Numerical Simulation of Stochastic Kuramoto-Sivashinsky Equation

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.


Introduction
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 The coefficients of α and β are real constants, which are a number of im- portant mathematical physics equations in many physical problems.The second and fourth order terms represent the dissipation and instability of the system  [1] in dissipative structure of reaction diffusion system and Sivashinsky [2] in flame combustion and fluid dynamics instability.However, in practical situations, we must consider the effect of small irregular random factors, for example, adding a random force term to the right of the equation.
Let's think about the k-s equation with a random term ( ) Here λ is the amplitude of the noise, ξ  is additive noise, and a real value gaussian process.Suppose that ( ) The initial condition and boundary condition ( ) The following is a mathematical definition of ξ  .Setting Then the temporal and spatial white noise ξ  is the derivative of W to the time, that is: In the same way, we can also define space related noise, giving a kernel k and a linear operator Φ : defining the Wiener process W W = Φ  , then its time derivative ξ   is the related noise of the time t s δ − , and its spatial correlation function c: In form, there are 2) can be written in the form of the following ( ) Literature [3] proves that the Equations (1.8), (1.3) and (1.4) have a unique solution.In this paper, the finite difference method is used to simulate the solutions of Equations (1.8), (1.3) and (1.4), and the results of the numerical analysis will be obtained.

Derivation of the Difference Scheme
Assuming ( ) First considering the above equation as the form of the K-S equation, replacing the [ ] n t j u with the first order difference, and replace the ( ) with the center difference, so that 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. ( 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, And the

1) can use the following formula to approximate
Substituting the previous (1.4) and (1.5) into the above equation, we can get then through orthogonalization, ( ) ( ) is independent random variable and obeys the standard normal distribution ( ) random variable that obeys the standard normal distribution.So for each time increment,

Numerical Simulation
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 we simulate the initial value problem (1.1), and the initial condition is ( ) ( ) , Figure 1 below shows an image of a numerical solution and an exact solution, Table 1 and Table 2 show the absolute error between different numerical solutions and exact solutions.The obtained numerical solution can well reflect the solution of the equation, indicating that the format described in this paper is valid.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, In order to further study the stability of the solitary wave, we increase the amplitude of the noise, noise is uniform in space namely ( ) ( ) , c x y c x y = − and P. Gao et al.DOI: 10.4236/jamp.2018.6111982365 Journal of Applied Mathematics and Physics the noise is temporal and spatial white noise, then there are ( ) then the Equation (1. in Figure2(a)made a time t in the interval [ ] 0, 3 image, the other parameters are the same as before, it can be seen that the solitary wave is not destroyed, and the noise does not stop the propagation of the wave.We can see the same phenomenon by choosing different noise tracks.

Table 1 .
The absolute error data table between the numerical solutions and the exact so-

Table 2 .
The absolute error data table between the numerical solutions and the exact so-