Flux form Semi-Lagrangian methods for parabolic problems

TL;DR

This paper introduces a flux-form semi-Lagrangian method for parabolic problems that ensures conservation and demonstrates its effectiveness through analysis and numerical validation.

Contribution

It extends previous semi-Lagrangian methods to achieve fully conservative flux-form discretization for linear and nonlinear diffusion equations.

Findings

Method is consistent and convergent.

Numerical examples validate the approach.

Potential for advection-diffusion and nonlinear problems.

Abstract

A semi-Lagrangian method for parabolic problems is proposed, that extends previous work by the authors to achieve a fully conservative, flux-form discretization of linear and nonlinear diffusion equations. A basic consistency and convergence analysis are proposed. Numerical examples validate the proposed method and display its potential for consistent semi-Lagrangian discretization of advection--diffusion and nonlinear parabolic problems.