Fourth order accurate compact schemes for variable coefficient convection-diffusion equations are considered. A sufficient condition for stability of the schemes have been derived using a difference equation based approach. The constant coefficient problems are considered as a special case, and the unconditional stability of compact schemes for such case is proved theoretically. The condition number of the amplification matrix is also analysed, and an estimate for the same is derived. In order to verify the derived conditions numerically, MATLAB codes are provided in Appendix of the manuscript. An example is provided to support the assumption taken to assure stability.

翻译：本文研究变系数对流扩散方程的四阶精度紧致格式。采用基于差分方程的方法推导了格式稳定的充分条件。将常系数问题作为特例考虑，理论上证明了该类问题下紧致格式的无条件稳定性。分析了放大矩阵的条件数并给出了相应估计。为数值验证所推导条件，在附录中提供了MATLAB代码。最后通过算例验证了保证稳定性所需假设的合理性。