In this paper, we present a class of high-order and efficient compact difference schemes for nonlinear convection diffusion equations, which can preserve both bounds and mass. For the one-dimensional problem, we first introduce a high-order compact Strang splitting scheme (denoted as HOC-Splitting), which is fourth-order accurate in space and second-order accurate in time. Then, by incorporating the Lagrange multiplier approach with the HOC-Splitting scheme, we construct two additional bound-preserving or/and mass-conservative HOC-Splitting schemes that do not require excessive computational cost and can automatically ensure the uniform bounds of the numerical solution. These schemes combined with an alternating direction implicit (ADI) method are generalized to the two-dimensional problem, which further enhance the computational efficiency for large-scale modeling and simulation. Besides, we present an optimal-order error estimate for the bound-preserving ADI scheme in the discrete $L_2$ norm. Finally, ample numerical examples are presented to verify the theoretical results and demonstrate the accuracy, efficiency, and effectiveness in preserving bounds or/and mass of the proposed schemes.

翻译：本文针对非线性对流扩散方程提出了一类高效的高阶紧致差分格式，该格式能够同时保持解的界与质量守恒性。对于一维问题，我们首先提出了一种高阶紧致Strang分裂格式（记为HOC-Splitting），其在空间上具有四阶精度、时间上具有二阶精度。随后，通过将拉格朗日乘子法与HOC-Splitting格式相结合，我们构建了两种额外的保界或/且保质量HOC-Splitting格式，这些格式无需过多计算成本即可自动保证数值解的一致有界性。结合交替方向隐式（ADI）方法，这些格式被推广至二维问题，从而进一步提升大规模建模与模拟的计算效率。此外，我们对保界ADI格式在离散$L_2$范数下给出了最优阶误差估计。最后，通过大量数值算例验证了理论结果，并展示了所提格式在保持解的界或/且质量守恒方面的精确性、高效性与有效性。