SCOUT: Semi-Lagrangian COnservative and Unconditionally sTable Schemes for Nonlinear Advection-Diffusion Problems

In this work, we propose a new semi-Lagrangian (SL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ext {SL}$$\end{document}) finite difference scheme for nonlinear advection-diffusion problems. To ensure conservation, which is fundamental for achieving physically consistent solutions, the governing equations are integrated over a space-time control volume constructed along the characteristic curves originating from each computational point. By applying Gauss theorem, all space-time surface integrals can be evaluated. For nonlinear problems, a nonlinear equation must be solved to find the foot of the characteristic, while this is not needed in linear cases. This formulation yields SL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ext {SL}$$\end{document} schemes that are fully conservative and unconditionally stable, as verified by numerical experiments with CFL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ext {CFL}$$\end{document} numbers up to 100. Moreover, the diffusion terms are directly incorporated within a conservative semi-Lagrangian framework, leading to the development of a novel characteristic-based Crank-Nicolson discretization in which the diffusion contribution is integrated along the characteristic. A broad set of benchmark tests demonstrates the accuracy, robustness, and strict conservation property of the proposed method, as well as its unconditional stability.