Convergence of finite element solution of stochastic Burgers equation

1.   Introduction

We consider the stochastic Burgers equation

where $D = [0, L]^{2}\subset\mathbb{R}^{2}$, $\mathit{\boldsymbol{u}} = (u_{1}, u_{2})$, $\nu > 0$, $L > 0, T > 0$. $\mathit{\boldsymbol{\Psi}}$ is a deterministic function. $B^{H}(t)$ is a fractional Brownian motion on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_{t}\}_{t\geq0})$, $H\in(1/4, 1/2)$ or $H\in(1/2, 1)$. $\mathit{\boldsymbol{u}}_{0}$ is a given random variable.

The stochastic Burgers equation (SBE) has applications in many areas^[1,2,3,4,5]. For $H = 1/2$ in (1.1), Bertini et al. ^[6], Brzeźniak et al. ^[7], and Kim ^[8] established the well-posedness results of the SBE. Catuogno and Olivera ^[9] proved the existence of a strong solution of the SBE. Twardowska and Zabczyk ^[10] and Goldys and Maslowski ^[11] obtained the ergodicity results of the SBE. In ^[12], E et al. established a stationary solution of the SBE. Zou and Wang ^[13] considered the existence result of a fractional-order SBE. Zhou et al. ^[14] obtained strong solutions for the SBE. The SBE is a typical evolution equation, and recently a series of numerical methods, like the two-grid method ^[15,16,17], ADI method^[18,19,20], finite difference method^[21,22], spectral method^[23], finite volume method^[24,25], and extrapolation method ^[26], have been developed to solve it. From a computational point of view, Hairer and Voss ^[27] devoted their research to the finite difference approximations of the SBE. Hairer and Matetski ^[28] obtained the optimal convergence rate of the SBE. Blömker and Jentzen ^[29] studied Galerkin approximations of the SBE. Jentzen et al. ^[30] proposed and analyzed explicit space-time discrete numerical approximations for the SBE. Uma et al. ^[31] developed an approximate solution method for solving the SBE.

Fractional Brownian motion (fBm for short) is a type of Gaussian process that exhibits long-range dependence. It is characterized by its self-similarity and its correlation structure which is determined by a parameter called the Hurst index $H$. It plays a crucial role in various fields. Its applications can be broadly categorized into two main areas: finance and image/signal processing. In finance, fBm is utilized for modeling and simulating price movements in financial markets. It is particularly useful in capturing the long-term dependency and self-similarity observed in asset prices. Furthermore, fBm has notable applications in image and signal processing. Due to its ability to represent fractal-like structures, fBm is employed for texture synthesis and texture modeling, enabling the generation of realistic textures. Moreover, fBm is utilized for noise reduction, interpolation, and denoising in image processing, providing improved image quality and enhanced feature extraction capabilities. These applications have motivated theoretical and numerical investigations of stochastic differential equations (SDE) driven by fBm. For examples of theoretical results of the SDE driven by fBm, see, Wang et al. ^[32], Jiang et al. ^[33], Hinz ^[34], Pei et al. ^[35], Yang et al. ^[36], Zou et al. ^[37], and the references therein. However, there is less literature on studying numerical approximations for the SDE driven by fBm. For example, Cao et al. ^[38] considered finite element approximations for the SDE driven by fBm. Tudor ^[39] investigated Wong-Zakai type approximations. Qi and Lin ^[40] obtained the optimal error bound. Hong et al. ^[41] investigated the super-convergence result for the stochastic heat equation driven by fBm. The challenge of analyzing the convergence of SDE driven by fBm is that the Burkholder-Davis-Gundy inequality is unavailable. Therefore, it is necessary to take a different strategy to solve this problem. In the existing literature, we are not aware of any numerical investigation of (1.1)–(1.3). This paper aims to fill the gap.

The main difficulty of equations (1.1)–(1.3) is that the noise term causes the solutions $\mathit{\boldsymbol{u}}$ to be very "rough". Thus, new techniques and skills need to be explored. First of all, we establish the following energy estimates

and

where $C_{H, T} > 0$. Next, we prove the Hölder regularity results, that is, if $H\in(1/4, 1/2)$, it holds that

and if $H\in(1/2, 1)$, we obtain

with the induced norms $\|\cdot\|_{\beta} = \|A^{\beta/2}\cdot\|$ for any $0\leq \beta\leq1$. For the detailed definitions of operator $A$ and operator $A^{\beta/2}$, we refer the readers to Section 2. A key tool used here is a very careful treatment of multiple integrals.

