Multi‐stage high order semi‐Lagrangian schemes for incompressible flows in Cartesian geometries

Efficient transport algorithms are essential to the numerical resolution of incompressible fluid‐flow problems. Semi‐Lagrangian methods are widely used in grid based methods to achieve this aim. The accuracy of the interpolation strategy then determines the properties of the scheme. We introduce a simple multi‐stage procedure, which can easily be used to increase the order of accuracy of a code based on multilinear interpolations. This approach is an extension of a corrective algorithm introduced by Dupont & Liu (2003, 2007). This multi‐stage procedure can be easily implemented in existing parallel codes using a domain decomposition strategy, as the communication pattern is identical to that of the multilinear scheme. We show how a combination of a forward and backward error correction can provide a third‐order accurate scheme, thus significantly reducing diffusive effects while retaining a non‐dispersive leading error term. Copyright © 2016 John Wiley & Sons, Ltd.