Speaker: Zhang Qiang , Professor and Doctoral Supervisor, Department of Mathematics, Nanjing University

Date: November 26, 2023

Time: 15:00-16:00

Location: B924, Zhixin Building, Shandong University

Sponsor: School of Mathematics, Shandong University

Abstract:

In this talk we shall take the fourth order in time Runge-Kutta discontinuous Galerkin method, as an example of high order schemes, to establish a sharp a priori L²-norm error estimates for sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard CFL condition, and the quasi-optimal and/or optimal order of accuracy in space are achieved for many widely-used numerical fluxes, no matter whether the exact solution contains sonic points or not, Note that the convergence order in space strongly depends on the relative upwind effect of the used numerical flux, which is related to the local flowing speed and the strength of the numerical viscosity provided by the used numerical flux. Two main tools are used in this talk. One is the matrix transferring process, based on the temporal differences of stage errors. It gives a useful energy equation and helps us to get the theoretical result under the acceptable temporal-spatial condition. The other is the generalized Gauss-Radau projection of the reference functions, which depends on the relative upwind effect and helps us to achieve the optimal order in space in many cases. Finally some numerical experiments are given to support the theoretical results.

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https://www.view.sdu.edu.cn/info/1020/185589.htm