Numerical study of the Benjamin-Bona-Mahony-Burgers equation

Transcription

1 Bol. Soc. Paran. Mat. (3s.) v (017): c SPM ISSN on line ISSN in press SPM: doi:10.569/bspm.v35i Numerical study of te Benjamin-Bona-Maony-Burgers equation M. Zarebnia and R. Parvaz abstract: In tis paper, te quadratic B-spline collocation metod is implemented to find numerical solution of te Benjamin-Bona-Maony-Burgers (BBMB) equation. Applying te Von-Neumann stability analysis tecnique, we sow tat te metod is unconditionally stable. Also te convergence of te metod is proved. Te metod is applied on some test examples, and numerical results ave been compared wit te exact solution. Te numerical solutions sow te efficiency of te metod computationally. Key Words: Quadratic B-spline, Benjamin-Bona-Maony-Burgers equation, Convergence analysis, Finite difference, Collocation metod. Contents 1 Introduction 17 Quadratic B-spline collocation metod 18 3 Stability and convergence analysis Numerical examples Conclusion Introduction In tis paper we consider te solution of te Benjamin-Bona-Maony-Burgers (BBMB) equation u t u xxt αu xx +βu x +uu x = 0, x [a,b], t [0,T], (1.1) wit te initial condition and boundary conditions u(x,0) = f(x), x [a,b], (1.) u(a,t) = u(b,t) = 0, (1.3) were α and β are positive constants. For α = 0, Eq. (1.1) is called te Benjamin- Bona-Maony (BBM) equation. Te BBMB equation as been proposed as a model for propagation of long waves. Tis equation incorporates dispersive and dissipative effects. Te dissipative term can be found in αu xx. For more details 000 Matematics Subject Classification: 65M10, 78A48 17 Typeset by B S P M style. c Soc. Paran. de Mat.

2 18 M. Zarebnia and R. Parvaz on tis topic, see [1,]. In recent years, many different metods ave been used to estimate te solution of te Benjamin-Bona-Maony-Burgers equation and te BBM equation, for example, see [3,4,5,7,11]. Te paper is organized as follows. In Section, quadratic B-spline collocation metod is explained. In Section 3, is devoted to stability analysis and convergence analysis of te metod. In Section 4, examples are presented. A summary is given at te end of te paper in Section 5.. Quadratic B-spline collocation metod Our numerical treatment for BBMB equation using te collocation metod wit quadratic B-spline is to find an approximate solution U(x,t) to te exact solution u(x,t) in te form N U(x,t) = c i (t)b i (x), (.1) i= 1 were c i (t) are time-dependent quantities to be determined from te boundary conditions and collocation form of te differential equations. Also B i (x) are te quadratic B-spline basis functions at knots, given by [6,1] B i (z) = 1 (z z i 1 ), z [z i 1,z i ), (z i+1 z) (z z i ), z [z i,z i+1 ), (z i+ z), z [z i+1,z i+ ), 0, oterwise. (.) Te solution domain a z b partitioned into a mes of uniform lengt = b a N, by te knots z j were j = 0,1,,...,N suc tat a = z 0 < z 1...z N 1 < z N = b and z j = z 0 +j. Te values of B i (z) and its first and second derivatives at te mid knots points are given in Table 1. Also numerical solutions are given at mid points. We note tat te mid points are x i = zi+zi+1. Table 1: B i, B i, B i at mid points. x x i x i 1 x i x i+1 x i+ B i B i B i By using approximate function (.1) and Table 1, we ave u(x i,t n ) U n i = 1 4 cn i cn i cn i+1, (.3)

