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.

翻译：本文提出了一种针对非线性对流扩散问题的新型半拉格朗日有限差分格式。为确保守恒性（这是获得物理一致解的基础），控制方程在沿每个计算点出发的特征曲线构建的时空控制体上进行积分。通过应用高斯定理，所有时空面积分均可被精确计算。对于非线性问题，需通过求解非线性方程确定特征线起点，而线性情形则无需此步骤。该框架构建的半拉格朗日格式具有完全守恒性与无条件稳定性，经CFL数高达100的数值实验验证。此外，扩散项首次被直接纳入守恒型半拉格朗日框架，由此发展出基于特征线的Crank-Nicolson离散新方法，其中扩散贡献在特征线起点处隐式求解。大量基准测试表明，所提方法具有高精度、强鲁棒性、严格守恒特性及无条件稳定性。