Numerically Neither Dissipative Nor Compressive Scheme for Linear Advection Equation and Its Application to the Euler System

  • Authors:

  • Hongxia Li;Zhigang Wang;De-Kang Mao

  • Affiliations:

  • College of Mathematics and Statistics, Zhejiang University of Finance and Economics, Zhejiang, People's Republic of China 310018;Department of Mathematics, Fuyang Normal College, Anhui, People's Republic of China 236042;Department of Mathematics, Shanghai University, Shanghai, People's Republic of China 200444

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2008

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Abstract

The numerical dissipation is always a big issue in the numerical simulation of hyperbolic equations. The problem is that on one hand one needs it for the stability of the scheme and on the other hand one wishes to get rid of it for obtaining good quality of the solution. In this paper we are going to present a new approach for tackling this problem by developing a new type of finite volume scheme for the linear advection equation. The scheme computes approximations to both the solution and entropy, which are then used in the reconstruction of solution in each cell. Ultra-bee limitation is performed in the solution reconstruction to eliminate the spurious oscillations near discontinuities. Designed in such a way, the scheme maintains the conservation of both the solution and entropy, and in this sense the scheme is numerically neither dissipative nor compressive. We then apply this method to the linearly degenerated second characteristic field of the Euler system to improve the resolution of numerical solution there. Numerical examples of both the linear advection equation and Euler system are displayed to show the efficiency of the method.