Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models

Analysis of a Legendre spectral element method (LSEM) for the two-dimensional system of a nonlinear stochastic advection-reaction-diffusion models

Mostafa Abbaszadeh, Mehdi Dehghan111Corresponding author. E-mail addresses: (M.Abbaszadeh),, (M. Dehghan), (A. Khodadadian), (T.Wick). , Amirreza Khodadadian,  Thomas Wick
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences,
Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran
Institute of Applied Mathematics, Leibniz University of Hannover

Welfengarten 1, 30167 Hanover, Germany
April 15, 2019

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.

1 Introduction

We consider the stochastic nonlinear system of advection-reaction-diffusion models [1, 2]


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 [2]. Several scholars investigated Eq. (1.1) for example using an improved finite element approach [2], meshless local approaches [3, 4], lattice Boltzmann technique [5], a front-tracking method [6], novel WENO methods [7], or a finite element method [8]. 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 [11].

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 [12]. The initial value is deterministic as well. Therefore, (1.2) has a continuous mild solution [13]


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 [2], new Krylov WENO methods [7], local radial basis function collocation method [14], etc. Also, the SEM is applied to solve some important problems such as the Schrödinger equations [15], Pennes bioheat transfer model [16], the shallow water equations [17], integral differential equations [18, 19, 20], hyperbolic scalar equations [21], predator-prey problem [22], 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

Theorem 2.1.

If , then relation (2.4) will be unconditionally stable.


Let and . We want to find such that


Let be an approximate solution of , then


where . Subtracting Eq. (2.7) for Eq. (2.6) , results



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

Theorem 2.2.

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 [25].

Lemma 3.1.

[25] 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 [26]. By dividing the computational region into non-overlapping elements

Now, we define the following projection operator.




and is the spectral element approximation space

Lemma 3.2.

[27] 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

Lemma 3.3.