Numerical method based on radial basis functions for

Numerical method based on radial basis functions for solving reaction-diffusion equations Ling-De Su

Zi-Wu Jiang

Tong-Song Jiang

North-Eastern Federal University Belinskogo, Yakutsk, Russia Email: [email protected]

Department of Mathematics Linyi University, P. R. China Email: [email protected]

Department of Mathematics Linyi University, P. R. China Email: [email protected]

Abstract—In this paper, we propose a numerical scheme to solve the time-dependent reaction-diffusion equations by using the meshfree method and approximating the solution using multiquadrics (MQ) Radial Basis Function (RBF). The scheme works in very similar fashion as finite difference methods. The results of numerical experiments are presented, and compared with analytical solutions to confirm the good accuracy of the presented scheme. Keywords—Reaction-diffusion equation; Meshfree method; Radial Basis Function (RBF); Numerical solution

I. I NTRODUCTION This paper is devoted to the numerical computation of the time-dependent reaction-diffusion equation, ∂u = αΔu(x) + βu(x), x ∈ Ω ⊂ Rd , 0 < t ≤ T, (I.1) ∂t with suitable initial and Dirichlet boundary conditions in some continuous domain and where u represent a group of physical or biological species, α is the diffusion constant, βu describes the chemical or biological reaction, Δ denotes the Laplace operator. The solution for such kind of equation is of fundamental importance in physics, biology, and chemistry[1, 2]. Reaction-diffusion equations are conventionally used in chemical physics in order to describe concentration and temperature distributions. In this case, heat and mass transfer are described by the diffusion term while the reaction term describes the rate of heat and mass production. Many phenomena in physics, biology and chemistry and other fields can be described by various reaction-diffusion equations. So it is very important for the theory and applied to find the practical numerical method for this equation. There has been many methods to solve the reactiondiffusion equation, the finite-difference methods is one of the most famous[3–5]. This article presents a numerical scheme to solve the linear reaction-diffusion equation using the collocation method and approximating directly the solution using MQ radial basis function. The results of numerical experiments are presented, and compared with analytical solutions to confirm the good accuracy of the presented √ scheme. The MQ-RBF takes the form: r2 + c2 , c > 0. It is very import to choose a suitable parameter c for the accuracy. The singular matrix will be produced by unsuitable parameter. Many researchers from the numerical study found that the parameter c in MQ have important effect on the accuracy[6]. ____________________________________

But how the shape parameters affect the accuracy has not been studied yet. The choice of this optimal value is still under intensive investigation. Recently, many works used the parameter of c which was found experimentally to minimize error or chose rather arbitrarily without apparent effort at optimization. In our computation, the shape parameter c in using the MQ-RBF also found experimentally. Moreover, the accuracy also be affected by how many nodes we choose. The theory of RBF difference method can see[7, 8]. II. T HE COLLOCATION METHOD A. Approximation using radial basis function The RBF interpolation function for u(x) is represented by a linear combination of these bases in the following form, u(x) 

N 

λj ϕ(x, xj ) + ψ(x), f or x ∈ Ω ⊂ Rd ,

(II.1)

j=1

where x = (x1 , x2 , ..., xd ), d is the dimension of the problem, N is the number of interpolation points, the λ, s are coefficients to be determined and ϕ is the radial basis function. Eq. (II.1) can be written without the additional polynomial ψ. In that case, ϕ must be unconditional positive definite to guarantee the solvability of the resulting system (e.g. Gaussian or inverse multiquadrics). However, when ϕ is conditionally positive definite, ψ is usually required. We will use the RBF, which defined as,  (II.2) MQ : ϕ(x, xj ) = ϕ(rj ) = rj2 + c2 , c > 0, where rj = x − xj  is the Euclidean norm. Since ϕ given by (II.2) is C ∞ continuous ,we can use it directly. If Pqd denotes the space of d-variate polynomial of order not exceeding than q, and letting the polynomials P1 , P2 , ..., Pm be the basis of Pdq in Rd , then the polynomial ψ(x) in Eq. (II.1), is usually written in the following form, ψ(x) =

m 

ζi Pi (xj ),

(II.3)

i=1

where m = (q − 1 + d)!/(d!(q − 1)!). The collocation method is used for getting the coefficients (λ1 , λ2 , ..., λN ) and (ζ1 , ζ2 , ..., ζm ). However, in addition to