To prove the convergence, the main problem here is caused by the subtle interplay of the nonlinear convection term and the stochastic forcing, which prevents a Gronwall argument in the context of expectations. We investigate the error estimates on a subset of $\Omega$ and use Markov's inequality to overcome the difficulties. Based on the regularity properties, and for a subset $\Omega_{k}\subset \Omega$, with $\mathbb{P}[\Omega_{k}]\rightarrow 1\; \mathrm{as}\; k\rightarrow 0$, if $H\in(1/4, 1/2)$, then

and if $H \in(1/2, 1)$, we have

where $\mathbb{E}_{\Omega_{k}}[\cdot] = \mathbb{E}[\mathbb{I}_{\Omega_{k}}\cdot]$, $\varepsilon > 0$ is arbitrarily small, and $k$ is the time step. For a subset $\Omega_{h}\subset \Omega$, with $\mathbb{P}[\Omega_{h}]\rightarrow 1\; \mathrm{as}\; h\rightarrow 0$, where $h > 0$ is the space step, if $H\in(1/4, 1/2)$, we have

and for $H\in(1/2, 1)$, we get

where $\Omega_{k, h} = \Omega_{k}\cap\Omega_{h}$, and both $\epsilon > 0$ and $\varepsilon > 0$ are arbitrarily small. It should be emphasized that the convergence analysis presented here requires various delicate error estimates, which is different from the case of the Wiener process.

Section 2 presents some preliminaries. The spatial and temporal regularity results of (1.1)–(1.3) are established in Section 3. Section 4 shows the convergence results for the time discretization scheme. Section 5 is devoted to the space-time discretization. We present the numerical experiments in Section 6.

2.   Notations and preliminaries

We assume that $L^{q}(D)$ and $W^{k, p}(D)$ are Lebesgue and Sobolev spaces. Let $\mathbb{L}^{q}(D): = [L^{q}(D)]^{2}$, $\mathbb{W}^{k, p}(D): = [W^{k, p}(D)]^{2}$. We define the space

We define the operator $A$ via $A{\boldsymbol{u}} = -\nu\Delta \mathit{\boldsymbol{u}}$, in the domain $D(A) = V\cap \mathbb{W}^{2, 2}(D)$. We define the operator $A^{\frac{s}{2}}, s\in \mathbb{R}$, by

where $\{(\gamma_j, e_j)\}^{\infty}_{j = 1}$ are the pairs of eigenvalues and eigenfunctions of $A$.

Let $\dot{V}^{s} = D(A^{\frac{s}{2}})$ be the Hilbert space endowed with the norm

We define the space $\mathcal{L}(V)$ by

with norm $\|\cdot\|_{\mathcal{L}(V)}$.

It is also well-known that (see ^[42])

and

Definition 2.1.^[1,43] A continuous Gaussian process $\{B^{H}(t), t\geq 0\}$ that satisfies

is called an fBm, where $H\in(0, 1)$ is the Hurst index. In particular, if $H = \frac{1}{2}$, then the process is in fact Brownian motion or a Wiener process.

Define $K_{H}(t, s)$ by

where $C_{H} = (\frac{2H\Gamma(\frac{3}{2}-H)}{\Gamma(H+\frac{1}{2})\Gamma(2-2H)})^{\frac{1}{2}}$ is a constant.

By (2.3) and (2.4), we have

Let $\mathcal{H}$ denote the reproducing kernel Hilbert space of fBm. For every $0 < H < 1, \; s < \tau$, we define the linear mapping $K_{\tau}^{*}:\mathcal{H}\rightarrow L^{2}([0, T])$^[43,44] by

For $1/2 < H < 1$, we have a simpler expression

Define a linear bounded covariance operator as $Q$ that satisfies $\mathrm{Tr} (Q) < \infty$. The corresponding eigenvalues and eigenfunctions are denoted by $\{(\lambda_k, e_k)\}^{\infty}_{k = 1}$. We denote by $\mathcal{L}_{2}^{0}: = \mathcal{L}_{2}^{0}(Y, X)$ the space of all $Q$-Hilbert-Schmidt operators $\phi:Y\rightarrow X$ endowed with the norm $\|\phi\|_{\mathcal{L}_{2}^{0}}^{2} = \sum\limits_{k = 1}^{\infty}\|\lambda^{1/2}_{k}\phi e_{k}\|^{2} < \infty$, where $Y$ and $X$ are real separable Hilbert spaces.

