SCOUT: Semi-Lagrangian COnservative and Unconditionally sTable schemes for nonlinear advection-diffusion problems

TL;DR

This paper introduces a novel semi-Lagrangian finite difference scheme for nonlinear advection-diffusion problems that is fully conservative, unconditionally stable, and incorporates diffusion directly within the framework, verified through extensive benchmarks.

Contribution

The paper develops the first conservative semi-Lagrangian scheme that directly incorporates diffusion and is unconditionally stable for nonlinear advection-diffusion equations.

Findings

Scheme is fully conservative and unconditionally stable up to CFL=100.

Successfully incorporates diffusion within a conservative semi-Lagrangian framework.

Demonstrates high accuracy and robustness through benchmark tests.

Abstract

In this work, we propose a new semi-Lagrangian (SL) 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 schemes that are fully conservative and unconditionally stable, as verified by numerical experiments with CFL numbers up to 100. Moreover, the diffusion terms are, for the first time, 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 implicitly evaluated at the foot of 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.