Semi-Lagrangian methods for parabolic problems in divergence form

TL;DR

This paper introduces semi-Lagrangian methods tailored for divergence form parabolic problems, addressing their application in computational fluid dynamics with a focus on mass conservation and improved numerical stability.

Contribution

It develops a new framework for semi-Lagrangian schemes in divergence form, including a detailed nonconservative discretization and extension to multi-dimensional nonlinear problems.

Findings

Numerical results show advantages over standard low-order methods.

The approach is effective for fluid dynamics applications.

The methods maintain stability and consistency in complex scenarios.

Abstract

Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the Semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analysed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low order time discretization approaches.