For the definition of the Wiener integral $\int_{0}^{t}\sigma(s)dB^{H}(s)$, we refer to references ^[43,45]. Then, the stochastic integral satisfies

and the inequality holds

where $\{B(t), t\in[0, T]\}$ is a Wiener process, the function $\sigma:[0, T]\rightarrow \mathcal{L}_{2}^{0}$, and the constant $C(p) > 0$.

We introduce the following assumptions:

$(S_{1})$ ${\boldsymbol{u}}_{0}\in L^{p}(\Omega, \mathcal{F}, \mathbb{P}, V), \; p\geq2$.

$(S_{2})$ For $\gamma\in [0, 3], \; \delta\in[0, \gamma/2]$, the mapping $\mathit{\boldsymbol{\Psi}}: [0, T] \rightarrow \mathcal{L}_{2}^{0}$ satisfies

and

where $C_{T}$ is a constant.

Definition 2.2. (Strong solution) Let $(S_{1})$–$(S_{2})$ be valid. An adapted $V$-valued process $\{{\boldsymbol{u}}(t)\}_{t\in [0, T]}$ is called a strong solution to (1.1)–(1.3) if ${\boldsymbol{u}}(\cdot, \omega)\in C([0, T];V)\cap L^{2}(0, T;\mathbb{W}^{2, 2}\cap V)$ $\mathbb{P}$-a.s., and for $\forall t\in[0, T], \; \forall \mathit{\boldsymbol{v}}\in V$,

The following properties are valid:

Using integration by parts, we have

From the above estimate, using the Gagliardo-Nirenberg-Sobolev inequality $\mathbb{W}^{1, 2}(D)\subset \mathbb{L}^{4}(D)$ leads to

and applying Ladyzhenskaya's inequality $\|{\boldsymbol{u}}\|_{\mathbb{L}^{4}}\leq C\|{\boldsymbol{u}}\|_{\mathbb{L}^{2}}^{1/2}\|\nabla{\boldsymbol{u}}\|_{\mathbb{L}^{2}}^{1/2}$ shows

More about the above properties of trilinear operator can be found in ^[46].

3.   The regularity results

We establish the regularity properties of (2.12) in the spatial and temporal directions.

Theorem 3.1. We assume that $(S_{1})$–$(S_{2})$ hold. If $H\in(1/4, 1/2)$ or $H\in(1/2, 1)$, then the solution $\mathit{\boldsymbol{u}}(t)$ to (1.1)–(1.3) satisfies

and

where $C(H, T) > 0$ is a constant.

Proof. $(i)$. By (1.1)–(1.3), we have

Applying (2.13), we deduce

Integrating (3.3) from $0$ to $t$ yields

Using Young's inequality and (2.9), we have

For $1/4 < H < 1/2$, by (2.6), the inequality (^[32])

and the application of Hölder's inequality and (2.5), (2.10), and (2.11) gives

where we use the fact $(s/r)^{1-2H}\leq 1$ for $s\leq r$ and $H\in(1/4, 1/2)$.

Similarly, for $H\in(1/2, 1)$, one can derive that

where we use the results $r^{2H-1}\leq t^{2H-1}$ for $H > 1/2$ and $r\leq t$, $(t-s)^{2H-1}\leq T^{2H-1}$ for $(s, t)\subset[0, T]$.

Therefore, (3.3)–(3.7) lead us to the proof of (3.1).

$(ii)$. From (1.1), we have

and by the Dirichlet condition, it can be concluded that

Integrating (3.8) from $0$ to $t$, we have

Using the inequality $|([\mathit{\boldsymbol{u}}(s)\cdot\nabla]\mathit{\boldsymbol{u}}(s), \Delta\mathit{\boldsymbol{u}}(s))|\leq C\|\nabla \mathit{\boldsymbol{u}}(s)\|^{2}\|\Delta \mathit{\boldsymbol{u}}(s)\|$ (see ^[47]) and Young's inequality, we have

For the stochastic integral term in (3.9), by (2.10), using the same methods as in the derivation of (3.6) and (3.7), we have

Let $\nu > 0$ be sufficiently large such that $\nu-C(H, T, \mathit{\boldsymbol{u}}_{0}) > 0$. Taking the expectation, from (3.8)–(3.11), we finish the proof of (3.2).

