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

Abstract

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, which we solve by Runge-Kutta-Gill method. A concrete example solved is used to examine the method’s feasibility.

1. Introduction

Most of the physics phenomenon are stated in terms of partial differential equations (PDEs). Convection-diffusion equation is a kind of PDE which 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 always attracted the attentions of the researchers for a long time.

The standard Galerkin finite-element method can solve the solution of the equations, but it is numerically unstable for small values of the diffusion parameter. So on the basis of this method, in [1] King and Krueger investigated the effect of a stabilized finite-element approximation and drew a conclusion that the stabilized system can provide accurate controllers. A novel multilevel particle methods and two complementary approaches are researched in [2]. In this paper, a new 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 two-dimensional convection-diffusion equation, Gupta et al. proposed a fourth-order nine-point compact finite-difference formulae, which is shown 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 wavelet method, Shi et al. solved the solution of covection-diffusion equations by Haar wavelet method in [5]. In short, convection-diffusion equations are studied by scholars via different methods.

Recently, Wavelet analysis as a new subject has attracted a lot of attention. As a mathematical tool, it has been widely used in numerical analysis, signal processing [6, 7], image processing, and so forth. Many years ago, wavelet methods were used for numerical analysis, particularly the numerical solution of PDEs. Up to now, researchers have utilized the simplest Haar wavelets to solve kinds of PDEs. Chen and Hsiao, in [8], established an operational matrix of integration based on Haar wavelets and used a procedure for applying the matrix to obtain wavelet solution of PDEs. In [9, 10], Cattani solved Poisson's problem and Fredholm type integral equations by Harmonic wavelet method. Other wavelets are also extensively used to solve the kinds of PDEs, in which Shannon wavelet is applied in the numerical solution of some equations, such as [11, 12]. Shannon scaling function and Sinc function combined with other methods (Galerkin, etc.) are used to solve some PDEs [13, 14]. In light of the above description, we are enlightened that Shannon wavelet is a useful tool to obtain the solution of convection-diffusion equations, which combined with Rugge-Kutta-Gill method.

In this paper, the content is assigned as follows. In Section 2, Shannon wavelet is introduced. We elaborate the concrete method solving convection-diffusion equation in Section 3. In Section 4 the viability of Shannon wavelet collocation is tested by a listed example.

2. Preliminaries

2.1. Shannon Wavelet

Wavelets are classified as families with names, such as Haar wavelet, Meyer wavelet, and Shannon wavelet. Shannon wavelets are the real part of harmonic wavelets. They have a slow decay in the time domain but a very sharp compact support in the frequency (Fourier) domain. This fact, together with the Parseval's identity, is used to compute the inner product and the expansion coefficients of the Shannon wavelets easily. A set of Shannon scaling functions in the subspace 𝑉_𝑗 is defined as 𝜑_𝑗,𝑘(𝑡)=2^𝑗/2sin𝜋2^𝑗𝑡−𝑘𝜋2^𝑗𝑡−𝑘,𝑘∈𝐙 (1) and the mother wavelets are 𝜓_𝑗,𝑘(𝑡)=2^𝑗/2sin𝜋2^𝑗𝑡−𝑘−1/2−sin2𝜋2^𝑗𝑡−𝑘−1/2𝜋2^𝑗,𝑡−𝑘−1/2𝑘∈𝐙. (2) In (1) and (2), the scaling function and mother wavelet for 𝑗=𝑘=0 (see Figure 1) are as follows: 𝜑(𝑡)=sin𝜋𝑡=𝑒𝜋𝑡^𝜋𝑖𝑡−𝑒^{−𝜋𝑖𝑡},𝜓(𝑡)=()()2𝜋𝑖𝑡sin𝜋𝑡−1/2−sin2𝜋𝑡−1/2𝜋().𝑡−1/2 (3)(4)

Figure 1 

Shannon scaling function 𝜑 and mother function 𝜓.

To some properties of Shannon scaling and wavelet functions, Cattani has detailedly researched in [15–18]. So in this paper, we will not narrate the properties again.

