Improved finite volume method for solving 1-D advection equation

Abstract

In the framework of the second-order finite volume method, a new improved finite volume method (FVM) for solving one-dimensional advection equations is proposed based on its conservation form. The new method first applies the scalar conservation law to the cells in the FVM, ensuring that it is conserved in time and space, and that the flat flow (ie, the transport physical quantity) is conserved. Secondly, the time integral values of adjacent grids boundary are equalized. Finally, by establishing an equation, numerical solution values are obtained. A strong discontinuity function was used in the paper to test the new method described in this paper and compare it to the central difference method (CDM) and traditional FVM. Without the limiter, the results show that the new method described in this paper has less dissipation and better stability than CDM and traditional FVM. In addition, after adjusting the convergence condition criterion number CFL to 2, the accuracy of the numerical solution can still be guaranteed.