Research Article A Shannon-Runge-Kutta-Gill Method for Convection-Diffusion Equations

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1 Mathematical Problems in Engineering Volume 203, Article ID 63734, 5 pages Research Article A Shannon-Runge-Kutta-Gill Method for Convection-Diffusion Equations Xiaoming Duan, Jinsong Leng, Carlo Cattani, 2 and Caiyun Li School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 673, China 2 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, Fisciano, Italy Correspondence should be addressed to Jinsong Leng; jslengjs@gmail.com Received 20 January 203; Accepted 5 February 203 Academic Editor: Shengyong Chen Copyright 203 Xiaoming Duan et al. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited. A Shannon-Rugge-Kutta-Gill method for solving convection-diffusion equations is discussed. This approach transforms convection-diffusion equations into one-dimensional equations at collocations points, hich e solve by Runge-Kutta-Gill method. A concrete example solved is used to examine the method s feasibility.. Introduction Most of the physics phenomenon are stated in terms of partial differential equations (PDEs). Convection-diffusion equation is a kind of PDE hich can be used in many science and technology fields, especially in image and signal procession such as image segmentation and the quickly stability of image processing. The numerical solution of convection-diffusion equations as an important subject has alays attracted the attentions of the researchers for a long time. The standard Galerkin finite-element method can solve thesolutionoftheequations,butitisnumericallyunstable for small values of the diffusion parameter. So on the basis of this method, in [] KingandKruegerinvestigatedthe effect of a stabilized finite-element approximation and dre a conclusion that the stabilized system can provide accurate controllers. A novel multilevel particle methods and to complementary approaches are researched in [2]. In this paper, a ne class of particle based on mapping functions is introduced, and particle remeshing is used as a key element in overlapping domains in the particle-amr method. For the to-dimensional convection-diffusion equation, Gupta et al. proposed a fourth-order nine-point compact finite-difference formulae, hich is shon to be computationally efficient and stable and yield highly accurate numerical solutions in [3, 4]. The resulting linear system is solved by classical iterative methods for large values of the Reynolds number in [4]. With the avelet method, Shi et al. solved the solution of covectiondiffusion equations by Haar avelet method in [5]. In short, convection-diffusion equations are studied by scholars via different methods. Recently, Wavelet analysis as a ne subject has attracted a lot of attention. As a mathematical tool, it has been idely used in numerical analysis, signal processing [6, 7], image processing, and so forth. Many years ago, avelet methods ere used for numerical analysis, particularly the numerical solution of PDEs. Up to no, researchers have utilized the simplest Haar avelets to solve kinds of PDEs. Chen and Hsiao, in [8], established an operational matrix of integration based on Haar avelets and used a procedure for applying thematrixtoobtainaveletsolutionofpdes.in[9, 0], Cattani solved Poisson s problem and Fredholm type integral equations by Harmonic avelet method. Other avelets are also extensively used to solve the kinds of PDEs, in hich Shannon avelet is applied in the numerical solution of some equations, such as [, 2]. Shannon scaling function and Sinc function combined ith other methods (Galerkin, etc.) areusedtosolvesomepdes[3, 4].Inlightoftheabove description, e are enlightened that Shannon avelet is a useful tool to obtain the solution of convection-diffusion equations, hich combined ith Rugge-Kutta-Gill method. In this paper, the content is assigned as follos. In Section 2, Shannon avelet is introduced. We elaborate the concrete method solving convection-diffusion equation in

