Optimal quadratic spline collocation (QSC) method has been widely used in various problems due to its high-order accuracy, while the corresponding numerical analysis is rarely investigated since, e.g., the perturbation terms result in the asymmetry of optimal QSC discretization. We present numerical analysis for the optimal QSC method in two space dimensions via discretizing a nonlinear time-fractional diffusion equation for demonstration. The L2-1\(_\sigma \) formula on the graded mesh is used to account for the initial solution singularity, leading to an optimal QSC–L2-1\(_{\sigma }\) scheme where the nonlinear term is treated by the extrapolation. We provide the existence and uniqueness of the numerical solution, as well as the second-order temporal accuracy and fourth-order spatial accuracy with proper grading parameters. Furthermore, we consider the fast implementation based on the sum-of-exponentials technique to reduce the computational cost. Numerical experiments are performed to verify the theoretical analysis and the effectiveness of the proposed scheme.

最优二次样条配置（QSC）方法因其高阶精度而在各种问题中得到广泛应用，但相应的数值分析却很少被研究，例如扰动项导致最优QSC离散化的不对称性。我们通过离散化非线性时间分数扩散方程来对二维空间中的最佳 QSC 方法进行数值分析以进行演示。分级网格上的L 2-1 \(_\sigma \)公式用于解释初始解奇异性，从而得出最佳 QSC– L 2-1 \( _ {\sigma }\)方案，其中非线性项通过外推法处理。我们提供了数值解的存在性和唯一性，以及具有适当分级参数的二阶时间精度和四阶空间精度。此外，我们考虑基于指数和技术的快速实现以降低计算成本。数值实验验证了理论分析和所提方案的有效性。