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

TL;DR

This paper presents a multi-stage correction method to enhance the accuracy of semi-Lagrangian schemes for incompressible flow simulations, achieving third-order precision with minimal additional computational complexity.

Contribution

It introduces a simple multi-stage correction approach that increases the order of accuracy of semi-Lagrangian schemes without complicating parallel implementation.

Findings

Achieves third-order accuracy in semi-Lagrangian schemes

Reduces diffusive effects in flow simulations

Maintains non-dispersive leading error term

Abstract

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 multi-linear 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 communications pattern is identical to that of the multi-linear 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.