Numerical approximation of time-fractional Burgers-type equation

In this work, we analyze and test a local discontinuous Galerkin method for solving the Burgers-type equation. The proposed numerical method, which is high-order accurate, is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We prove that the scheme is unconditionally stable and convergent. Some numerical tests are provided to illustrate the accuracy and capability of the scheme.

In this paper, we consider the following time-fractional Burgers-type equation: where 0 < α < 1 is the order of the fractional derivatives, f , u 0 are given smooth functions, λ 1 ≥ 0 is a real parameter. The solution is considered to be either periodic or compactly supported in this paper. The Caputo fractional derivative ∂ α u(x,t) ∂t α is defined as follows: where Γ (·) is the gamma function. The paper is organized as follows. In Sect. 2, some notations and auxiliary results are described. In Sect. 3, we present the local discontinuous Galerkin method for fractional equation (1.1) and prove that the scheme is unconditionally stable and the numerical solution is convergent. Some numerical examples, given in Sect. 4, are presented to illustrate the accuracy of the method, and concluding remarks are provided in Sect. 5.

Notations and auxiliary results
In this section we introduce some notations, projections, and the numerical flux.

Notations and projections
Denote by V k h the space of piecewise discontinuous polynomials of the degree up to k: The following two projections in one dimension [a, b] will be used when proving error estimate: projection P and projection P ± , and ). (2. 3) The approximation results for the projections P and P ± hold [2,15,18,20] ϑ The positive constant C, solely depending on ω, is independent of h. Here τ h is the union of all element interface points, and the L 2 -norm on τ h is defined by In the present paper the usual notation of norms in Sobolev spaces is used. Let the scalar inner product on L 2 (Ω) be denoted by (·, ·) Ω , and the associated norm by · Ω .

A quantity related to the numerical flux
g(w -, w + ) is a given monotone numerical flux which is related to the discontinuous Galerkin discretization in space. It depends on the two values of the function g at the point x j+ 1 2 , that is, w ± (i) locally Lipschitz continuous; (ii) consistent with the flux g(w), that is, g(w, w) = g(w); (iii) a nondecreasing function of its first argument and a nonincreasing function of its second argument.

Fully discrete LDG scheme
Set t = T/M to be the time mesh size, M is a positive integer, t j = j t, j = 0, 1, . . . , M, is a mesh point. First we estimate the time-fractional derivative ∂ α u(x,t) ∂t α at t n as follows [6,17]: where By some analysis, we know [16] γ n (x) ≤ C( t) 2-α , (3.3) where C is a constant depending on u, T, α.
Rewrite Eq. (1.1) as a first-order system: . Then a fully discrete local discontinuous Galerkin method is given: is a monotone flux as described in (2.2). In order to simplify the computation, we can choose the flux Remark that the crucial part for flux (3.6) is taking u n h and p n h from opposite sides. Because the problem is nonlinear, an iterative method should be used when computing: where m is the iterative step. Without loss of generality, we consider the case f = 0 in numerical analysis. For the stability of scheme (3.5), we have the following result.

Theorem 3.1 Suppose that u n
h is the solution of numerical scheme (3.5), if the numerical flux (3.6) is used, it will hold that Proof We prove Theorem 3.1 by mathematical induction. First, for notational convenience, we introduce the following notations: (3.9) We consider the case n = 1 in (3.5) and take the test Next we deal with the term G(u 1 h ). We define Based on the monotonicity of flux g and the mean value theorem, we have By some manual calculation, we have that is, and Suppose that Considering the case n = P + 1 in scheme (3.5) and taking the test Analogous to the proofs of inequalities (3.11) and (3.12), we know Then the inequality follows This finishes the proof of the stability result.
In order to simplify the analysis, the linear case g(u) = u is studied.

18)
where C is a constant depending on u, T, α.
Proof Consider the separation of numerical error in the form e n u = u(x, t n )u n h = ξ n uη n u , ξ n u = Pe n u , η n u = Pu(x, t n )u(x, t n ), e n p = p(x, t n )p n h = ξ n pη n p , ξ n p = Pe n p , η n p = Pp(x, t n )p(x, t n ).

(3.19)
Here η n u and η n p have been estimated by inequality (2.4). In what follows we are going to estimate ξ n u and ξ n p .
With flux (3.6), we can get the error equation Using (3.19), error equation (3.20) can be written as follows: Taking the test functions v = ξ n u , w = χλ 1 ξ n p in (3.21), using properties (2.1) and (2.3), then the following equality holds: Error estimates (3.18) will be proved by mathematical induction. First, we consider the case n = 1, that is, We know and based on (2.4), we have (3.24) Choosing small ε, we know Then suppose that (3.25) We can obtain By using the triangle inequality and the interpolating property (2.4), we have Theorem 3.2.

Numerical examples
In this section, we present some numerical experiments for the proposed local discontinuous Galerkin method to illustrate its capability. f (x, t) = 24t 4-α Γ (5α) + π 2 t 4 + 1 sin(πx) + π sin(πx) cos(πx) t 4 + 1 2 , (4.1) and the exact solution is u(x, t) = (t 4 + 1) sin(πx). The L 2 and L ∞ errors and the numerical orders of accuracy at time T = 1 for several α are contained in Tables 1-2. We can see that the convergence rates are well displayed and justify our theoretical results.

Conclusion
In this paper we have presented a fully discrete local discontinuous Galerkin method for solving a class of fractional Burgers-type equations. Numerical experiments show that  our scheme is very effective. The advantages of the method are its flexibility in terms of mesh and shape functions, and it can achieve a high order of convergence. In future we would study a local discontinuous Galerkin method and an alternating direction implicit/iterative scheme to solve the two-or higher-dimensional problems.