Numerical Solution of two dimensional coupled viscous Burgers Equation using the Modified Cubic B-Spline Differential Quadrature Method

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1 umercal Soluton of two dmensonal coupled vscous Burgers Equaton usng the odfed Cubc B-Splne Dfferental Quadrature ethod H. S. Shukla 1, ohammad Tamsr 1*, Vneet K. Srvastava, Ja Kumar 3 1 Department of athematcs & Statstcs, DDU Gorakhpur Unversty, Gorakhpur-73009, Inda ISRO Telemetry, Trackng and Command etwork (ISTRAC), Bangalore , Inda 3 ISRO Satellte Center (ISAC), Bangalore , Inda ABSTRACT In ths paper, a numercal soluton of the two dmensonal nonlnear coupled vscous Burgers equaton s dscussed wth the approprate ntal and boundary condtons usng the modfed cubc B-splne dfferental quadrature method (CB-DQ). In ths method, the weghtng coeffcents are computed usng the modfed cubc B-splne as a bass functon n the dfferental quadrature method. Thus, the coupled Burgers equatons are reduced nto a system of ordnary dfferental equatons (ODEs). An optmal fve stage and fourth-order strong stablty preservng Runge Kutta (SSP-RK54) scheme s appled to solve the resultng system of ODEs. The accuracy of the scheme s llustrated va two numercal examples. Computed results are compared wth the exact solutons and other results avalable n the lterature. umercal results show that the CB-DQ s effcent and relable scheme for solvng the two dmensonal coupled Burgers equaton. Keywords: Burgers equaton; Reyonlds number; odfed cubc B-splne functon; CB- DQ; SSP-RK Introducton Consder the two dmensonal nonlnear unsteady coupled vscous Burgers equatons: u 1 u u v u u u, (1.1) t x y Re x y u v t x y Re x y v v v 1 v v wth the ntal condtons:, (1.)

2 ,,0 1, ;,,,0, ;, u x y x y x y, v x y x y x y (1.3) and Drchlet boundary condtons: u x, y, t x, y, t; x, y vx, y, t x, y, t; x, y where, :, u x, y, t and,,, t 0 (1.4) x y a x b c x d s the computatonal doman and s ts boundary, v x y t are the velocty components to be determned, 1,, and are u u known functons, s unsteady term, u t s the nonlnear convecton term, 1 u u x Re x y s the dffuson term, and Re s the Reynolds number. Coupled vscous Burgers equaton s a more approprate form of the aver-stokes equatons havng the exact solutons. It has the same convectve and dffuson form as the ncompressble aver-stokes equatons, and s a smple model for understandng of varous physcal flows and problems, such as hydrodynamc turbulence, shock wave theory, wave processes n thermo-elastc medum, vortcty transport, dsperson n porous meda 1-3. umercal soluton of Burgers equaton s a natural and frst step towards developng methods for the computaton of complex flows. Thus, t has become customary to test new approaches n computatonal flud dynamcs by mplementng novel and new approaches to the Burgers equaton yeldng n varous fnte-dfferences, fnte volume, fnte-element and boundary element methods etc. Analytc soluton of two dmensonal coupled Burgers equatons was frst gven by Fletcher 4 usng the Hopf-Cole transformaton. The numercal soluton of coupled Burgers equatons are numercally solved by many researchers In recent years, varous researchers 16- proposed varant of dfferental quadrature method for the numercal soluton 3, 4 of the one and two dmensonal lnear/nonlnear dfferental equatons. Korkmaz & Dag proposed cubc B-splne and snc dfferental quadrature methods. Arora & Sngh 5 proposed the modfed cubc B-splne dfferental quadrature method (CB-DQ) and appled on one dmensonal Burgers equaton to checked ts effcency and accuracy. They found that CB- DQ s very powerful and effcent scheme as compared to other exstng numercal methods. Recently, an extenton of the CB-DQ s proposed by Jwar & Yuan 6 to show the computatonal modelng of two-dmensonal reacton dffuson Brusselator model wth approprate ntal and eumann boundary condtons. Researchers 7-9 developed an optmal