To (3), its Fourier transform 𝜑(𝜔)=𝜒_[−1/2,1/2). It is very easy to see that ^∞_𝑛=−∞𝜑(𝜔+𝑛)²=1. (5)

According to this equation, the sequence of function {𝜑(𝑥−𝑛)}^∞_𝑛=−∞ is orthonormal. A reproducing kernel is generated [19] as follows: 𝐾𝑥−𝑦=sin𝜋𝑥−𝑦𝜋𝑥−𝑦. (6) Recomposing (6), we obtain a new reproducing kernel 𝑤=()𝑥−𝑦sin𝜋/Δ𝑥−𝑦(𝜋/Δ)𝑥−𝑦, (7) where Δ is the spatial mesh size.

In one-dimensional function 𝑓(𝑥), we make the domain [𝑎,𝑏] be discrete and set the grid size Δ=𝑏−𝑎2^𝑗 (8) So we obtain collocation points 𝑥_𝑖=𝑖Δ,𝑖=0,1,2,…,2^𝑗, (9) where 2^𝑗 is a number of nodes, which used in the discretization and also is the maximum wavelet index number. Now, a basis function 𝑤_𝑗(𝑥−𝑥_𝑛) of Shannon wavelet will be constructed by (7) 𝑤_𝑗𝑥−𝑥_𝑛=sin(𝜋/Δ)𝑥−𝑥_𝑛(𝜋/Δ)𝑥−𝑥_𝑛,𝑛=0,1,2,…,2^𝑗. (10) It has some properties as follows.(i)To the random 𝑥_𝑘(𝑘=0,1,2,…,2^𝑗), the function 𝑤_𝑗(𝑥−𝑥_𝑛) fulfills interpolation property: 𝑤_𝑗𝑥_𝑘−𝑥_𝑛=𝛿_𝑘𝑛=1𝑘=𝑛,0𝑘≠𝑛. (11)(ii)We have noticed that the constructed basis functions are orthogonal to each other as follows: ^∞_−∞𝑤_𝑗𝑥−𝑥_𝑘𝑤_𝑗𝑥−𝑥_𝑛𝑑𝑥=Δ𝛿_𝑘𝑛. (12)(iii)If we make the integral with the basis functions and their derived functions, we obtain. ^∞_−∞𝑤_𝑗𝑥−𝑥_𝑘𝑑^𝑚𝑤_𝑗𝑥−𝑥_𝑛𝑑𝑥^𝑚𝑑𝑑𝑥=Δ^𝑚𝑤_𝑗𝑥_𝑘−𝑥_𝑛𝑑𝑥^𝑚. (13)

Both 𝑤_𝑗(𝑥−𝑥_𝑘) and its associated wavelet play an important part in signal processing. Unfortunately, when 𝑥→∞, the reduction of 𝑤_𝑗(𝑥−𝑥_𝑘) is very slow. So our paper only researches the case which 𝑥 belongs to finite interval.

2.2. Function Approximation

According to Shannon's sampling theorem, any function 𝑓(𝑥)∈𝐵²_𝜋 can be denoted as [19] 𝑓(𝑥)=_𝑛∈𝑍𝑓𝑥_𝑛𝑤_𝑗𝑥−𝑥_𝑛, (14) where the coefficients 𝑓(𝑥_𝑛) is the value of the function 𝑓(𝑥) at the point 𝑥_𝑛. 𝐵²_𝜋 is the Paley-Wiener reproducing kernel Hilbert space which is a subspace of the Hilbert space 𝐿²(𝑅). In 𝑉_𝑗, 𝑓(𝑥) can be approximated by 𝑓_𝑗(𝑥)∈𝑉_𝑗. So we get 𝑓(𝑥)≈𝑓_𝑗(𝑥)=^2𝑗_𝑛=0𝑓_𝑗𝑥_𝑛𝑤_𝑗𝑥−𝑥_𝑛. (15)

