomes U 0 c for j 1,, c, j 1,, j j j are the solutons to the sstem j, cj f, for 1,,. j1 Wrtten n matrx form we obtan BU GU b,, a, and W, aw, B G b. j j j j t Applng the Crank-colson method to dscretze wth respect to tme, we have the new sstem of equato ns gven b or n1 n n1 n U U U U B G b t t n1 t n B GU B GU tb. (18) It can be shown that ths method s uncondtonall stable [8]. In order to determne whether the teratve processes could produce a convergng soluton for (18), we analze the spectral rad of the Jacob and the Gauss-Sedel teraton matrces. For the numercal computatons, we consder our equaton on a specfc doman, such that ut uxx u ux u, x,,0t T (19) wth the followng boundar condtons: u, 0, ux, 0, x ux, f x, x f f 0. For ever computaton we take 1, 1. We wll take the stead state soluton to be our ntal condton and we wll nterpolate the boundar condton so that we have W x,, t U, 1, 1 x, 1 U, 1 f x and, 1 are the pecewse blnear basc functo ns on the boundar fo r x. So our Galerkn approxmaton takes the form w,,,, 1, 1,. U x t U t x U x 1 1 Henc e, we have the lnear sstem t n1 t n B GU B G U tb here j j,, j a j, U j, 1 a j, 1,. b j1 B G and Usng separaton of varables [8], t s eas to show that the tme dependent soluton wll tend to a stead state soluton of the form: 1, x u x exp n1 f sexp s cosn 1sds (0) n x an1 snh a cos 1 snh a If f x cos x u x, n n1 4n 4. wth 0, then (0) reduces to x 4 snh exp b n x an1 snh a cos 1 snh n1 n1 n1 (1) b n 4 4 n n1 4 n1 otce that when 0 we get a stead state equaton smlar to (5), so we are nterested n the case 0. For our numercal computatons we wll choose f x such that the stead state so luton conssts of onl the frst two terms of the seres soluton (1). ow, the stead state soluton takes the form. Coprght 013 ScRes.

7 86 H. C. GODIEZ, V. S. MAORAJA u x, 4 x snh exp cosxsnh a1 b 1 snh a1 b 3 cos3xsnh a3 snh a3 gven th at the boundar condton at f x 4 exp snh bcos x bcos 3 x. s We wll take t, and h for all our computa- 5 0 tons. Fgures 6 and 7 show the egenspectra of the Jacob and the Gauss-Sedel teraton matrces respectvel when 10. We can clearl see that the spectral rad are less than one and so, we are sure to have convergence. The graphs n Fgure 8 show the stead state solutons when α = 10, γ = 1, and α = 1, γ = 10 respectvel. Fgures 9 and 10 show the errors n the computed solutons when usng the Gauss-Sedel teraton method wth α = 10, γ = 1, and α = 1, γ = 10 respectvel. The computed solutons wth the Jacob method are exactl the same; the onl dfference s that the Gauss-Sedel method converges a lot faster. Also, the computed solutons do not produce an unphscal oscllatons when usng ether the Gauss-Sedel or the Jacob teraton process. 4. Concluson In ths paper, we are able to demonstrate that exponental Fgure 6. Egenspectrum of the Jacob teraton matrx for α = 10 and γ = 10. Coprght 013 ScRes.

8 H. C. GODIEZ, V. S. MAORAJA 87 Fgure 7. Egenspectrum of the Gauss-Sedel teraton matrx for α = 10 and γ = 10. (a) Fgure 8. Analtcal stead state solutons wth (a) α = 10, β = 1, γ = 1, δ = 1 and (b) α = 1, β = 1, γ = 10, δ = 1. (b) upwndng s an extremel useful technque n convecton dffuson computatons. Even, f convecton s domnant, emplong exponental upwndng helps one to compute the soluton wthout an dffcult. In partcular, we have shown that one could easl use ether the Jacob or the Gauss-Sedel teraton process on the lnear sstem of equatons resultng from an exponentall upwnded scheme and obtan a converged soluton that s free of unwanted oscllatons. Ths s possble because, n the exponentall upwnded case, the spectral rad of the Coprght 013 ScRes.

9 88 H. C. GODIEZ, V. S. MAORAJA correspondng teraton matrces are alwas found to be less than one. Thus, satsfng the condton for convergence of the chosen teraton process. Fgure 9. Error n the computed soluton when usng Gauss- Sedel method wth α = 10, β = 1, γ = 1, δ = 1. REFERECES [1] D. F. Grffths and A. R. Mtchell, On Generatng Upwnd Fnte Element Methods, Fnte Element Methods for Convecton Domnated Flows, Vol. 34, 1979, pp [] G. H. Golub and C. F. Van Loan, Matrx Computatons, The Johns Hopkns Unverst Press, Baltmore, [3] R. S. Varga, Matrx Iteratve Analss, Prentce-Hall, Upper Saddle Rver, 196. [4] V. S. Manoranjan and R. Drake, A Spectrum Envelopng Technque for Convecton-Dffuson Computatons, IMA Journal of umercal Analss, Vol. 13, o. 3, 1993, pp do: /manum/ [5] R. Wat and A. R. Mtchell, Fnte Element Analss and Applcatons, John Wle and Sons, ew York, [6] A. R. Mtchell, Computatonal Methods n Partal Dfferental Equatons, John Wle and Sons, London, [7] K. W. Morton and D. F. Maers, umercal Soluton of Partal Dfferental Equatons, Cambrdge Unverst Press, Cambrdge, 001. [8] V. S. Manoranjan and M. O. Gomez, Alternatng Drecton Implct Method wth Exponental Upwndng, Computers & Mathematcs wth Applcatons, Vol. 30, o. 11, 1995, pp do: / (95)00163-s Fgure 10. Error n the computed soluton when usng Gauss- Sedel method wth α = 1, β = 1, γ = 10, δ = 1. Coprght 013 ScRes.