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the N equations resulting from collocating Eq. (II.1) at the N points, an extra m equations which are ensured by the m conditions for Eq. (II.1) are required, N 

λj Pi (xj ) = 0, i = 1, 2, ..., m.

where [u]n = [un1 un2 · · · unN −d 0 · · · 0]T , [λ]n = [λn1 λn2 · · · λnN +1 ] and A = [aij , 1 ≤ i, j ≤ N + 1] is given by ⎞ ⎛

(II.4)

In a similar representation as Eq. (II.1), for any linear partial differential operator L, Lu can be approximated by, Lu(x) 

N 

λj Lϕ(x, xj ) + Lψ(x).

(II.5)

where X = [x1 1; x2 1; · · · ; xN −d 1],

j=1

B. Reaction-diffusion equations

⎛

Considering the reaction-diffusion equations Eq. (I.1) with following boundary condition, u(x, t) = g(x),

x ∈ ∂Ω, 0 < t ≤ T,

u(x, 0) = u0 (x),

x ∈ Ω,

(II.7)

where g(x) and u0 (x) are given functions, the function u(x, t) is unknown. First, discretizing Eq. (I.1) using the following θ-weighted scheme, u

  (x) − u (x) = θ · α · Δun+1 (x) + β · un+1 (x) dt   +(1 − θ) · α · Δun (x) + β · un (x) , n

(II.8)

where 0 ≤ θ ≤ 1, dt is the time step size and Δ denotes the Laplace operator, un = u(x, tn ) where tn = tn−1 + dt, rearranging Eq. (II.8), we get, un+1 − θ · dt · (α · Δun+1 +β · un+1 ) = un + (1 − θ)·dt · (α · Δun + β · un ).

N −d 

λnj ϕ(rj ) +

j=1

d 

λnN −d+k xk + λnN +1 .

(II.10)

k=1

un (xi ) ≈

λnj ϕ(rij ) +

j=1

d 

λnN −d+k xik + λnN +1 , (II.11)

k=1

where rij = xi − xj . Due to Eq. (II.4) additional conditions are written as, N −d  j=1

λnj

=

N −d 

λnj xjk

= 0, k = 1, 2, · · · , d.

ϕ12 .. .

ϕ(N −d)1

ϕ(N −d)2

··· .. . ···

ϕ1(N −d) .. .

⎞ ⎟ ⎠.

ϕ(N −d)(N −d)

Ad =[aij f or (1 ≤ i ≤ p, 1 ≤ j ≤ N + 1) and 0 elsewhere], Ab =[aij f or (p + 1 ≤ i ≤ N − d, 1 ≤ j ≤ N + 1) and 0 elsewhere], Ae =[aij f or (N − d + 1 ≤ i ≤ N + 1, 1 ≤ j ≤ N + 1) and 0 elsewhere].

Using the notation LA to designate the matrix of the same dimension as A and containing the elements a ˆij where a ˆij = Laij , 1 ≤ i, j ≤ N , then Eq. (II.9) writing in matrix form together with boundary conditions Eq. (II.6), B[λ]n+1 = C[λ]n + [G]n+1 ,

(II.14)

where C = (1 − θ) · dt · (α · ΔAd + β · Ad ) + Ad ,

To determine coefficients (λn1 , λn2 , · · · , λnN ), the collocation method is used by applying Eq. (II.10) at every point xi , i = 1, 2, · · · , N − d. Thus we have, N −d 

ϕ11 .. .