2 2 Mathematical Problems in Engineering Section 3. InSection4 the viability of Shannon avelet collocation is tested by a listed example. 2. Preliminaries 2.. Shannon Wavelet. Wavelets are classified as families ith names, such as Haar avelet, Meyer avelet, and Shannon avelet. Shannon avelets are the real part of harmonic avelets. They have a slo decay in the time domain but a very sharp compact support in the frequency (Fourier) domain. This fact, together ith the Parseval s identity, is used to compute the inner product and the expansion coefficients of the Shannon avelets easily. A set of Shannon scaling functions in the subspace V j is defined as φ j,k (t) =2 j/2 sin π(2j t k), k Z () π(2 j t k) and the mother avelets are ψ j,k (t) =2 j/2 sin π(2j t k /2) sin 2π (2 j t k /2), π(2 j t k /2) k Z. In () and(2),thescalingfunctionandmotheraveletfor j=k=0(see Figure ) are as follos: ψ (t) = φ (t) = sin πt πt (2) = eπit e πit, (3) 2πit sin π (t /2) sin 2π (t /2). (4) π (t /2) To some properties of Shannon scaling and avelet functions, Cattani has detailedly researched in [5 8]. So in this paper, e ill not narrate the properties again. To (3), its Fourier transform φ(ω) = χ [ /2,/2).Itisvery easy to see that φ (ω+n) 2 =. (5) n= According to this equation, the sequence of function {φ(x n)} n= is orthonormal. A reproducing kernel is generated [9] as follos: sin π(x y) K(x y)=. (6) π(x y) Recomposing (6), e obtain a ne reproducing kernel sin (π/δ) (x y) (x y)=, (7) (π/δ) (x y) here Δ is the spatial mesh size. In one-dimensional function f(x), emakethedomain [a, b] be discrete and set the grid size So e obtain collocation points Δ= b a 2 j (8) x i =iδ, i=0,,2,...,2 j, (9) here 2 j is a number of nodes, hich used in the discretization and also is the maximum avelet index number. No, abasisfunction j (x x n ) of Shannon avelet ill be constructed by (7) j (x x n )= sin (π/δ) (x x n),,,2,...,2 j. (0) (π/δ) (x x n ) It has some properties as follos. (i) To the random x k (k = 0,,2,...,2 j ),thefunction j (x x n ) fulfills interpolation property: j (x k x n )=δ kn ={ k=n, 0 k=n. () (ii) We have noticed that the constructed basis functions are orthogonal to each other as follos: j (x x k ) j (x x n )dx=δδ kn. (2) (iii) If e make the integral ith the basis functions and their derived functions, e obtain. j (x x k ) dm j (x x n ) dx m dx=δ dm j (x k x n ) dx m. (3) Both j (x x k ) and its associated avelet play an important part in signal processing. Unfortunately, hen x,thereductionof j (x x k ) is very slo. So our paper only researches the case hich x belongs to finite interval Function Approximation. According to Shannon s sampling theorem, any function f(x) B 2 π can be denoted as [9] f (x) = f(x n ) j (x x n ), (4) n Z here the coefficients f(x n ) is the value of the function f(x) at the point x n. B 2 π is the Paley-Wiener reproducing kernel Hilbert space hich is a subspace of the Hilbert space L 2 (R).

3 Mathematical Problems in Engineering 3 φ Shannon scaling function t ψ Shannon mother function t Figure : Shannon scaling function φ and mother function ψ N=6, t = 0.00 N=32, t = u(x, t) 5 u(x, t) x x Wavelet solution Analytical solution Wavelet solution Analytical solution Figure 2: Comparison of the analytical solution and Shannon avelet solution. In V j, f(x) canbeapproximatedbyf j (x) V j.soeget f (x) f j (x) = 2 j f j (x n ) j (x x n ). (5) ith initial condition and boundary conditions: u (x, 0) =f(x), 0 x b, u (0, t) =g 0 (t), u(2, t) =g (t), 0<t T. (7) 3. Method of Solution of Convection-Diffusion Equation In this section, let us consider the one-dimensional convection-diffusion equation ith constant coefficients: u +a u x =α 2 u x 2 0<x<2, 0<t<T (6) Like (8) and(9), e ill also divide the interval [0, 2] into N=2 j equal parts of length Δ=2/Nand denote x i =iδ, i= 0,, 2,..., N. Weknothatu(x, t) can be approximated by u j (x, t) V j expanded in terms of the constructed basis functionas formula (5) as follos: u (x, t) u j (x, t) = N u j (x n,t) j (x x n ). (8)

4 4 Mathematical Problems in Engineering We multiply formula (6) ith the constructed basis function j (x x k ),theneobtain u j (x, t) j (x x k ) = a u j (x, t) x j (x x k )+α 2 u j (x, t) x 2 j (x x k ). (9) Integrate that formula (9) ithrespecttox from to as follos: u j (x, t) j (x x k )dx a u j (x, t) x j (x x k )dx = + α 2 u j (x, t) x 2 j (x x k )dx. The left expression of formula (20)isasfollos: u j (x, t) j (x x k )dx N u j (x n,t) j (x x n ) j (x x k )dx = = N u j (x n,t) j (x x n ) j (x x k )dx = N u j (x n,t) j (x k x n ) Δ = u j (x k,t) Δ. The right expression of formula (20)is as follos: a N u j (x n,t) j (x x n ) x j (x x k )dx + (20) (2) α 2 N u j (x n,t) j (x x n ) x 2 j (x x k )dx = anu j (x n,t) +α Nu j (x n,t) = aδnu j (x n,t) j (x k x n ) +αδnu j (x n,t) j (x k x n ). j (x x n ) x j (x x k )dx 2 j (x x n ) x 2 j (x x k )dx (22) Via (2)=(22), e obtain the folloing expression: u j (x k,t) = N u j (x n,t)[ a j (x k x n )+α j (x k x n )] (23) To any x k,eillgetoneequation.son+equations ill be obtained. In order to simplify the N+equations, e define the matrices U and V as follos: U=[u j (x 0,t),u j (x,t),u j (x 2,t),...,u j (x N,t)], V kn = a j (x k x n )+α j (x k x n ), V=(V kn ) (N+) (N+). (24) Combining ith (24), the formula (23) isevolvedintoa matrix equation: U =VU. (25) No e ill use Runge-Kutta-Gill method to solve formula (25) as follos: U i+ =U i + Δt 6 [K +(2 2) K 2 +(2+ 2) K 3 +K 4 ], K =VU i, K 2 =V(U i + 2 K ), K 3 =V(U i K K 2 2 ), K 4 =V(U i 2 2 K K 2 3 ), (26) here Δt is the time interval. From (9) and(7), the initial value U 0 is obtained, and then e can evaluate the numerical solution at any collocation point ithin the different parameter t. 4. Test of Example A concrete convection-diffusion equation has knon exact solution ill be considered, and e observe ho ell the Shannon avelet solution approximates the exact solution. We assume that α = 0., a =, f (x) =e (x 2)2 /8, g 0 (x) = t e (5+4t) /0(t+20), g (t) = t e 2(5+2t) /5(t+20), (27)

