| This article studies the numerical scheme by using weighted numerical flux for solving evolution equation. The evolution equation this article studies is heat equation, and the discontinuous Galerkin methods is used to develop the numerical scheme for solving the heat equation.Time-dependent equaiton is a class of partial differential equation, which plays a significant role in scientific studies and engineering applications. Heat equation, among many kinds of time-dependent equations, is a type of parabolic equation, that describes the distribution of heat or particle in a given region over time. Discontinuous Galerkin methods(DG) were first proposed and analyzed in 1973 as a numerical method to solve partial differential equation. Discontinuous Galerkin methods combine features of the finite element and the finite volume framework, they are developed into a numerical method with high accuracy and have been successfully applied to hyperbolic, elliptic, parabolic equations.As the basic framework of discontinuous Galerkin methods comes from finite element and finite volume, so the numerical flux is a vital item of the methods, that it is the key for the solvability and properties of scheme. Upwind and central fluxes are normal choices, they are available for solving many equations and have good properties. The weighted flux(up-wind biased flux) this article studies is a special form of numerical fluxes. Unlike upwind and central flux taking the value of one side or average of two side at the nodes, it takes the weighted value of two side, which made the scheme be more adaptive for complex equations and have better convergence. So weighted flux has better properties than others in some ways.This article develops a numerical scheme to solve heat equation by discontinuous Galerkin methods applies the weighted flux, and studies the stability analysis and error estimate. Meanwhile, numerical examples are consistent with the error estimate, which proves the validity of scheme. |
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