Assuming that there are p < N − d internal points and N − d − p boundary points, then the (N + 1) × (N + 1) matrix A can be split into: A = Ad + Ab + Ae , where

(II.9)

Assuming that there are N − d interpolation points, based on the radial basis function approximation, u(x, tn ) can be approximated by, un (x) ≈

⎜ A1 = ⎝

(II.6)

and initial condition,

n+1

A1 X ⎟⎟⎟ ⎟, ⎟ T X 0 ⎟⎠

⎜ ⎜ ⎜ A=⎜ ⎜ ⎜ ⎝

j=1

(II.12)

B = Ad − θ · dt · (α · ΔAd + β · Ad ) + Ab + Ae , n+1 n+1 n+1 T gp+2 · · · gN and [G]n+1 = [0 0 · · · 0 gp+1 −d 0 · · · 0] , we can get all the λ’s together with the initial condition Eq. (II.7), then we get the numerical solutions. Remark. Although in Eq. (II.14) the θ can be choose for any value of θ ∈ [0, 1], we will use the famous Crank-Nicolson scheme which θ = 1/2.

III. N UMERICAL EXAMPLES A. Example 1 In this example, we consider an one-dimensional linear reaction-diffusion equation ∂u ∂2u = − 3u, 0 ≤ x ≤ 3, 0 < t, ∂t ∂x2 with the analytical solution u(x, t) = exp(−2t + x), and the initial condition

j=1

Writing Eq. (II.11) together with Eq. (II.12) in a matrix form we have, (II.13) [u]n = A[λ]n ,

u(x, 0) = exp(x). The boundary conditions we can get from the analytical solution. We use MQ (c = 0.09) as the RBF for the discussed



scheme. These results are obtained for dx = 0.01, dt = 0.001. Table I presents the L∞ , L2 error and Root-Mean-Square (RMS) of errors for t = 0.1, 0.5, 1.0, 1.5 and 2.0. We can see the errors are very small from the Table I. TABLE I NUMERICAL ERRORS AT DIFFERENT TIMES FOR EXAMPLE

t 0.1 0.5 1.0 1.5 2.0

L∞ − error 1.607×10−6 7.272×10−6 2.678×10−6 9.855×10−7 3.626×10−7

L2 − error 7.132×10−6 4.190×10−6 1.677×10−6 6.360×10−7 2.364×10−7

B. Example 2 In this example, we consider a two-dimensional linear reaction-diffusion equation ∂2u ∂2u ∂u = 0.2 · ( 2 + 2 ) + 0.1u, (x, y) ∈ [0, 2π] × [0, 2π]. ∂t ∂x ∂y

1.

The analytical solution of the equation is

RMS − error 4.097×10−6 2.407×10−6 9.637×10−7 3.653×10−7 1.358×10−7

u(x, y, t) = exp(−0.1t) · (cos x + cos y).

The graph of analytical and numerical solution for t = 0.1, 0.3, 0.5, 0.7 and 1.0 are given in Fig 1, it shows the very good accuracy and efficiency of the new approximate scheme.

We can find initial and boundary conditions from the analytical solution. We use MQ (c = 2.3) as the radial basis function with dx = dy = 0.1π and dt = 0.001 for the discussed scheme. Table II presents the L∞ and L2 error and Root-Mean-Square (RMS) of errors for different times.

TABLE II NUMERICAL ERRORS AT DIFFERENT TIMES FOR EXAMPLE

2.

The exact solution and numerical solution at different times 18

14 12

u(x,t)

t 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Exact Solution t=0.1 Numerical Solution t=0.1 Exact Solution t=0.3 Numerical Solution t=0.3 Exact Solution t=0.5 Numerical Solution t=0.5 Exact Solution t=0.7 Numerical Solution t=0.7 Exact Solution t=1.0 Numerical Solution t=1.0

16

10 8 6 4 2 0

Fig. 1.

0

0.5

1

1.5 x

2

2.5

3

Analytical and numerical solution at different times for Example 1.

We also give the space-time graph of numerical results for T = 1 in Fig 2.

