A weak Galerkin finite element method for solving time-fractional coupled Burgers' equations in two dimensions

Abstract

In this paper, we present a continuous and discrete time weak Galerkin finite element method (WG-FEM) for solving two dimensional time fractional coupled Burgers' equations. The stability is proved for discrete time WG-FEM and the optimal order error in $L^2$-norm is obtained based on fractional derivative definition, fractional integral definition and dual argument technique for continues and discrete time WG-FEM. The numerical example is to illustrate the theoretical analysis with polynomial mixture $\{P_k(K),P_{k-1}(\partial K),{[P_{k-1}(K)]}^2\}$.

Introduction

In this paper, we consider nonlinear two dimensional time fraction coupled Burgers' problem [1].

$${\mathit{u}}_t^{\mathit{\beta }}-ϵ{\mathrm{\nabla }}^2\mathit{u}+(\mathit{u}\cdot \mathrm{\nabla })\mathit{u}=\mathit{f}(x,y,t),(x,y,t)\in \mathrm{\Omega }\times (0,T],$$

with Dirichlet boundary conditions

$$\mathit{u}(x,y,t)=\mathit{\eta }(x,y,t),(x,y,t)\in \partial \mathrm{\Omega }\times (0,T],$$

and initial conditions

$$\mathit{u}(x,y,0)={\mathit{u}}^0(x,y),(x,y)\in \mathrm{\Omega },$$

where $\mathit{\beta }=\{({\beta }_1,{\beta }_2),0<{\beta }_1,{\beta }_2<1\}$, $\mathrm{\Omega }=\{x\in [a,b],y\in [b,c]\}\subset {\mathbb{R}}^2$ is the computational domain and ∂Ω its boundary, $\mathit{u}=(u,v)$, u and v are the velocity components, ${\mathit{u}}^{\mathbf{0}}=(u^0,v^0)$, $\mathit{\eta }=({\eta }_u,{\eta }_v)$ are known functions, $ϵ{\mathrm{\nabla }}^2\mathit{u}$ is the diffusion term, and $ϵ=\frac{1}{R_e}$ is diffusion constant and $R_e$ is the Reynolds number. $\mathit{f}=(f_1,f_2)$ are the source terms.

The Burgers' equation is a vital partial differential equation (PDE) in fluid dynamics. It exists in numerous areas of applied mathematics, such as acoustic waves, heat conduction and modelling of dynamics [6]. Because of the wide variety of uses of the Burgers' equation, it has been intensively studied, and many numerical techniques have been proposed to solve it. Most of these numerical techniques fall into either the finite difference method or the finite element and spectral method ([11], [8], [3], [18], [5], [17]). Recently, significant development has been reported in the field of fractional calculus (FC). FC has been applied in nearly every branch of science, engineering, economics and mathematics. This is because the fractional derivatives offer more precise models of real-world problems than the integer-order derivatives. This is also the same reason why several models of fractional Burgers' equations (FBEs) have been suggested and studied lately. These models are obtained by replacing integer-order space/time derivatives and ordinary initial/boundary conditions with their fractional counterparts ([19], [14], [15], [9], [12], [7], [22]).

The goal of this paper is to obtain the optimal order error by applying the WG-FEM with configuration $(P_k(K),P_{k-1}(\partial K),{[P_{k-1}(K)]}^2)$ and stabilization term for solving two dimensional time fraction coupled Burgers' equations for continuous and discrete time WG-FEM.

For continuous time WG-FEM we use the generalized fractional derivative and fractional integral of order β which are defined as (see [2]):

$$D_t^{\beta }u(t)=\frac{{(z(t))}^{1-\beta }}{z^{\prime }(t)}{u}_t(t),0<\beta <1,$$

$$I^{\beta }u(t)=\underset{0}{\overset{t}{\int }}\frac{z^{\prime }(s)}{{(z(s))}^{1-\beta }}u(s)ds,$$

where $z(t):[0,T]\to \mathbb{R}$ is a continuous nonnegative mapping such that $z(t),z^{\prime }(t)\ne 0$ and $u:[0,T]\to \mathbb{R}$ is a differentiable function; there are some properties for above equations:

$$D_t^{\beta }(f(t).g(t))=f(t)D_t^{\beta }g(t)+g(t)D_t^{\beta }f(t)\text{(product rule)}.$$

$$I^{\beta }{D}_t^{\beta }u(t)=u(t)-u(0).$$

For discrete time WG-FEM we use the Caputo fraction derivative of order β with respect to t.

$$D_t^{\beta }u(t)=\{\begin{array}{cc}\frac{1}{\mathrm{\Gamma }(1-\beta )}{\int }_0^t{(t-s)}^{-\beta }{u}_t(s)ds,\hfill & 0<\beta <1;\hfill \\ u_t,\hfill & \beta =1.\hfill \end{array}$$

And the Riemann Liouville integral $I^{\beta }$ is defined as follows:

$$I^{\beta }u(t)=\frac{1}{\mathrm{\Gamma }(\beta )}\underset{0}{\overset{t}{\int }}{(t-s)}^{\beta -1}u(s)ds,$$

also, we have

$$D_t^{\beta }u(t)=I^{(1-\beta )}{u}_t(t),$$

where Γ is Gamma function.

Rest of the paper is organized as follows. In Section 2, we introduce the definition of discrete weak gradient, weak finite element spaces and some lemmas which are necessary in error estimate. Section 3 is devoted to variational form and weak variational form for continuous and discrete time WG-FEM. Section 4 is devoted to the stability of the discrete time WG-FES. In Section 5 we derive the optimal order error for both continuous and discrete time WG-FES in $L^2$-norm. Finally, in Section 6 numerical experiments are presented to show the efficacy of the WG-FEM and confirm our theoretical analysis.