In this paper, the random Kuramoto-Sivashinsky equation with additive noise is studied numerically, using the finite difference method to simulate the effect of different amplitude of noise on the solitary wave. And numerical experiments show that the white noise does not affect the propagation of the solitary wave, but can increase the amplitude of the solitary wave.
In recent years, many scholars have studied deterministic k-s equations and made important achievements, but there are relatively few studies on stochastic Kuramoto-Sivashinsky equations, and studying the numerical solution of the equation is a new field. In general, there is no analytic solution to stochastic Kuramoto-Sivashinsky equation, so numerical analysis becomes an important tool to develop its properties. Moreover, it has high computational efficiency, low computational complexity and good reliability. In this paper, its accuracy can be seen by comparing the numerical solution with the exact solution. Moreover we can also discover some phenomena about solution properties directly by numerical analysis.
We consider the following form of nonlinear evolution equation
u t + u u x + α u x x + β u x x x x = 0 (1.1)
The coefficients of α and β are real constants, which are a number of important mathematical physics equations in many physical problems. The second and fourth order terms represent the dissipation and instability of the system respectively, and the second one represents the convective nonlinear effect. Equation (1.1) is called the Kuramoto-Sivashinsky equation, hereinafter referred to as k-s equation; it is independently obtained in the analysis of Kuramotol [
Let’s think about the k-s equation with a random term
u t + u u x + α u x x + β u x x x x = λ ξ ˙ , ( x , t ) ∈ I × R (1.2)
Here λ is the amplitude of the noise, ξ ˙ is additive noise, and a real value gaussian process. Suppose that u ( x , t ) is defined in the region R : [ − L , L ] × [ 0, t ] .
The initial condition
u ( 0 , x ) = u 0 ( x ) , − L < x < L ; (1.3)
and boundary condition
u x ( 0 , x ) = 0 , t > 0 (1.4)
The following is a mathematical definition of ξ ˙ .
Setting ( W ( t ) ) t ≥ 0 be a cylinder wiener process on L 2 ( R n , R ) , for the arbitrary orthogonal basis ( e i ) i ∈ N on the L 2 ( R n , R ) space, setting
β i ( t ) = ( W ( t ) , e i ) , i ∈ N , t ≥ 0 ,
so ( β i ) i ∈ N is a column of independent random Brownian motion, the column of Brownian motion β i ( t , ω ) , t ≥ o , ω ∈ Ω is stochastic process which is defined at random base ( Ω , Ϝ , P , ( Ϝ ( t ) ) t ≥ 0 ) , as long as t ≥ s is Ϝ t -measurable gaussian random independent variable of Ϝ s for each i, β i ( t ) − β i ( s ) . Therefore, W can be written as:
W ( t , x , ω ) = ∑ i ∈ N β i ( t , ω ) e i ( x ) , t ≥ 0 , x ∈ R , ω ∈ Ω . (1.5)
Then the temporal and spatial white noise ξ ˙ is the derivative of W to the time, that is:
ξ ˙ = d W d t = W t (1.6)
In the same way, we can also define space related noise, giving a kernel k and a linear operator Φ :
Φ f ( x ) = ∫ R n k ( x , y ) f ( y ) d y , f ∈ L 2 ( R n ) , (1.7)
defining the Wiener process W ˜ = Φ W , then its time derivative ξ ˜ ˙ is the related noise of the time δ t − s , and its spatial correlation function c:
c ( x , y ) = ∫ R n k ( x , z ) k ( y , z ) .
In form, there are E ( ξ ˜ ˙ ( x , t ) , ξ ˜ ˙ ( y , s ) ) = c ( x , y ) δ t − s . If k ( x , y ) = k ( x − y ) is a convolution kernel, noise is uniform in space namely c ( x , y ) = c ( x − y ) and the noise is temporal and spatial white noise, then there are k ( x , y ) = δ x − y and Φ = I d , then the Equation (1.2) can be written in the form of the following
d u + ( u u t + α u x x + β u x x x x ) d t = λ d W ˜ (1.8)
Literature [
Assuming that u ( x , t ) is defined on region R : [ − L , L ] × [ 0, t ] , the following partition is made to R
− L = x 0 < x 1 < ⋅ ⋅ ⋅ < x J − 1 < x J = L ,
0 = t 0 < t 1 < ⋅ ⋅ ⋅ < x N − 1 < x N = t ,
remember u j n = u ( j h , n τ ) , then in point ( j h , n τ ) there are
[ u t ] j n + 1 2 [ ( u 2 ) x ] j n + α [ u x x ] j n + β [ u x x x x ] j n = γ f j n + 1 2 (2.1)
First considering the above equation as the form of the K-S equation, replacing the [ u t ] j n with the first order difference, and replace the [ ( u 2 ) x ] j n [ u x x ] j n and [ u x x x x ] j n with the center difference, so that
u j n + 1 − u j n + τ 4 ( [ ( u 2 ) x ] j n + 1 + [ ( u 2 ) x ] j n ) + τ α 2 ( [ u x x ] j n + 1 + [ u x x ] j n ) + τ β 2 ( [ u x x x x ] j n + 1 + [ u x x x x ] j n ) = 0 (2.2)
If the partial derivative of (2.2) with respect to x is simply substituted by the difference quotient, the problem of solving nonlinear equations will be encountered, in order to overcome this difficulty, we did Taylor expansion for nonlinear terms.