L∞ − error 4.887×10−6 9.751×10−6 1.458×10−5 1.936×10−5 2.407×10−5 2.872×10−5 3.328×10−5 3.775×10−5 4.214×10−5 4.643×10−5

L2 − error 1.553×10−5 3.092×10−5 4.614×10−5 6.115×10−5 7.593×10−5 9.046×10−5 1.047×10−4 1.187×10−4 1.324×10−4 1.458×10−4

RMS − error 1.315×10−6 2.618×10−6 3.906×10−6 5.177×10−6 6.429×10−6 7.659×10−6 8.867×10−6 1.005×10−5 1.121×10−5 1.235×10−5

The graph of analytical and numerical results for T = 0.1 is given in Fig 3. The results obtained show the very good accuracy of the new approximate scheme. Note that we can’t distinguish the exact solution from the estimated solution in Fig 3.

Exact

Estimated

SpaceíTime Estimated

u(x,t)

15

1

0

í2 10

5 3 2 0.8

0.6

0.2

0

0

5

y

x

The space-time graph of numerical results for Example 1.

Fig. 3.



0 í1 í2 10

10

5

1

0.4 t

Fig. 2.

2

1

í1

10

0 1

2 Estimated

Exact

20

0 0

x

10

5 y

5 0 0

x

The numerical and analytical solutions for Example 2.

C. Example 3

Exact at (t,z)=(1,π)

In this example, we consider a three-dimensional linear reaction-diffusion equation

0.5

where Ω = {(x, y, z)|0 ≤ x ≤ π, 0 ≤ y ≤ π, 0 ≤ z ≤ π}. The exact solution has the form

L∞ − error

RMS − error

0.1

1.102×10−4

2.928×10−5

0.3

1.938×10−4

5.949×10−5

0.5

2.332×10−4

7.312×10−5

0.7

2.485×10−4

7.991×10−5

1.0

2.521×10−4

8.461×10−5

3.

0.5

Estimated

Exact

1

0.5 0 í0.5 í1

í1 í1.5

0

0 2

0 2

4 y

Fig. 4.

0 í0.5

í1.5

4 6

6

x

2

0 2

4 y

4 6

6

í2 í2.5 0 0 2 4 6

6

x

2

0 2

4 y

4 6

6

x

The numerical and analytical solutions at (t, z) = (1, π).

IV. C ONCLUSION In this paper, we proposed a numerical scheme to solve the time-dependent reaction-diffusion equation using the collocation method and approximating the solution using MQRBF. The results of numerical experiments are given with onedimensional two-dimensional and three-dimensional equations respectively and compared with analytical solutions in the previous section confirm the good accuracy of the presented scheme. This work was supported by the National Natural Science Foundation of China (Grant nos. 11301252, 11301529 and 11201212) and Applied Mathematics Enhancement Program of Linyi University.

1.5

1

í2 í2.5

y

Fig. 5.

í1 í1.5

ACKNOWLEDGMENT

Estimated at (t,z)=(1,0.5π)

1.5

í1 í1.5

4

The graph of analytical and numerical results at (t, z) = (1, 0.5π) and (t, z) = (1, π) are given in Fig 4 and Fig 5. Note that we can’t distinguish the exact solution from the estimated solution in the figures.

Exact at (t,z)=(1,0.5π)

í0.5

2

The initial condition in the simulations is taken from the exact solution at t = 0. The boundary conditions can be found from the exact solution. We use MQ (c = 1.9) as the radial basis function for the discussed scheme. These results are obtained for dx = dy = dz = 0.1π and dt = 0.001. Table III presents the L∞ -error and Root-Mean-Square (RMS) of errors for T = 0.1, 0.3, 0.5, 0.7 and 1.0. We can see that the errors in Table III are very small.

t

0

í0.5

0

u(x, y, z, t) = exp(−0.1t) · (cos x + cos y + cos z).

TABLE III

0.5

0 Estimated

Exact

∂u ∂2u ∂2u ∂2u = 0.2( 2 + 2 + 2 ) + 0.1u, (x, y, z) ∈ Ω, ∂t ∂x ∂y ∂z

NUMERICAL ERRORS AT DIFFERENT TIMES FOR EXAMPLE

Estimated at (t,z)=(1,π)

x

The numerical and analytical solutions at (t, z) = (1, 0.5π).

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