Discontinuous Galerkin method with quadrature-free formulation

In the calculation process of using discontinuous Galerkin finite element method, the corresponding integral expression needs to be constructed as the starting point of numerical solving method, then the volume integral and surface integral are introduced. Under normal circumstances, the value of these integral items is acquired by using numerical integration method. When high-order discontinuous Galerkin finite element method needs to be used, the demand for numerical integration calculation accuracy increases accordingly. This demand results in large calculation amount, and the numerical integration calculation amount largely determines the computational efficiency of discontinuous Galerkin finite element method. In order to solve this problem, an explicit semi-discretization of quadrature-free discontinuous Galerkin finite element method was structured by establishing the relationship between Lagrange interpolation polynomial basis function and Jacobi orthogonal polynomial basis function. An explicit semi-discretization was applied to the direct numerical simulation of linear and nonlinear one-dimensional, two-dimensional conservation laws with different condition, and ideal numerical results were obtained. Using this method, there's no need to calculate the integral items of each element by numerical integration, and the high precision of discontinuous Galerkin finite element method can be achieved effectively. The method has very significant meaning for structuring high efficient high-order discontinuous Galerkin finite element method.