Abstract
In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.
Highlights
The discontinuous Galerkin (DG) method was first proposed by Reed and Hill [1] to solve the neutron problems in 1973
We study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes
The error estimates obtained by Liu can reach to order k +1 for the linear and nonlinear convection diffusion equations by using the direct discontinuous Galerkin (DDG) method in [5]
Summary
The discontinuous Galerkin (DG) method was first proposed by Reed and Hill [1] to solve the neutron problems in 1973. With the development of DG, the direct discontinuous Galerkin (DDG) method [2] was proposed by Liu to solve the second order partial differential equations. The error estimates obtained by Liu can reach to order k +1 for the linear and nonlinear convection diffusion equations by using the DDG method in [5]. 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 k +1 in semi-discrete and fully-discrete scheme in 2016 [7]. 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.
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