3 Numerical study of te BBMB equation 19 Table : B i and B iat node points. z z i z i 1 z i z i+1 z i+ B i B i u x (x i,t n ) (U ) n i = c n i 1 +c n i+1, (.4) u xx (x i,t n ) (U ) n i = c n i 1 4c n i +c n i+1, (.5) were c n i := c i(t n ). In tis step by using te finite difference metod, we can write u n+1 u n un+1 xx u n xx t α un+1 xx +u n xx +β un+1 x +u n x + (uu x) n+1 +(uu x ) n = 0. t (.6) Te nonlinear term in (.6) can be approximated by using te following formula [9]: (uu x ) n+1 = u n+1 u n x +u n u n+1 x (uu x ) n. (.7) Substituting te approximate solution U for u and putting te values of te mid values U, its derivatives using (.3),(.4) and (.5) at te knots in (.6) yield te following difference equation wit te variables c i, i = 0,1,...,N, were ác n+1 i 1 + bc n+1 i +ćc n+1 i+1 = Ψn i, i = 0,1,...,N 1, (.8) Ψ n i = Un i +( 1+α t ) n )(U i β t ) n (U i, (.9) and á := ȧ 4 ḃ + ċ, b := 3ȧ 4ċ, ć := ȧ 4 + ḃ + ċ, wit ȧ := 1+ t ) n t (U i, ḃ := β + t Un i, ċ := 1 α t, d := α t 1, é := β t. Te system (.8) consists of N linear equations in N + unknowns {c 1,c 0,..., c N 1,c N }. To obtain a unique solution for c = {c 1,c 0,...,c N 1,c N }, we must use te boundary conditions. From te boundary conditions and Table, we can write c n+1 1 +cn+1 0 = 0, (.10) c n+1 N 1 +cn+1 N = 0. (.11) Associating (.10) and (.11) wit (.8), we obtain a (N +) (N +) system of equations in te following form AC = Q, (.1)

4 130 M. Zarebnia and R. Parvaz were A := á b ć , (.13) 0... á b ć Q:= C := (c n+1 1,cn+1 0,...,c n+1 N 1,cn+1 N )T, (.14) ( 0,u(x 0,t)+( 1+α t )u xx(x 0,t) β t u x(x 0,t),..., T,if u(x N 1,t)+( 1+α t )u xx(x N 1,t) β t u x(x N 1,t),0) t = t, ( ) T, 0,Ψ n 0,...,Ψ n N 1,0 if t > t. (.15) 3. Stability and convergence analysis We present te stability of te quadratic B-spline approximation (.8) using te Von Numann metod [8,10]. According to te Von-Neumann metod, we ave c n i = ξn exp(λki), λ = 1, (3.1) were k is te mode number and is te element size. To apply tis metod, we ave linearized te nonlinear term uu x by consider u as a constant in term (.6). We obtain te equation were âξ n+1 exp(λk(i 1))+ bξ n+1 exp(λk(i))+ĉξ n+1 exp(λk(i+1)) = dξ n exp(λk(i 1))+êξ n exp(λk(i))+ f ξ n exp(λk(i+1)), (3.) â := x y, b := 3 4x, ĉ := x + y, d := z + y, ê := 3 4z, f := z y, wit x := 1 α t, y := β t + t, z := 1+ α t. Dividing bot sides of (3.) by exp(iλk), we can obtain ξ n+1( ) âexp( λk)+ b+ĉexp(λk) = ξ n( dexp( λk)+ê+ ) f exp(λk), (3.3)

5 Numerical study of te BBMB equation 131 were (3.3) can be rewritten in a simple form as X := ( 1 + 4z )cos(k)+( 3 4z ), X 1 := ( 1 + 4x )cos(k)+( 3 4x ), Y := ( y )sin(k). X and X 1 can be rewritten in te form: ξ = X λy X 1 +λy, (3.4) X = ( 1 4 )cos(k)+( ) 4α t (1 cos(k)), X 1 = ( 1 4 )cos(k)+( )+ 4α t (1 cos(k)). We note tat X X 1, so ξ = ξ ξ = X +Y X 1 +Y 1. Terefore, te linearized numerical sceme for te BBMB equation is unconditionally stable. Now we discuss te convergence of te collocation metod. Teorem 3.1. Suppose tat f(x) C 4 [a,b] and f 4 (x) L for x [a,b]. Let be a partition = {a = x 0 < x 1 < < x N = b} be te equally spaced partition of [a,b] wit step size. If S(x) be te unique spline function interpolate f(x) at knots x 0,x 1,,x N ten tere exist a constant λ j suc tat f (j) S (j) λ j L 4 j, j = 0,1,. (3.5) Proof: For te proof see [14]. Lemma 3.. Te B-splines {B 1,,B N } satisfy te following inequality: N i= 1 Proof: At any nodal point x = x i, we can write B i (x) 7, (a x b). (3.6) N B i (x) = B i 1 (x) + B i (x) + B i+1 (x) =, i= 1 also for any x [x i 1,x i ] we ave N B i (x) 7. i= 1