3. Method of Solution of Convection-Diffusion Equation

In this section, let us consider the one-dimensional convection-diffusion equation with constant coefficients: 𝜕𝑢𝜕𝑡+𝑎𝜕𝑢𝜕𝜕𝑥=𝛼²𝑢𝜕𝑥²0<𝑥<2,0<𝑡<𝑇 (16) with initial condition and boundary conditions: 𝑢()(𝑥)𝑢()𝑥,0=𝑓,0≤𝑥≤𝑏,0,𝑡=𝑔₀(𝑡),𝑢(2,𝑡)=𝑔₁(𝑡),0<𝑡≤𝑇. (17) Like (8) and (9), we will also divide the interval [0,2] into 𝑁=2^𝑗 equal parts of length Δ=2/𝑁 and denote 𝑥_𝑖=𝑖Δ,𝑖=0,1,2,…,𝑁. We know that 𝑢(𝑥,𝑡) can be approximated by 𝑢_𝑗(𝑥,𝑡)∈𝑉_𝑗 expanded in terms of the constructed basis function as formula (15) as follows: 𝑢(𝑥,𝑡)≈𝑢_𝑗(𝑥,𝑡)=^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗𝑥−𝑥_𝑛. (18) We multiply formula (16) with the constructed basis function 𝑤_𝑗(𝑥−𝑥_𝑘), then we obtain 𝜕𝑢_𝑗(𝑥,𝑡)𝑤𝜕𝑡^𝑗𝑥−𝑥_𝑘=−𝑎𝜕𝑢_𝑗(𝑥,𝑡)𝑤𝜕𝑥^𝑗𝑥−𝑥_𝑘𝜕+𝛼²𝑢_𝑗(𝑥,𝑡)𝜕𝑥²𝑤_𝑗𝑥−𝑥_𝑘. (19)Integrate that formula (19) with respect to 𝑥 from −∞ to ∞ as follows: ^∞_−∞𝜕𝑢_𝑗(𝑥,𝑡)𝑤𝜕𝑡^𝑗𝑥−𝑥_𝑘𝑑𝑥=^∞_−∞−𝑎𝜕𝑢_𝑗(𝑥,𝑡)𝑤𝜕𝑥^𝑗𝑥−𝑥_𝑘+𝑑𝑥_∞−∞𝛼𝜕²𝑢_𝑗(𝑥,𝑡)𝜕𝑥²𝑤_𝑗𝑥−𝑥_𝑘𝑑𝑥. (20) The left expression of formula (20) is as follows: ^∞_−∞𝜕𝑢_𝑗(𝑥,𝑡)𝑤𝜕𝑡^𝑗𝑥−𝑥_𝑘=𝑑𝑥_∞−∞𝜕∑^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗𝑥−𝑥_𝑛𝑤𝜕𝑡^𝑗𝑥−𝑥_𝑘=𝑑𝑥_𝑁𝑛=0𝜕𝑢_𝑗𝑥_𝑛,𝑡𝜕𝑡^∞_−∞𝑤_𝑗𝑥−𝑥_𝑛𝑤_𝑗𝑥−𝑥_𝑘=𝑑𝑥𝜕∑^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗𝑥_𝑘−𝑥_𝑛Δ=𝜕𝑡𝜕𝑢_𝑗𝑥_𝑘,𝑡𝜕𝑡Δ. (21) The right expression of formula (20) is as follows: ^∞_−∞−𝑎𝜕∑^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗𝑥−𝑥_𝑛𝑤𝜕𝑥^𝑗𝑥−𝑥_𝑘+𝑑𝑥_∞−∞𝛼𝜕²∑^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗𝑥−𝑥_𝑛𝜕𝑥²𝑤_𝑗𝑥−𝑥_𝑘𝑑𝑥=−𝑎_𝑛=0𝑁𝑢_𝑗𝑥_𝑛,𝑡^∞_−∞𝜕𝑤_𝑗𝑥−𝑥_𝑛𝑤𝜕𝑥^𝑗𝑥−𝑥_𝑘𝑑𝑥+𝛼_𝑛=0𝑁𝑢_𝑗𝑥_𝑛,𝑡^∞_−∞𝜕²𝑤_𝑗𝑥−𝑥_𝑛𝜕𝑥²𝑤_𝑗𝑥−𝑥_𝑘𝑑𝑥=−𝑎Δ_𝑛=0𝑁𝑢_𝑗𝑥_𝑛,𝑡𝑤^′_𝑗𝑥_𝑘−𝑥_𝑛+𝛼Δ_𝑛=0𝑁𝑢_𝑗𝑥_𝑛,𝑡𝑤_𝑗^′′𝑥_𝑘−𝑥_𝑛. (22) Via (21) = (22), we obtain the following expression: 𝜕𝑢_𝑗𝑥_𝑘,𝑡=𝜕𝑡^𝑁_𝑛=0𝑢_𝑗𝑥_𝑛,𝑡−𝑎𝑤^′_𝑗𝑥_𝑘−𝑥_𝑛+𝛼𝑤_𝑗^′′𝑥_𝑘−𝑥_𝑛 (23)