Theorem 3.2. If $(S_{1})$–$(S_{2})$ are valid and $\beta \in [0, 1]$, the strong solution $\mathit{\boldsymbol{u}}(t)$ is Hölder continuous. Furthermore, if the Hurst index satisfies $H\in(1/4, 1/2)$, it holds that

and for $H\in(1/2, 1)$, it holds that

Proof. Referring to some results available in ^[47], we can represent the strong solution to (1.1)–(1.3) as

For any $t_{1} < t_{2}$, from (3.12), we have

Application of (2.1) and (2.2) leads to

Using the Sobolev embedding theorem, we can obtain

Furthermore,

For $I_{b1}$ in (3.16), by (2.1), (2.2), (3.15), and (3.2), we have

For the term $I_{b2}$, we obtain

The term $I_{c}$ also can be rearranged as

For $1/4 < H < 1/2$, by (2.9), we get

Using (2.1), (2.2), (2.10), and Hölder's inequality, one can obtain

Similarly, we get

where we use the fact $(s/r)^{1-2H}\leq1$ for $r > s$ and $1/4 < H < 1/2$. In a similar manner, we have

For the term $I_{c2}$, we obtain

For the term $I_{c21}$, we have

where we use the inequality $t_{2}^{\alpha}-t_{1}^{\alpha}\leq (t_{2}-t_{1})^{\alpha}$ for $0 < \alpha < 1$ and $t_{2} > t_{1}$.

Similarly, we have

and

Now, we consider $H\in(1/2, 1)$ in (3.19). By Lagrange's mean value theorem, $\exists \; \zeta\in(s, t)\subset[0, T]$ such that $t^{4H-1}-s^{4H-1} = (4H-1)\zeta^{4H-2}(t-s)\leq (4H-1)t^{4H-2}(t-s)$. Then, we have

For $I_{c2}$, owing to the fact $s^{1-2H}\leq t_{1}^{1-2H}$ with $s\geq t_{1}$ and $H > 1/2$, we can then obtain

Combining with (3.13)–(3.29), the proofs of Theorem 3.2 are finished.

4.   The implicit Euler scheme

We let $t_{n} = nk, \; k = T/N,$ for $n = 0, 1, \cdots, N, \; N\in \mathbb{N}_{+}$. Furthermore, we let ${\boldsymbol{u}}^{0}: = {\boldsymbol{u}}_{0}$. We define an implicit Euler scheme by seeking a random variable ${\boldsymbol{u}}^{n}$ in $V$ such that $\mathbb{P}$-a.s.

for any $\mathit{\boldsymbol{v}}\in V$, where $\Delta B_{n}^{H} = B^{H}(t_{n})-B^{H}(t_{n-1})$ denotes the fractional Brownian increments.

Lemma 4.1. We assume that $(S_{1})$–$(S_{2})$ hold. For ${\boldsymbol{u}}_{0}\in V$ and $0\leq s < t\leq T$, we have

Proof. Setting $\mathit{\boldsymbol{v}} = {\boldsymbol{u}}^{n}$ in (4.1), we have

By $(a-b)a = \frac{1}{2}[a^{2}-b^{2}+(a-b)^{2}]$ and $([{\boldsymbol{u}}^{n}\cdot\nabla]{\boldsymbol{u}}^{n}, {\boldsymbol{u}}^{n}) = 0$, we can reformulate (4.3) as

Using (2.9) and Young's inequality, we have

For the case of $1/4 < H < 1/2$, we have

Similarly, for $H\in(1/2, 1)$, one can derive that

Summing up (4.4) from $l = 1$ to $n$, taking expectations and applying (4.5)–(4.7) gives

Taking the maximum of (4.8) over $1\leq n\leq N$, and using Gronwall's inequality, we finish the proof.

Theorem 4.1. We assume that $(S_{1})$–$(S_{2})$ hold. We let ${\boldsymbol{u}}(t_{n})$ and ${\boldsymbol{u}}^{n}$ be the solutions to (2.12) and (4.1), respectively. For the Hurst index $H \in(1/4, 1/2)$, it holds that

and for $H \in(1/2, 1)$, we have

where $\Omega_{k}\subset \Omega$ such that $\mathbb{P}[\Omega_{k}]\rightarrow 1\; \; \mathrm{as}\; \; k\rightarrow 0$, $\varepsilon > 0$ is arbitrarily small.