5 Mathematical Problems in Engineering 5 For hich the exact solution is u (x, t) = t e (x 2 t) /0.4(t+20). (28) In the course of the experiment, e got t max = 0.0 and setted Δt = Wegottheapproximatechartsinthe case of N=32and N=6and obtained the conclusion that the avelet solution is approximate to the exact solution more precisely (see Figure 2). 5. Conclusion In this paper, the theory of Shannon avelet combined ith Runge-Kutta-Gill method is used to solve the approximation of convection-diffusion equations. It has been shon that the key idea of shannon avelet collocation method is to transform convection-diffusion equations into one-dimensional equations at collocations points and to solve the problem via Rugge-Kutta-Gill method. Acknoledgment The authors are thankful to the referees for their valuable comments and suggestions that improved the presentation of this paper. This ork as supported by the National Natural Science Foundation of China (2700). References [] B. B. King and D. A. Krueger, The -D convection diffusion equation: galerkin least squares approximations and feedback control, in Proceedings of the 43rd IEEE Conference on Decision and Control,pp ,2004. [2] M. Bergdorf, G.-H. Cottet, and P. Koumoutsakos, Multilevel adaptive particle methods for convection-diffusion equations, Multiscale Modeling and Simulation, vol.4,no.,pp , [3] M.M.Gupta,R.P.Manohar,andJ.W.Stephenson, Afourth order, cost effective and stable finite difference scheme for the convection-diffusion equation, in Numerical Properties and Methodologies in Heat Transfer, pp , Hemisphere, 983. [4]M.M.Gupta,R.P.Manohar,andJ.W.Stephenson, Asingle cell high order scheme for the convection-diffusion equation ith variable coefficients, International Journal for Numerical Methods in Fluids,vol.4,no.7,pp.64 65,984. [5] Z. Shi, L. Y. Deng, and Q. J. Chen, Numerical solution of differential equations by using Haar avelets, in Proceedings of the International Conference on Wavelet Analysis and Pattern Recognition, pp. 2 4, Beijing, China, [6] C.Cattani, Waveletbasedapproachtofractalsandfractalsignal denoising, Transactions on Computational Sciences Journal, vol. 5730, pp , [7]J.S.Leng,T.Z.Hung,Y.F.Jing,andW.Jiang, Astudyon conjugate quadrature filters, EURASIP Journal on Advances in Singnal Processing, vol. 20, Article ID 23754, 7 pages, 20. [8] C. F. Chen and C. H. Hsiao, Haar avelet method for solving lumped and distributedparameter systems, IEE Proceedings Control Theory and Applications,vol.44,pp.87 94,997. [9] C. Cattani, Harmonic avelet solution of Poisson s problem, Balkan Geometry and Its Applications, vol.3,no., pp , [0] C. Cattani and A. Kudreyko, Harmonic avelet method toards solution of the Fredholm type integral equations of the second kind, Applied Mathematics and Computation, vol. 25, no. 2, pp , 200. [] C. Cattani, Shannon avelets for the solution of integrodifferential equations, Mathematical Problems in Engineering,vol. 200,ArticleID40848,22pages,200. [2] C. Cattani, Second order Shannon avelet approximation of C 2 -functions, Politehnica University of Bucharest Scientific Bulletin A,vol.73,no.3,pp.73 84,20. [3]V.G.Koures, Solving the Coulomb Schrödinger equation in d=2+viasinccollocation, Computational Physics, vol.28,no.,pp. 5,996. [4] F. Stenger, A sinc-galerkin method of solution of boundary value problems, Mathematics of Computation, vol.33,no.45, pp , 979. [5] C. Cattani, Connection coefficients of Shannon avelets, Mathematical Modelling and Analysis,vol.,no.2,pp.7 32, [6] C. Cattani, Shannon avelets theory, Mathematical Problems in Engineering,vol.2008,ArticleID64808,24pages,2008. [7] C. Cattani, Signorini cylindrical aves and Shannon avelets, Advances in Numerical Analysis,vol.202,ArticleID7359,24 pages, 202. [8] C. Cattani, Fractional calculus and Shannon avelets, Mathematical Problems in Engineering, vol. 202, Article ID 50282, 26 pages, 202. [9] G. W. Wei, Quasi avelets and quasi interpolating avelets, Chemical Physics Letters,vol.296,pp ,998.

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