6 13 M. Zarebnia and R. Parvaz wic completes te proof. Teorem 3.3. Suppose tat u(x, t) be te exact solution of (1.1) and assume tat L and U(x,t) be te approximate solution of BBMB (1.1) given by our approac, ten u(x,t) U(x,t) ( + t), (3.7) 4 u(x,t) x 4 were is a constant and independent of. Proof: At te (n+1)t time step, we assume tat S be te unique spline interpolate to te exact solution u of (1.1)-(1.3) given by S(x) = N c B i (x), (3.8) i= 1 To continue, we note tat matrix A is strictly diagonally dominant matrix, and from [13] we can find Ḿ independent of, suc tat A 1 Ḿ. Also from Teorem 3.1, we can write u(x) S(x) λ 0 L 4, (3.9) we substituting S(x) in (.6), we ave te following result AC = Q. (3.10) Subtracting (.1), (3.10) and taking te infinity norm, we can write From (.9) and using Teorem 3.1, we get C C = A 1 Q Q. (3.11) Q Q M, (3.1) were M = λ 0 L + β t λ 1 L+ 1+ α t λ L. Ten we ave Applying (3.13) and Lemma 3., we get te result as were M = 7 MḾ/6. From (3.9) and (3.14),we ave C C MḾ. (3.13) U S M, (3.14) u U M. (3.15) In te next step, suppose tat ε i = u(t i ) U(t i ) be te local truncation error for (.6) at te it level of time. By using te truncation error, we get ε i v i t, i 1, (3.16)

7 Numerical study of te BBMB equation 133 were v i is some finite constant. We can write te following global error estimate at n+1 level E n+1 = n ε i, ( t T/n), (3.17) i=1 tus wit te elp of (3.16), we can write n n E n+1 = ε i υ i t nυ t nυ T t = ρ t, (3.18) n i=1 i=1 were ρ = υt and v = max{v 1,...,v n }. Wic completes te proof. 4. Numerical examples In order to illustrate te performance of te quadratic B-spline collocation metod in solving BBMB equation and justify te accuracy and efficiency of te present metod, we consider te following examples. To sow te efficiency of te present metod for our problem in comparison wit te solution, we report L and L using formulae L = max i U(x i,t) u(x i,t), L = ( i U(x i,t) u(x i,t) ) 1, were U is numerical solution and u denotes exact solution. Note tat we ave computed te numerical results by Matematica-9 programming. Example 4.1. Consider te BBMB equation wit α = 0 and β = 1 in te interval [ 40,60], wit te solution u(x,t) = 3csec (k(x vt x 0 )). We ave taken c = 0.03,0.1, v = 1+c, x 0 = 0 and k = c 4v. Te initial condition is taken from te solution. Also te solution satisfy tree conservation laws: C 1 = udx = i un i, C = (u +u x)dx = i( (u n i ) +((u x ) n i )), C 3 = u3 +3u dx = i( (u n i ) 3 +3(u n i )). Table 3 and Table 6 give C 1,C,C 3,L and L found by our metod in different times for c = 0.1, 0.03 and Table 4 and Table 7 give numerical results from metod in [4,5]. Figure. 1 and Figure. sow tat solution obtained by our metod is closed to te solutions. In addition, in Table 5 and Table 8 we see tat L decreases wit decreasing in t or increasing in N. Also from Figure 3 we can see tat numerical solutions sow te same beavior as solutions.

8 134 M. Zarebnia and R. Parvaz Table 3: Numerical results for Example 4.1 wit = 0.1, N = 1000 and c = 0.1. Time C 1 C C 3 L 10 3 L Table 4: Numerical results for Example 4.1 wit = 0.1, N = 1000 and c = 0.1 wit te algoritim of [4,5]. Time C 1 C C 3 L 10 3 L Table 5: Comparison of L for Example 4.1 at different t wit c = 0.1. P artition\t ime N = 1000, t = N = 1000, t = N = 1000, t = Example 4.. Consider te BBMB equation in te interval [ 10, 10] wit α = 1 and β = 1 and te initial condition u(x,0) = exp( x ). Table 9 and Table 10 give numerical results found by our metod in different times. Also Figure. 4 sows approximate solution graps. In addition, we can see tat te grap sows te same beavior as in [7].

