Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Yuan Xu; Qiang Zhang

doi:10.1007/s42967-020-00116-z

Communications on Applied Mathematics and Computation ›› 2021, Vol. 4 ›› Issue (1) : 319 -352. DOI: 10.1007/s42967-020-00116-z

Original Paper

Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation

Yuan Xu ¹
, Qiang Zhang ¹^,b

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Abstract

In this paper, we shall establish the superconvergence properties of the Runge-Kutta discontinuous Galerkin method for solving two-dimensional linear constant hyperbolic equation, where the upwind-biased numerical flux is used. By suitably defining the correction function and deeply understanding the mechanisms when the spatial derivatives and the correction manipulations are carried out along the same or different directions, we obtain the superconvergence results on the node averages, the numerical fluxes, the cell averages, the solution and the spatial derivatives. The superconvergence properties in space are preserved as the semi-discrete method, and time discretization solely produces an optimal order error in time. Some numerical experiments also are given.

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Yuan Xu, Qiang Zhang. Superconvergence Analysis of the Runge-Kutta Discontinuous Galerkin Method with Upwind-Biased Numerical Flux for Two-Dimensional Linear Hyperbolic Equation. Communications on Applied Mathematics and Computation, 2021, 4(1): 319-352 DOI:10.1007/s42967-020-00116-z