[ ( u 2 ) x ] j n + 1 = [ ( u 2 ) x ] j n + [ ( u 2 ) t x ] j n τ + o ( τ 2 ) = [ ( u 2 ) x ] j n + [ ( 2 u u t ) x ] j n τ + o ( τ 2 ) = [ ( u 2 ) x ] j n + 2 [ ( u j n ) ( u j n + 1 − u j n τ + o ( τ ) ) τ ] x + o ( τ 2 ) = [ ( u 2 ) x ] j n + 2 [ ( u j n ) ( u j n + 1 − u j n ) ] x + o (τ2)
thus
[ ( u 2 ) x ] j n + [ ( u 2 ) x ] j n + 1 = 2 [ ( u j n ) ( u j n + 1 − u j n ) ] x + 2 [ ( u 2 ) x ] j n + o ( h 2 ) = 2 [ u j n u j n + 1 ] x + o ( h 2 ) = u j + 1 n u j + 1 n + 1 − u j − 1 n u j − 1 n + 1 h + o ( h 2 ) (2.3)
You can get the following difference scheme
a u j − 2 n + 1 + b j n u j − 1 n + 1 + e u j n + 1 + c j n u j + 1 n + 1 + a u j + 2 n + 2 = d j n (2.4)
among
a = τ β 2 h 4 , b j n = τ α 2 h 2 − 2 τ β h 4 − τ 4 h u j − 1 n ,
e = 1 + 3 τ β h 4 − τ α h 2 , c j n = τ α 2 h 2 − 2 τ β h 4 + τ 4 h u j + 1 n ,
d j n = − τ β 2 h 4 u j − 2 n − ( τ α 2 h 2 − 2 τ β h 4 ) u j − 1 n + ( 1 − 3 τ β h 4 + τ α h 2 ) u j n − ( τ α 2 h 2 − 2 τ β h 4 ) u j + 1 n − τ β 2 h 4 u j + 2 n , n = 0 , 1 , ⋯ , N , j = 0 , 1 , ⋯ , J .
For the difference scheme (2.4), the value of each node is required, we need to solve a large linear system of linear equations with a matrix order of J at every step of time t, according to the supposition of the boundary conditions, u − 1 = u 0 = u 1 and u J + 2 = u J + 1 = u J .
And the f j n + 1 2 of (2.1) can use the following formula to approximate
1 h τ ∫ ( j − 1 2 ) h ( j + 1 2 ) h ∫ t n t n + 1 ξ ˙ d s d x , j = 0 , ⋯ , J .
Substituting the previous (1.4) and (1.5) into the above equation, we can get
f j n + 1 2 = 1 h τ ∫ ( j − 1 2 ) h ( j + 1 2 ) h ∫ t n t n + 1 ∑ i ∈ N e i ( x ) d β i ( s ) d x = 1 h τ ∑ i ∈ N ( ∫ ( j − 1 2 ) h ( j + 1 2 ) h ∫ t n t n + 1 e i ( x ) d x ) ( β i ( t n + 1 ) − β i ( t n ) )
if the orthogonal basis ( e i ) i ∈ N on L 2 ( − L , L ) is taken as the following form
e j = 1 h 1 [ ( j − 1 2 ) h , ( j + 1 2 ) h ] , j = − J + 1 , ⋯ , J − 1
e − J = 1 h / 2 1 [ − J h , ( − J + 1 2 ) h ] , e J = 1 h / 2 1 [ ( J − 1 2 ) h , J h ] ,
then through orthogonalization, ∫ ( j − 1 2 ) h ( j + 1 2 ) h e i ( x ) d x = 0 if i ≠ j , j = 0 , ⋯ , J , i ∈ N . Further
f j n + 1 2 = 1 τ h ( β i ( t n + 1 ) − β i ( t n ) ) , j = − J + 1 , ⋯ , J − 1 ,
f − J n + 1 2 = 1 τ h / 2 ( β − J ( t n + 1 ) − β − J ( t n ) ) , f J n + 1 2 = 1 τ h / 2 ( β J ( t n + 1 ) − β J ( t n ) )
Due to ( β j ( t n + 1 ) − β j ( t n ) ) / τ is independent random variable and obeys
the standard normal distribution N ( 0,1 ) , selecting ( χ j n + 1 / 2 ) n ≥ 0 , j = − J , ⋯ , J is a
random variable that obeys the standard normal distribution. So for each time increment, f n + 1 2 can use the vector ( χ − J n + 1 / 2 , ⋅ ⋅ ⋅ , χ J n + 1 / 2 ) to simulate.
