The discontinuous Galerkin (DG) method was first proposed by Reed and Hill  to solve the neutron problems in 1973. With the development of DG, the direct discontinuous Galerkin (DDG) method  was proposed by Liu to solve the second order partial differential equations. The main idea of DDG is to direct force solve the higher order equation so as to avoid the reduction of the equation.
For the error analysis of DDG, we first got the linear result of the error estimates can reach to order k in . In , a series of special precision analyses were made for the numerical solution by using Fourier transform. The error estimates obtained by Liu can reach to order for the linear and nonlinear convection diffusion equations by using the DDG method in . In 2016, Cao  discussed the superconvergence of DDG method and obtained that the projection superconvergence at some points can achieve order .
In this article, we use first order numerical fluxes to the diffusion term and use the upwind-biased fluxes to the convective term. The upwind-biased fluxes was first proposed by Meng and Shu, they proved that the optimal error estimates of the linear hyperbolic conservation equations can obtain order in semi-discrete and fully-discrete scheme in 2016 . Meng extended the upwind-biased fluxes to the generalized alternating fluxes in .
The main content of this paper: In Section 2, we introduce the semi-discrete scheme of second-order partial differential equation and solve the error estimates problems by using the upwind-biased fluxes and first order numerical fluxes. In Section 3, we use the third-order RK time discretization methods for completing numerical experiments and obtain that the error estimates can reach order .
2. The Method of DDG
This paper considers the following convection diffusion equation
For the convenience, we take the periodic boundary condition into discussion.
2.1. The Meshes of DDG
Let us denote the computational interval , consisting of cells , where .
We define and , and then use and to denote the left and right limits at the discontinuity point. In what follows, we define and . The following piecewise polynomials space is chosen as the finite element space
where denotes the polynomials of degree up to defined on cell .
2.2. Function Spaces and Norms
Define the broken Sobolev spaces as
The norms of the broken Sobolev spaces with are given by: and .
In the case , we have .
2.3. The Semi-Discrete DDG Scheme
The DDG scheme is defined as follows: find both and in , by integration by parts and need some interface corrections, the Equations (1) can be written as
Summing j we have
Here is the upwind-biased fluxes as: , where .
Following  we take
We define two operators
So the Equation (3) can be written as
We define energy norm and introduce a quantity
where and .
According to  there exists such that
Lemma 1 For a quadratic entropy flux, it holds that
A quadratic entropy flux satisfies 
Firstly we figure out that . Then using Equation (11) we get
2.4. The Stability of DDG
Theorem 1 Consider the semi-discrete of DDG, it satisfies the following properties:
1) Conservation of mass: .
2) There exists such that
3) The scheme is stable: .
1) Taking into Equation (6) we have . Combining with Equation (2) with leads to the mass conservation.
2) Taking into Equation (6), we obtain
According Equation (9) and combining with Lemma 1 together prove the Equation (12).
3) It follows from Equations (12) and (2) that
2.5. The Global Projections
For the DDG method using the upwind-biased fluxes, we need to construct a globally projection P. For , the projection P is defined as
We quote the lemma as follows 
Lemma 2 For and, the projection P holds that
where C is independent of h and depends on.
2.6. The Error Estimates of DDG
Theorem 1 Assume that u are the exact solutions, we take the upwind-baised fluxes and the finite element space, there hold the following error estimates
where C is independent of h and depends on.
Firstly we set
Since both the exact and numerical solutions satisfy the weak solution form, we have
For the left side we use to obtain
And for the right side using the definition of projection we have
Thus, we get
Summing and to obtain
According Equation (13) the highest order is. We have. And by the properties of projection we obtain
So the right side of Equation (16) can be written as
Combining Equation (17), Equation (21) and Lemma 2, we have
Finally by using the Gronwall inequality, we obtain Theorem 1.
3. Numerical Experiments
We present numerical experiments to validate the error estimates of DDG method based on upwind-biased fluxes. We adopt elements on the uniform mesh, with. In order to reduce time errors, we use the third order Runge-Kutta method and compute until.
For time discretization, we use TVD type third-order Runge-Kutta method 
Consider the equation
The exact solution of the equation is.
Table 1 shows that the error estimates of the convection diffusion equation by using the DDG method and the upwind-biased fluxes can reach to the order, With the coefficients changes, the results change together, so we can choose the best error results.
Table 1. The error estimates until.
Based on the idea of DDG method and the upwind-biased fluxes, this paper proves the stability of numerical solutions and the error estimates of convection diffusion equation can reach to the order. Numerical experiments show that the scheme is stability and the error estimates is accurate.
 Zhang, M. and Yan, J. (2012) Fourier Type Error Analysis of the Direct Discontinuous Galerkin Method and Its Variations for Diffusion Equations. J. Sci. Comput, 638-655. https://doi.org/10.1007/s10915-011-9564-5
 Cao, W., Liu, H. and Zhang, Z.M. (2017) Superconvergence of the Direct Discontinuous Galerkin Method for Convection-Diffusion Equations. Numerical Methods for Partial Differential Equations. https://doi.org/10.1002/num.22087
 Meng, X., Shu, C.W. and Wu, B. (2016) Optimal Error Estimates for Discontinuous Galerkin Methods Based on Upwind-Biased Fluxes for Linear Hyperbolic Equations. Math. Comp, 1225-1261. https://doi.org/10.1090/mcom/3022
 Cheng, Y., Meng, X. and Zhang, Q. (2016) Application of Generalized Gauss-Radau Projections for the Local Discontinuous Galerkin Method for Linear Convection-Diffusion Equations. Math.Comp, 1233-1267. https://doi.org/10.1090/mcom/3141