9 Numerical study of te BBMB equation 135 ux,t spacex t 1 An t 1 Nu t 10 An t 10 Nu t 0 An t 0 Nu Figure 1: Numerical and analytical solutions for Example 4.1 wit c = 0.1,N = 1000, and t = 0.1. ux,t spacex t 1 An t 1 Nu t 10 An t 10 Nu t 0 An t 0 Nu Figure : Numerical and analytical solutions for Example 4.1 wit c = 0.03, N = 1000, and t = spacex t 5 0 Figure 3: Analytical solution (left) and numerical solution (rigt) using t = 0.1 and N = 500 wit c = 0.03 of Example 4.1.

10 136 M. Zarebnia and R. Parvaz Table 6: Numerical results for Example 4.1 wit = 0.1, N = 1000 and c = Time C 1 C C 3 L 10 3 L Table 7: Numerical results for Example 4.1 wit = 0.1, N = 1000 and c = 0.03 wit te algoritim of [4,5]. Time C 1 C C 3 L 10 3 L Table 8: Comparison of L for Example 4.1 at different N wit c = 0.1. P artition\t ime N = 00, t = N = 400, t = N = 1000, t = Table 9: Numerical results for Example wit t = 0.1 and N = 00. x\t

11 Numerical study of te BBMB equation 137 ux,t t 0 t 0.5 t 1 t t 3 t 4 t 5 t 6 t 7 t 8 t 9 t x Figure 4: Approximate solution graps of Example 4. in different times wit t = 0.1 and N = 00. Table 10: Numerical results for Example wit t = 0.1 and N = 400. x\t Conclusion Te quadratic B-spline collocation metod is used to solve te Benjamin-Bona- Maony-Burgers(BBMB) equation wit initial and boundary conditions. We study te stability analysis and te convergence analysis of te metod. Te numerical results given in te previous section demonstrate te good accuracy and stability of te proposed sceme in tis researc. References 1. J.A. Tenreiro Macado, D. Baleanu, A.C. J. Luo, Discontinuity and Complexity in Nonlinear Pysical Systems, Springer Intetnational Publising Switzerland,(014).. R.T. Benjamin, J.L. Bona, J.J. Maony, Model equations for long waves in nonlinear dispersiv systems, Pilos. Trans. Roy. Soc. London. 7, 47 78, (197). 3. A.O. Celebi, V.K. Kalantarov, M. Polat, Attractors for te generalized Benjamin- Bona-Maony equation, Commun. Nuner. Met. En. 157, , (1999). 4. L. R. T. Gardner, G. A. Gardner, I. Dag, A B-spline finite element metod for te regularized long wave equation, Commun. Nuner. Met. En. 11, 59 68, (1995). 5. P. C. Jain, R. Sankar, T. V. Sing, Numerical solution of regularized long-wave equation, Commun. Nuner. Met. En. 9, , (1993). 6. D. Kincad, W. Ceny, Numerical Analysis, Brooks/COLE, 1991.

12 138 M. Zarebnia and R. Parvaz 7. K. Omrani, M. Ayadi, Finite difference discretization of te Benjamin-Bona-Maony- Burgers equation, Numer. Metods. Partial. Differ. Equ. 1, 39 48, (007). 8. R.D Rictmyer, K.W. Morton, Difference Metods for Initial-Value Problems, Inter science Publisers (Jon Wiley), New York, (1967). 9. S.G. Rubin,R.A. Graves, Cubic Spline Approximation for Problems in Fluid Mecanics, NASA TR R-436, Wasington, DC; G.D. Smit, Numerical Solution of Patial Differential Metod, Second Edition, Oxford University Press, M. Stanislavovaa,A. Stefanova, B. Wang, Asymptotic smooting and attractors for te generalized BenjaminŰBonaŰMaony equation on R 3, Differ. Equ. 19, , (005). 1. J. Stoer, R. Bulirsc, Introduction to Numerical Analysis, tird edition,springer-verlg, J. Vara, A lower bound for te smallest singular value of matrix, Linear. Algebra. appl. 11, 3 5, (1975). 14. W. J. Kammerer, G. W. Reddien, and R. S. Varga, Quadratic Interpolatory Splines, Numer. Mat., 41-59, (1974). M. Zarebnia (Corresponding autor) Department of Matematics, University of Moageg Ardabili, Ardabil, Iran. address: zarebnia@uma.ac.ir and R. Parvaz Department of Matematics, University of Moageg Ardabili, Ardabil, Iran. address: rparvaz@uma.ac.ir