Although our purpose is to simulate the solution of K-S equation and study its properties, there is a very important problem that we need to verify whether the format described above is effective. First in the interval I × R = [ 0,1 ] × [ − 5,5 ] we simulate the initial value problem (1.1), and the initial condition is
u 0 ( x ) = − c k + 60 19 k ( − 38 β k 2 + α ) tanh k x + 120 β k 3 tanh 3 k x (3.1)
among k = 11 α 76 β , this problem has the following solitary wave solution [
u ( x , t ) = − c k + 60 19 k ( − 38 β k 2 + α ) tanh ( k x + c t ) + 120 β k 3 tanh 3 ( k x + c t ) (3.2)
Taking α = 0.1 , β = 0.1 , space step h = 0.1 , and time step τ = 0.01 ,
Now we have the numerical simulation Equation (1.7), using the methods described above and the initial conditions.
When the amplitude of noise is small, λ = 10 − 3 , as shown in
In order to further study the stability of the solitary wave, we increase the amplitude of the noise, λ = 0.5 × 10 − 2 , the impact will be strengthened. The initial conditions are the same as before, As shown in
t\x | −5 | −4 | −3 | −2 | −1 |
---|---|---|---|---|---|
0.2 | 1.795014E−03 | 1.528259E−03 | 1.463254E−03 | 1.117276E−03 | 1.017013E−03 |
0.4 | 2.379896E−03 | 2.459518E−03 | 2.538881E−03 | 2.617961E−03 | 2.696738E−03 |
0.6 | 2.650232E−03 | 2.584384E−03 | 2.546679E−03 | 2.455604E−03 | 2.339724E−03 |
0.8 | 5.060148E−03 | 5.174104E−03 | 5.415453E−03 | 5.793879E−03 | 5.324566E−03 |
1.0 | 6.568193E−03 | 6.504965E−03 | 6.490092E−03 | 7.487003E−03 | 7.521187E−03 |
t\x | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
0.2 | 1.015736E−03 | 1.128104E−03 | 1.448107E−03 | 1.721008E−03 | 1.920104E−03 |
0.4 | 2.795016E−03 | 2.028257E−03 | 2.463255E−03 | 2.117272E−03 | 2.017014E−03 |
0.6 | 3.450235E−03 | 3.684386E−03 | 3.146678E−03 | 3.855605E−03 | 3.839722E−03 |
0.8 | 5.060142E−03 | 5.174102E−03 | 5.415453E−03 | 5.793874E−03 | 5.824569E−03 |
1.0 | 6.168193E−03 | 6.204967E−03 | 7.290094E−03 | 7.427002E−03 | 7.621184E−03 |
increases through the propagation of the solitary wave, which makes it clear that the noise will enhance the amplitude of the wave.
Increasing the amplitude of the noise again, λ = 0.01 as shown in Figure2(c), in this case, the whole solitary wave is still intact, in order to further study this phenomenon, we use another representation method to represent the image of the solution, as shown in
In this paper, the finite difference method is used to carry out numerical experiments on the solution of the random K-S equation. The results show that the noise does not affect the propagation of the solitary wave, but it can enhance the amplitude of the solitary wave. This is similar to the phenomenon observed in random Kdv equations [
The authors declare no conflicts of interest regarding the publication of this paper.
Gao, P., Cai, C.J. and Liu, X.Y. (2018) Numerical Simulation of Stochastic Kuramoto-Sivashinsky Equation. Journal of Applied Mathematics and Physics, 6, 2363-2369. https://doi.org/10.4236/jamp.2018.611198