Mean Square Consistency on Numerical Solutions of Stochastic Wave Equation with Cubic Nonlinearities on 2D Rectangles

Abstract. In this article we study the mean square consistency on numerical solutions of stochastic wave equations with cubic nonlinearities on two dimensional rectangles. In [8], we proved that the strong Fourier solution of these semi-linear wave equations exists and is unique on an appropriate Hilbert space. A linear-implicit Euler method is used to discretize the related Fourier coefficients. We prove that the linear-implicit Euler method applied to a solution of nonlinear stochastic wave equations in two dimensions is mean square consistency under the geometric condition.


I. INTRODUCTION
In this article we study the linear-implicit Euler method for the numerical solution of semi-linear stochastic wave equations utt = σ 2 u + A(u, ut) + B (u, ut) dW dt with cubic nonlinearities in two dimensions in terms of all systems parameters, i.e., with non-global Lipschitz continuous nonlinearities.
Our study focuses on numerical solution using linear-implicit Euler method (LIEM) under the geometric condition (Note that lx and ly are the dimension parameters of a vibrating plate or a membrane). We shall impose the following boundary conditions: (note that t > 0)  Also the initial conditions are u(x, y, 0) = f (x, y) with f ∈ L 2 (initial position) and ut(x, y, 0) = g(x, y) with g ∈ L 2 (initial velocity). Recall that where μ is the Lebesgue measure in two dimensions.
There is a little information about the numerical solution of nonlinear stochastic wave equations. However, Schurz [17] proved that the linear-implicit method for the nonlinear stochastic heat equation just in R is mean square consistent with rate r0 = 1.5. Also, Higham [11] studied the mean square stability of stochastic theta method and plotted the mean square stability when the test equation has real parameters. Higham, D.J., Mao, X., Stuart, A.M. [12] proved that strong convergence results but they took less restrictive conditions. To my information and since there are a few researcher work on mean square consistent of LIEM of nonlinear stochastic wave equations in two dimensions, which make me interested in working that.
Consider the stochastic nonlinear wave equation with both additive and multiplicative noise The form of Fourier solutions u and its approximate Fourier solutions uN given by with its coefficients c i,j n,m satisfying where λn = Note that equation (4) is equivalent to the following systeṁ The solution of system (5) is exist and unique [8]. Now, we introduce the following standard definitions. Definition 1: For k ∈ N, take the partition 0 = t0 < t1 < t2 < ... < t k = T of [0, T ] with current step sizes h k = t k+1 − t k > 0, then , as in Schurz [18], the linear-implicit Euler-type method (LIEM) is governed by the iterative scheme where In [10], we proved that the explicit representation of LIEM as in the following theorem Theorem 1: Assume that c i,j n,m (t k ) < ∞, v i,j n,m (t k ) < ∞, and a2 ≥ 0, κ ≥ 0, and ∀ n, m ∈ N : − σ 2 (λn + βm) + a1 h 2 k < 1 + h k κ. Then the method (LIEM) governed by equation (6) and equation (7) has the non-exploding explicit representation and where fn,m(u(t k )), gn,m(u(t k )), k W i,j n,m , and h k as above. Proof 2: (Theorem 1) See [10] II. MEAN SQUARE CONSISTENCY OF NUMERICAL METHODS Definition 2: A numerical method with the schemê where and f (u(t)) and g(u(t)) as above applied to SDE (5) is said to be mean square consistent with Lemma 3: (The Burkholder-Davis-Gundy Inequality) Proof 4: (Lemma 1) See Karatzas and Shreve [13].
Proof 6: (Lemma 2) See [9]. Theorem 7: The method (LIEM) given by is locally mean square consistent with rate r2 ≥ 1, where f (u(t k )) and g(u(t k )) as above and Proof 8: (Theorem 2) We want to prove that To prove inequality (14), we know from [10] that Thus, First, we will find the first part of the right hand side of inequality (16), Step 1: We know that dvn,m(r) by using Hölder and Burkerholder-Davis-Gundy inequalities, Lemma 3, then substituting Step 2: Also, we know that, which gives F (u(t)) ≥ 0. Now, if we pulling the expectation over the squared norm of the latter identity, we find that for large N and take the conditional expectation, then we get Therefore, Form inequalities (20) and (21), we have where K7 = K5 + K6. Now, the second part of inequality (16) is using Lemma 5.1, Fc(u(t)) ≥ 0, and (a+b+c) 4 = ((a+b+c) 2 ) 2 ≤ (3(a 2 +b 2 +c 2 )) 2 ≤ 27(a 4 +b 4 +c 4 ), Hence, by redoing above steps for the conditional expectation, we arrive at because we know from the Young's inequality that if p = 2 and q = 2, that a · b ≤ a 2 2 where Thus inequality (16) equivalent to 2) To prove inequality (15), from [10] we know that,

fv(u(s))−fv(u(t)) ds
and we know that
Pulling the conditional expectation over the last identity, we find that To complete the proof, we have to simplify the second part of the right side of inequality (24) which is t+h t gv(u(s))−ĝv(u(t)) dW (s) 2 N ≤CB t+h t gv(u(s))−ĝv(u(t)) 2 N ds but from [10] we know that We start with the first part of the right side of inequality (35) which is t+h t gv(u(s))−gv(u(t)) 2 N d s