Proof. We define the error ${\boldsymbol{e}}^{n} = {\boldsymbol{u}}(t_{n})-{\boldsymbol{u}}^{n}$. From Eqs (2.12) and (4.1), we obtain

Setting $\mathit{\boldsymbol{v}} = {\boldsymbol{e}}^{n}$ in (4.9), and using $(a-b)a = \frac{1}{2}(a^{2}-b^{2})+\frac{1}{2}(a-b)^{2}$, it is easy to get that

By means of Young's inequality, we have

For the case $1/4 < H < 1/2$, we have

and for $1/2 < H < 1$, we have

For the convection term in (4.9), we get

Using (2.14) and Ladyzhenskaya's inequality, we have

For $1/4 < H < 1/2$, by (2.14) and Young's inequality, we have

and for $1/2 < H < 1$, it holds that

For the term $L_{3}$ and if $1/4 < H < 1/2$, we have

and for $1/2 < H < 1$, we get

To bound the stochastic integral, using (2.9) and Young's inequality, we have

For $1/4 < H < 1/2$, by $(S_2)$, we have

For the case $1/2 < H < 1$, we have

By summing up (4.9) with $\mathit{\boldsymbol{v}} = {\boldsymbol{e}}^{n}$ from $n = 1$ to $N$, as well as considering the estimates (4.10)–(4.22), for $1/4 < H < 1/2$, we have

and for $1/2 < H < 1$, we get

Due to the expectations of $\{\|\nabla{\boldsymbol{u}}^{n}\|^{2}\}_{n = 1}^{N}$, we may not apply the discrete Gronwall inequality. We consider a subset

where $k > 0$, and application of Markov's inequality leads to

We define the indicator function $\mathbb{I}_{\Omega_{k}}$ by

Using the discrete Gronwall inequality, we have

and

For any $\varepsilon > 0$, $C_{T}^{-1}\ln (k^{-\varepsilon}) > 0$, and we denote by $\mathbb{E}_{\Omega_{k}}[\cdot] = \mathbb{E}[\mathbb{I}_{\Omega_{k}}\cdot]$, such that

and then if $1/4 < H < 1/2$, we obtain

and for $1/2 < H < 1$, it holds that

The proofs are finished.

5.   Space-time discretization

We let $\{\mathcal{T}_{h}\}_{0 < h < 1}$ be a regular family of triangulations of $D$ with the maximal mesh size of $h$. We define the finite element spaces $V_{h}$ by

$P_{h}$ is defined as the standard $L^{2}$-projection operator, i.e.,

we have (see ^[48])

The fully discrete scheme of (1.1)–(1.3) is to seek ${\boldsymbol{u}}_{h}^{n}\in L^{2}(\Omega, V_{h})$ such that

for any $\mathit{\boldsymbol{v}}_{h}\in V_{h}$, where $\Delta B_{n}^{H}$ denotes the fractional Brownian increments.

Theorem 5.1. We assume that $(S_{1})$–$(S_{2})$ hold. Let ${\boldsymbol{u}}_{h}^{n}$ and ${\boldsymbol{u}}^{n}$ be the solutions to (5.2) and (4.1), respectively. For $H\in(1/4, 1/2)$, it holds that

and if $H\in(1/2, 1)$, one can arrive at

where $\Omega_{h}\subset \Omega$ such that $\mathbb{P}[\Omega_{h}]\rightarrow 1\; \; \mathrm{as}\; \; h\rightarrow 0$, $\epsilon > 0$ is arbitrarily small.

Proof. Setting $\mathbf{E}^{n} = {\boldsymbol{u}}^{n}-{\boldsymbol{u}}_{h}^{n}$, from (4.1) and (5.2), we get

Taking $\mathit{\boldsymbol{v}}_{h} = \mathbf{E}^{n}$, we reformulate (5.3) as

For the convection term in (5.4), we have

and

For the stochastic integral, if $1/4 < H < 1/2$, utilizing (5.1) and (2.9), similar to the derivation of (4.6), we then have

and if $1/2 < H < 1$, similar to the derivation of (4.7), it follows that

Due to the result $\|{\boldsymbol{E}}^{0}\| = \|{\boldsymbol{u}}_{0}-P_{h}{\boldsymbol{u}}_{0}\|\leq Ch\|{\boldsymbol{u}}_{0}\|_{1}$, by summing up (5.4) from $n = 1$ to $N$, as well as considering the estimates (5.5)–(5.8), and for $1/4 < H < 1/2$, we have

and if $1/2 < H < 1$, we have

We consider a subset

which satisfies Markov's inequality, together with

Denote by $\mathbb{E}_{\Omega_{h}}[\cdot] = \mathbb{E}[\mathbb{I}_{\Omega_{h}}\cdot]$, from (5.3)–(5.10), using the discrete Gronwall inequality, for $1/4 < H < 1/2$, we have

and for $1/2 < H < 1$, then

The proofs are finished.

