Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models
In this work, we develop a Legendre spectral element method (LSEM) for solving the stochastic nonlinear system of advection-reaction-diffusion models. The used basis functions are based on a class of Legendre functions such that their mass and diffuse matrices are tridiagonal and diagonal, respectively. The temporal variable is discretized by a Crank–Nicolson finite difference formulation. In the stochastic direction, we also employ a random variable based on the Wiener process. We inspect the rate of convergence and the unconditional stability for the achieved semi-discrete formulation.
Then, the Legendre spectral element technique is used to obtain a full-discrete scheme. The error estimation of the proposed numerical scheme is substantiated based upon the energy method. The numerical results confirm the theoretical analysis.
Keywords: Nonlinear system of advection-reaction-diffusion equation, error estimate, spectral element method (SEM), stochastic PDEs, .
AMS subject Classification: 65M70, 34A34.
where , and denote the concentrations of the main ground substance, aqueous solution electrolyte and microorganism, respectively [1, 2]. In the above model is a known function, is the advection coefficient, is the diffusion coefficient, and are constant, respectively. Also, is a -Wiener process with respect to a filtered probability space . The nonlinear terms are
Predictions of solute transport in aquifers generally have to rely on mathematical models based on groundwater flow and convection-dispersion equations. The groundwater model is employed to prevent and control the groundwater contaminant with the microbiological technology . Several scholars investigated Eq. (1.1) for example using an improved finite element approach , meshless local approaches [3, 4], lattice Boltzmann technique , a front-tracking method , novel WENO methods , or a finite element method . The interested readers can refer to [9, 10] to get more information for Eq. (1.1).
In the past, the groundwater models have been based only on deterministic considerations. In practice, aquifers are generally heterogeneous, i.e., their hydraulic properties (e.g., permeability) change in space. These variations are irregular and characterized by length scales significantly larger than the pore scale. These spatial fluctuations cause the flow variables such as concentration to change in space in an irregular manner. Therefore, a reliable description of the groundwater model can be explained only in a stochastic form .
The first stochastic equation can be rewritten as
where is a linear, self-adjoint, positive definite operator where the domain is dense in and compactly embedded in (i.e., ) and the semigroup is generated by . Additionally, we assume that satisfies the linear growth condition and is twice continuously Frechet differentiable with bounded derivatives up to order 2 . The initial value is deterministic as well. Therefore, (1.2) has a continuous mild solution 
where for and . Regarding the expected value of the solution, we can assume that . The same mild solutions can be employed for and .
The deterministic case of Eq. (1.1) has been studied by some scholars for example a new finite volume method , new Krylov WENO methods , local radial basis function collocation method , etc. Also, the SEM is applied to solve some important problems such as the Schrödinger equations , Pennes bioheat transfer model , the shallow water equations , integral differential equations [18, 19, 20], hyperbolic scalar equations , predator-prey problem , some problems in the finance mathematics [23, 24] and so forth.
The main aim of the current paper is to propose a new high-order numerical procedure for solving the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models. The used technique is based on the modified Legendre spectral element procedure. The coefficient matrix of the employed technique is more well-posed than the traditional Legendre spectral element method. The structure of this article is as follows. In Section 2, we propose and analysis the time-discrete scheme. In Section 3, we develop the new numerical technique and analysis it. We check the numerical results to solve the considered model in Section 4. Finally, a brief conclusion of the current paper is written in Section 5.
2 Temporal discretization
First of all, we briefly review some important notations used in the paper. Considering , we define the following functional spaces
and the derivative
The corresponding inner products for and are as follows
and the associated norms are
Furthermore, associated norm for the space is as
To discretize the time variable, we define
where is the step size. We introduce additionally
The Crank-Nicolson scheme for problem (1.1) is as follows
where is a positive constant such that Discretizing relation (2.1) yields
The vector-matrix configuration of Eq. (2.3) is
where is the identity matrix and
and also the unknown vector is .
2.1 Error analysis of the semi-discrete formulation
If , then relation (2.4) will be unconditionally stable.
Let and . We want to find such that
Let be an approximate solution of , then
Setting in Eq. (2.8) yields
Applying the Cauchy-Schwarz inequality for Eq. (2.9), results
There exists constant such that
By simplification we have
So, from the following assumption and the definition of matrices and , we have
Now, we can get
Using the below relation
Eq. (2.1) is changed to
By summing Eq. (2.13) for from 0 to , gives
Thus, we have
Considering Gronwall’s inequality for Eq. (2.14) yields
So, we have
The convergence order of relation (2.4) is .
Let us assume . We set
where . Then, we have
According to the Crank-Nicolson idea, we have
Similar to Theorem 2.1, we obtain
which completes the proof. ∎
3 Error estimation for full-discrete plane
In this section, we employ a new class of Legendre polynomial functions which were developed in .
 Consider the following relations
in which and are the Legendre polynomials. Let us denote
The SEM as a combination of the finite element method and spectral polynomials has been developed by Patera . By dividing the computational region into non-overlapping elements
Now, we define the following projection operator.
and is the spectral element approximation space
 Let (), therefore
In the special cases and we get
We aim to find a such that
The spectral element formulation is: find a such that