Based on the combination of block-centered and compact difference methods, fourth order compact block-centered finite difference schemes for the numerical solutions of one-dimensional and two-dimensional elliptic and parabolic problems with variable coefficients are derived and analyzed. Stability and optimal fourth-order error estimates are proved for both the solution and flux. Numerical experiments for model problems are presented to confirm the theoretical results and superior performance of the proposed schemes.

中文翻译：

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椭圆和抛物线问题的高阶紧凑块中心有限差分方案

在以块为中心和紧致差分方法相结合的基础上，推导并分析了一维和二维变系数椭圆和抛物线问题数值解的四阶紧致以块为中心的有限差分格式。证明了溶液和通量的稳定性和最佳的四阶误差估计。进行了模型问题的数值实验，以确认理论结果和所提出方案的优越性能。