A numerical scheme for advection dominated problems based on a Lagrange interpolation

Abstract Applying conventional numerical techniques for solving advection dominated problems, especially in the presence of sharp gradients, produces inaccurate results. This paper presents a robust numerical scheme for approximating the advection equation. The proposed scheme is based on the method of characteristics and Lagrange interpolation. The accuracy and validity of the developed model have been verified through several test cases. The comparison between numerical outcomes and the exact solutions revealed the great capability of the proposed scheme to simulate the sharp gradient of the initial distribution and the rotating velocity field. Besides, it indicated deploying higher-order polynomials can lead to obtaining an accurate solution for the advection dominated problem. The stability analysis of different Lagrange interpolation polynomials has been carried out by comparing both amplification factor and relative phase error of the polynomials under different Courant numbers. The results can serve as a prominent indication for choosing a perfect algebraic polynomial interpolation to solve advection dominated problems.