A Triangular Spectral Element Method for the 2-D Viscous Burgers Equation

A triangular spectral element method is established for the two-dimensional viscous Burgers equation. In the spatial direction, a new type of mapping is applied. We splice the local basis functions on each triangle into a global basis function. The second-order Crank-Nicolson/ leap-frog (CNLF) method is used for discretization in the time direction. Due to the use of a quasi-interpolation operator, the nonlinear term can be handled conveniently. We give the fully discrete scheme of the method and the implementation process of the algorithm. Numerical examples verify the effectiveness of this method.


Introduction
The spectral method is one of the important numerical methods for solving partial differential equations. Its main advantage is high accuracy, especially for infinitely smooth problems, the convergence rate can reach infinite order. Because of its advantages, the spectral method is widely used in scientific and engineering computations. In recent years, the triangular element spectral method has been developed [1,2,3], that is, the spectral method is used in the triangular element while maintaining the high accuracy. Through the realization of the triangular element spectral method, due to the flexibility of the triangle itself, the application of triangulation to divide the region boundary into triangles can make the spectral method better applied to complex regions.
Recently, a new type of mapping is proposed in [4]. This mapping is a one-to-one mapping, which maps a vertex of a quadrilateral to the midpoint of the hypotenuse of the triangle, making the distribution of nodes more uniform than frequently used Duffy mapping [5]. This new type of mapping is applied in elliptic equation and a class of irrational basis functions is constructed on a triangular element in [6]. In [7], comparing Duffy mapping with this new type of mapping. By introducing auxiliary variables, a mixed triangular spectral element method is constructed in complex regions.
The Burgers equation is a classic nonlinear second-order partial differential equation which describes dissipation on the energy balance. It originates from the turbulence model [8]. It is often used in engineering computations [9] and as a model equation for testing numerical methods. .
where T and  are the positive constants. F(u) is a function of a certain smoothness class. Under the above conditions, the solution of (1) is unique.If the Dirichlet boundary condition is nonhomogeneous, we can convert the boundary condition to homogeneous, and then consider the above situation.
In this paper, we present a triangular spectral element method for (1) and give its semi-discrete scheme and fully discrete scheme. We use polynomials of different degrees and triangulations of differrrent numbers to approximate the unknown functions. Numerical results show the effectiveness of this method.

Triangular Spectral Element Method
In this section, we discrete spatial direction through triangular spectral element method and discrete time through the second-order Crank-Nicolson/leap-frog (CNLF) scheme to obtain the corresponding fully discrete scheme. The new type of mapping :

Preliminaries
We respectively chose ( , ), ( , ), ( , ) as the three vertices of k . How-ever, this kind of mapping will bring singularity [4] in calculation. We divide the domain in this way and arrange the singular point at the midpoint of the shared side of the two triangles, making it easier to calculate. , ( , ) .

Approximation Space and Quasi-Interpolation Operator
As defined in [10], we chose the spectral approximation space with the consistency condition as follow The second-order Crank-Nicolson/leap-frog (CNLF) scheme is applied to discretize in the time direction. Let , we can give the fully discrete scheme to (1), which is: . ,, .They are the mass matrix and the stiffness matrix, respectively.

Conclusion
In this paper, a triangular spectral element method for the two-dimensional viscous Burgers equation is developed. Some numerical examples show the effectiveness of this method. We compare it with other method to show the superiority.We will continue the above research, verify the stability and convergence of this method. And we are about to apply the method to discontinuous problem.