High‐order 𝒞1 finite‐element interpolating schemes—Part I: Semi‐Lagrangian linear advection

This paper is devoted to the development of accurate high‐order interpolating schemes for semi‐Lagrangian advection. The characteristic‐Galerkin formulation is obtained by using a semi‐Lagrangian temporal discretization of the total derivative. The semi‐Lagrangian method requires high‐order interpolators for accuracy. A class of ??¹ finite‐element interpolating schemes is developed and two semi‐Lagrangian methods are considered by tracking the feet of the characteristic lines either from the interpolation or from the integration nodes. Numerical stability and analytical results quantifying the amount of artificial viscosity induced by the two methods are presented in the case of the one‐dimensional linear advection equation, based on the modified equation approach. Results of test problems to simulate the linear advection of a cosine hill illustrate the performance of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.