3 strong stablty preservng (SSP) hgh order tme dscretzaton schemes. Strong stablty propertes of SSP methods s preserved n any norm, sem norm or convex functonal of the spatal dscretzaton coupled wth the frst order Euler tme steppng. A descrpton of the optmal explct and mplct SSP Runge-Kutta and multstep methods s also dscussed by the authors. In ths paper, we study the numercal smulaton of the two-dmensonal unsteady nonlnear coupled vscous Burgers equatons for dfferent Reynolds number. The effcacy and accuracy of the method s confrmed by takng two test problem wth sutable ntal and boundary condtons. Ths study shows that the CB-DQ results are acceptable and n good agreement wth the exact solutons and earler results avalable n the lterature. Rest of the artcle s prepared as: In Secton, the modfed cubc B-splne dfferental quadrature method s ntroduced. In Secton 3, the mplementaton procedure for the problem (1.1) (1.) together wth the ntal condtons (1.3) and boundary condtons (1.4) s llustrated; In Secton 4, two test problems are gven to establsh the applcablty and accuracy of the method, whle the Secton 5 concludes our study.. odfed cubc B-splne dfferental quadrature method In 197, Bellman et al. 16 ntroduced dfferental quadrature method (DQ). Ths method approxmates the spatal dervatves of a functon usng the weghted sum of the functonal values at the certan dscrete ponts. In DQ, the weghtng coeffcents are determned usng several knds of test functons such as splne functon 3, snc functon 4, Lagrange nterpolaton polynomals, Legendre polynomals 17- etc. Ths secton revsts the CB- DQ 5-6 n order to complete our problem n two dmenson. It s assumed that the and grd ponts: a x,... 1 x x b and c y,... 1 y y d are unformly dstrbuted wth the spatal step sze x x 1 x and y y 1 y n x and y drectons, respectvely. (keepng The frst and second order spatal partal dervatves of u x, y, t wth respect to x y as fxed) and wth respect to y (keepng x as fxed), approxmated at x and y, respectvely, are defned as:,, 1 k k u x y t x k1 w u x, y, t, 1,,..., (.1)

4 ,, k k u x y t x k1,, 1 k u x y t y k1 w u x, y, t, 1,,..., w u x, y, t, 1,,..., k (.) (.3),, k u x y t y k1 w u x, y, t, 1,,...,. k (.4) In the same way, the frst and second order spatal partal dervatves of vx, y, t wth respect to x and wth respect to y are approxmated as:,, 1 k k v x y t x k1 w v x, y, t, 1,,...,,, k k v x y t x k1,, 1 k v x y t y k1,, k v x y t y k1 w v x, y, t, 1,,..., w v x, y, t, 1,,..., k w v x, y, t, 1,,...,, k (.5) (.6) (.7) (.8) where w r and r w, r 1, are the weghtng coeffcents of the rth-order spatal partal dervatves wth respect to x and y. The cubc B-splne bass functons at the knots are defned as: m x x 3 xm, x xm, xm 1 x 3 3 xm x xm 1 x xm1 xm 3,, 1 4,, 0, otherwse, 3 h xm x x xm 1 xm (.9) where,,...,, s chosen n such a way that t forms a bass over the doman x, y : a x b; c y d. The values of cubc B-splnes and ts dervatves at the nodal ponts are depcted n Table 1.

5 Table 1: Coeffcents of the cubc B-splne m and ts dervatves at the node x m. m x xm xm 1 x m xm 1 xm m x 0 3/h 0 3/h 0 m x 0 6/h 1/ h 6/h 0 ow, to get a dagonally domnant system of the lnear equatons, the cubc B-splne bass functons are modfed as 5 : x x x x x x m 1 x 1 x 1 x x x x m x x for m 3,..., -, 1 where forms a bass over the doman,,..., 1 m x, y : a x b; c y d. (.10) In Eq. (.1), substtutng the values of ( x ), m 1,,...,, we get a system of lnear equatons: (1) x w x, 1,,...,. (.11) m k m k k1 Wth the help of Eq. (.9), (.10) and Table 1, Eq. (.11) reduces nto a trdagonal system of equatons: 1 Aw R, for 1,,...,, where w w, w,..., w 1 T (.1) s the weghtng coeffcent vector correspondng to x, 1,,,,......, 1,,, T R, and the coeffcent matrx A s gven by: 1,1 1,,1,,3 3, 3,3 3,4 A, 3,, 1 1, 1, 1 1,, 1,

6 Here, we pont out that the coeffcent matrx A s nvertble. The trdagonal system of lnear equatons (.1) s solved for each usng Thomas algorthm, whch gves the weghtng coeffcents w 1 k of the frst order partal dervatve. In a smlar way, the weghtng coeffcents w,1, are determned. Weghtng coeffcents w,1,, can be computed as : r1 r 1 r1 w w r w w, for and 1,,3,..., ; r,3,..., 1 x x r r w w, for, 1, (.13) where r 1 w and r w are the weghtng coeffcents of the th r 1 and th r order partal dervatves wth respect to x. Smlarly, the weghtng coeffcents w 1 k of the frst order partal dervatves wth respect to y usng the modfed cubc B-Splne functons n Eq. (11) s obtaned. Weghtng coeffcents w,1, for the second dervatves can be computed from the formule: r1 r 1 r1 w w r w w, for and 1,,3,..., ; r,3,..., 1 x x r r w w, for, 1, (.14) where r 1 w and r w are the weghtng coeffcents of the th r 1 and th r order partal dervatves wth respect to y. 3. CB-DQ for two-dmensonal coupled Burgers equaton On substtutng the approxmate values of the spatal dervatves computed by the CB- DQ, Eq. (1.1) can be wrtten as: u x, y, t t (1) (1),,,, k k k k u x y w u x y v x y w u x y k1 k1 1 () (),,,,, 0, 1,,...,, 1,,...,. k k k k Re w u x y w u x y x y R t k1 k1 (3.1) Smlarly, Eq. () can be wrtten as:

7 u x, y, t t (1) (1),,,, k k k k u x y w v x y v x y w v x y k1 k1 1 () (),,,,, 0, 1,,...,, 1,,...,. k k k k Re w v x y w v x y x y R t k1 k1 (3.) Eq. (3.1) and Eq. (3.) reduce nto a system of ordnary dfferental equatons: du x, y, t dt dv x, y, t dt 1 F u x, y, t, 1,,..., and 1,,...,. (3.3) F u x, y, t, 1,,..., and 1,,...,. (3.4) Eqs. (3.3) and (3.4) together wth the ntal condtons (1.3) and Drchlet boundary condtons (1.4) are solved by SSP-RK54 scheme. 4. Results and dscusson Here we consder two test problems of two dmensonal coupled Burgers equaton as gven n the ntroducton part to provde the CB-DQ numercal solutons. The accuracy and consstency of the scheme s measured n terms of error norms L and L, defned as: n n exact computed : exact computed,, 0 0 exact computed uexact ucomputed u, u,, L u u u u L := max where uexact and u computed represent the exact and computed solutons, respectvely., (4.1) problems. umercal solutons to Eqs. (1) and () wll be tested for the followng two test 4.1. Problem: The analytcal solutons of Eqs. (1.1) and (1.) can be generated as 4 : 3 1 u x, y, t 4 4 1exp4x 4y tre/ 3 x,y, 3 1 v x, y, t exp4x 4y tre/ 3 (4.)

8 The square doman x, y: 0 x 1,0 y 1 and the ntal and boundary condtons for,, s consdered as the computatonal doman, u x y t and vx, y, tare taken from the analytcal solutons Eq. (4.). 4.. Problem: In ths problem, we take the computatonal doman x, y :0 x 0.5,0 y 0.5 wth the ntal condtons:,,0 sn cos,,0 u x y x y x,y, v x y x y (4.3) and Drchet boundary condtons: u u v v u u v v 0, y, t cos y, 0.5, y, t y 0, yt, y, 0.5, yt, 0.5 y, 1 cos, x,0, t x x,0.5, t sn x, x,0, t x, x,0.5, t x0.5, 1 sn, 0 y 0.5, t 0, 0 x 0.5, t 0, (4.4) (4.5) For the test problem 4.1, we have taken a grd sze 0 0 wth tme step t and Re 100. Computed and exact values of the results gven by Srvastava et al u and v are shown n Tables and 3 along wth and Bahadr 9 at some typcal grd pont. The tabulated results show that the proposed scheme produces better result than Bahadr 9. Tables 4 and 5 show the errors L and L, and also the rate of convergence of u and v components, respectvely, at Re 100, t 1.0 for t From Tables 4 and 5, t can be observed that the CB-DQ performs better than Srvastava et al. 15 and gves more than quadratc rate of convergence. The CB-DQ computed solutons of u and v for Re 100 att 0.5 are depcted n Fg. 1 whle Fg. shows analytcal solutons of u and v, respectvely. For the test problem 4., numercal computatons are carred out wth the parameters: Re 50, 100, and 0 0 grd, t 0.65 for t n order to compare the computed results wth those gven by Jan & Holla 5, Bahadr 9, and Srvastava et al Table 6 shows

9 the comparsons of numercal results obtaned usng the CB-DQ scheme at t 0.65, wth the methods of Jan & Holla 5, Bahadr 9, and Srvastava et al From Table 6, t can be notced that the computed CB-DQ results are n good agreement wth Jan & Holla 5, Bahadr 9, and Srvastava et al Fg. 3 depcts the CB-DQ computed u and v solutons correspondng to Re =50, 100 and 500 at t Table : Comparson of CB-DQ and exact solutons of u for Re 100, 0 0 t grd and Grd xy, CB- DQ t 0.5 t.0 Exact I-LFD 14 Expo- Bahadr 9 CB- FD 15 DQ Exact I- LFD 14 Expo- Bahadr 9 FD 15 (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3, 0.3) (0.7, 0.3) (0.1, 0.5) (0.5, 0.5) (0.9, 0.5) (0.3, 0.7) (0.7, 0.7) (0.1, 0.9) (0.5, 0.9) (0.9, 0.9) Grd xy, Table 3: Comparson of CB-DQ and exact solutons of v for Re 100, 0 0 grd and t t 0.5 t.0 CB-DQ Exact I- Expo- Bahadr 9 CB-DQ Exact I- Expo- Bahadr 9 LFD 14 FD 15 LFD 14 FD 15 (0.1,0.1) (0.5,0.1) (0.9,0.1) (0.3, 0.3) (0.7, 0.3) (0.1, 0.5) (0.5, 0.5) (0.9, 0.5)