We set $\Omega_{k, h} = \Omega_{k}\cap\Omega_{h}$. By using the triangle inequality, then the global error estimates for fully discrete method are given as follows:

Theorem 5.2. We assume that $(S_{1})$–$(S_{2})$ hold. Let ${\boldsymbol{u}}(t_{n})$ and ${\boldsymbol{u}}_{h}^{n}$ be the solutions to (2.12) and (5.2), respectively. For the Hurst index $H\in(1/4, 1/2)$, there holds

and for $H\in(1/2, 1)$, we have

where $\Omega_{k, h}\subset \Omega$ such that $\mathbb{P}[\Omega_{k, h}]\rightarrow 1\; \mathrm{as}\; k\rightarrow 0\; \mathrm{and} \; h\rightarrow 0$, both $\epsilon > 0$ and $\varepsilon > 0$ are arbitrarily small.

Remark. The absence of a nonlinear convective term $[\mathit{\boldsymbol{u}}\cdot\nabla]\mathit{\boldsymbol{u}}$ leads to $\Omega_{k, h}\equiv\Omega$, that is $\varepsilon = \epsilon = 0$ in Theorem 5.2, since the Gronwall inequality may now be utilized directly. We use Newton's iterative algorithm when calculating the nonlinear term in numerical calculations.

6.   Numerical experiments

We present the approximation of fBm $B^{H}(t) = \int_{0}^{t}K_{H}(t, s)dB(s)$, where $B(t)$ is a Brownian motion (see ^[1]). Let $t_{n} = nk$ for $n = 0, 1, \ldots, N$ and $k = T/N$, then we approximate $B^{H}(t)$ by

It should be pointed out that the integral $\int_{t_{i}}^{t_{i+1}}K_{H}(t_{n}, s)ds$ is approximated by $K_{H}(t_{n}, \frac{t_{i+1}+t_{i}}{2})$.

The first example provided is to show the convergence rates, and the second example is used to show the numerical simulations about the impacts of fBm on the Burgers equation.

6.1.   Numerical test 1

where $\Omega = \{(x, y)|0\leq x, y\leq 2\}$.

We take the temporal interval $[0, 1]$ and the viscosity coefficient $\nu = 0.2$. The errors $\mathbb{E}[\|{\boldsymbol{e}}^{n}\|]: = (\mathbb{E}[\|{\boldsymbol{u}}_{h}^{n}-{\boldsymbol{u}}(t_{n})\|^{2}])^{1/2}$ in the sense of the $L_{2}$-norm are computed by Monte Carlo method over 200 samples, where the "true" solution ${\boldsymbol{u}}(t_{n})$ is approximated by a solution computed by small time step $k = \frac{1}{160}$ and space step $h = \frac{1}{200}$.

In Table 1, if the Hurst index $H$ satisfies $1/4 < H < 1/2$, the convergence order is close to the theoretical convergence order $\mathcal{O}(k^{\min\{2H-1/2, H\}-\varepsilon})$, and for Hurst index $1/2 < H < 1$, the orders are near to $\mathcal{O}(k^{\min\{3/4, H\}-\varepsilon})$. Table 2 shows that the optimal order of spatial error estimation is consistent with the theoretical result of $\mathcal{O}(h^{1-\epsilon})$.

6.2.   Numerical test 2

Here, let us consider the two-dimensional stochastic Burger equations with $\mathit{\boldsymbol{\Psi}}(t) = t$ in (1.1) on the spatial domain $\Omega = [0, 2]\times[0, 2]$ and temporal interval $[0, 1]$:

We choose the viscosity coefficient $\nu = 0.1$, $h = 1/40$, and $k = 0.05$. Figure 1 (A1 and B1) shows the numerical results of the corresponding deterministic equation ($\mathit{\boldsymbol{\Psi}}(t) = 0$). Figure 1 (A2 and B2) and Figure 2 (A3 and B4) exhibit the mean values of $u$ and $v$ to the stochastic Burger equation with Hurst indexes $H = 0.3, 0.6, 0.8$, respectively.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work is supported by R & D Program of Beijing Municipal Education Commission (KM202310853001), Research Project on Youth Foundation of Beijing Polytechnic College (BGY2021KY-05QT), Key Research Project of Beijing Polytechnic College (BGY2023KY-47Z), Research Project of Beijing Polytechnic College (BGY2023KY-50) and Beijing Natural Science Foundation (No.1224036).

Conflicts of interest

The authors declare no conflict of interest.