To any 𝑥_𝑘, we will get one equation. So 𝑁+1 equations will be obtained. In order to simplify the 𝑁+1 equations, we define the matrices 𝑈 and 𝑉 as follows: 𝑈=𝑢_𝑗𝑥₀,𝑡,𝑢_𝑗𝑥₁,𝑡,𝑢_𝑗𝑥₂,𝑡,…,𝑢_𝑗𝑥_𝑁,𝑡^⊤,𝑣_𝑘𝑛=−𝑎𝑤^′_𝑗𝑥_𝑘−𝑥_𝑛+𝛼𝑤_𝑗^′′𝑥_𝑘−𝑥_𝑛,𝑉=𝑣_{𝑘𝑛(𝑁+1)×(𝑁+1)}. (24) Combining with (24), the formula (23) is evolved into a matrix equation: 𝜕𝑈𝜕𝑡=𝑉𝑈. (25) Now we will use Runge-Kutta-Gill method to solve formula (25) as follows: 𝑈_𝑖+1=𝑈_𝑖+Δ𝑡6𝐾₁+2−√2𝐾₂+2+√2𝐾₃+𝐾₄,𝐾₁=𝑉𝑈_𝑖,𝐾₂=𝑉𝑈_𝑖+12𝐾₁,𝐾₃=𝑉𝑈_𝑖+√2−12𝐾₁+2−√22𝐾₂,𝐾₄=𝑉𝑈_𝑖−√22𝐾₂+2+√22𝐾₃, (26) where Δ𝑡 is the time interval. From (9) and (17), the initial value 𝑈₀ is obtained, and then we can evaluate the numerical solution at any collocation point within the different parameter 𝑡.

4. Test of Example

A concrete convection-diffusion equation has known exact solution will be considered, and we observe how well the Shannon wavelet solution approximates the exact solution. We assume that 𝛼=0.1,𝑎=0.8, 𝑓(𝑥)=𝑒^{−(𝑥−2)2/8},𝑔₀(𝑥)=20𝑒20+𝑡^{−(5+4𝑡)2/10(𝑡+20)},𝑔₁(𝑡)=20𝑒20+𝑡^{−2(5+2𝑡)2/5(𝑡+20)}, (27) For which the exact solution is 𝑢(𝑥,𝑡)=20𝑒20+𝑡^{−(𝑥−2−0.8𝑡)2/0.4(𝑡+20)}. (28)

In the course of the experiment, we got 𝑡_max=0.01 and setted Δ𝑡=0.00001. We got the approximate charts in the case of 𝑁=32 and 𝑁=16 and obtained the conclusion that the wavelet solution is approximate to the exact solution more precisely (see Figure 2).

Figure 2 

Comparison of the analytical solution and Shannon wavelet solution.

5. Conclusion

In this paper, the theory of Shannon wavelet combined with Runge-Kutta-Gill method is used to solve the approximation of convection-diffusion equations. It has been shown that the key idea of shannon wavelet 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.

Acknowledgment

The authors are thankful to the referees for their valuable comments and suggestions that improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (11271001).