10 (0.3, 0.7) (0.7, 0.7) (0.1, 0.9) (0.5, 0.9) (0.9, 0.9) Table 4. L, L errors and rate of convergence for the u -component for Re 100, t at t 1.0. Grd L Expo-FD 15 CB-DQ Expo-FD 15 CB-DQ L e e e e-003 Rate e e e e e e e e e e e-003.0e Rate e e Table 5. L, L errors and rate of convergence for the v -component for Re 100, t at t 1.0. Grd L Expo-FD 15 CB-DQ Expo-FD 15 CB-DQ L Rate e e e e e e e e e e e e e e e-003.0e Rate e e Table 6: Comparson of computed results of u and v for Re 50, grd sze 0 0 and t at t Grd Computed values of u Computed values of v xy, CB- I- Expo- DQ LFD 14 FD 15 Bahadr 9 Jan CB- I- Expoand DQ LFD 14 FD 15 Bahadr 9 Jan and Holla 5 Holla 5 (0.1, 0.1) (0.3, 0.1) (0., 0.) (0.4, 0.)

11 (0.1, 0.3) (0.3, 0.3) (0., 0.4) (0.4, 0.4) Fg. 1: umercal soluton at t 0.5 wth t , Re 100 and grd sze 0 0. for the test problem 4.1. Fg.. Exact soluton at t 0.5 wth t , Re 100 and grd sze 0 0. for the test problem 4.1.

12 Fg. 3. umercal solutons at t 0.65 wth t and grd sze problem for the 5. Conclusons A modfed cubc B-splne dfferental quadrature method s presented for the numercal solutons of two dmensonal nonlnear coupled vscous Burgers equatons. The computed

13 results show that the soluton obtaned by ths scheme s hghly accurate and very close to the exact solutons. We also notce that the scheme has more than quadratc rate of convergance. The obtaned results show that the CB-DQ s a promsng numercal scheme for solvng the hgher dmensonal nonlnear physcal problems governed by partal dfferental equatons. References 1 Cole J. D., On a quaslnear parabolc equatons occurrng n aerodynamcs, Quart Appl ath 9, 5 (1951). Espov S. E., Coupled Burgers equatons: a model of poly-dspersve sedmentaton, Phys Rev 5, 3711 (1995). 3 J. D. Logan, An ntroducton to nonlnear partal dfferental equatons, Wly-Interscence, ew York (1994). 4 C. A. J. Fletcher, Generatng exact solutons of the two-dmensonal Burgers equaton, Int. J. umer. eth. Fluds 3, 13 (1983). 5 P. C. Jan, D.. Holla, umercal soluton of coupled Burgers equatons, Int. J. umer. eth.eng. 1, 13 (1978). 6 C. A. J. Fletcher, A comparson of fnte element and fnte dfference of the one- and twodmensonal Burgers equatons, J. Comput. Phys. 51, 159 (1983). 7 F.W. Wubs, E.D. de Goede, An explct mplct method for a class of tme-dependent partal dfferental equatons, Appl. umer. ath. 9,157 (199). 8 O. Goyon, ultlevel schemes for solvng unsteady equatons, Int. J. umer. eth. Fluds, 937 (1996). 9 Bahadr A. R., A fully mplct fnte-dfference scheme for two-dmensonal Burgers equaton, Appled athematcs and Computaton 137, 131 (003). 10 V. K. Srvastava,. Tamsr, U. Bhardwa, Y. Sanyasrau, Crank-colson scheme for numercal solutons of two dmensonal coupled Burgers equatons, IJSER (5), 44 (011). 11. Tamsr, V. K. Srvastava, A sem-mplct fnte-dfference approach for twodmensonal coupled Burgers equatons, IJSER (6), 46 (011). 1 V. K. Srvastava,. Tamsr, Crank-colson sem-mplct approach for numercal solutons of two-dmensonal coupled nonlnear Burgers equatons, Int. J. Appl. ech. Eng. 17 (), 571 (01).

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