基于块广义Adams方法的卷积求积公式及其应用

【作者】 刘玲；

【导师】 王锦荣；

【作者基本信息】 贵州大学 ， 数学， 2025， 博士

【摘要】 卷积积分在反常扩散,流行性传染病,通信系统,粘弹性等问题的建模过程中具有广泛应用,其数值计算的研究对这些领域的发展具有重要推动意义.卷积求积公式是一种用于高效计算卷积积分的数值算法,能够有效处理已知核函数Laplace变换的情形,且继承了底层离散方法的收敛性,具有良好稳定性,高精度及易于分析等优点.本文利用块广义Adams方法构造了一类新型卷积求积公式,并将其应用于求解三种具体的分数阶扩散方程.主要研究内容如下:首先,针对卷积积分设计一类高性能卷积求积公式.与经典的线性多步法及Runge-Kutta方法不同,在均匀网格上采用块广义Adams方法离散底层初值问题,所提出的卷积求积公式无需改变网格点就可以调整收敛阶,增强了算法灵活性.通过合理选择局部插值多项式与块大小,该方法在双曲型核函数的卷积积分计算中实现了高阶收敛.对所提出的卷积求积公式建立了收敛性分析,并通过数值实验验证了理论结果的准确性.其次,研究时间分数阶次扩散方程的数值解.在时间方向采用基于块广义Adams方法的卷积求积公式进行逼近.通过引入修正项,给出了时间半离散格式的最优收敛阶,并分析了卷积求积公式的稳定性.空间方向利用谱配置方法进行离散,得到全离散格式.数值实验证实,该算法在时间方向采用均匀网格划分仍保持了理论预期的高阶收敛性.再次,研究向后分数阶Feynman-Kac方程的时间步进格式.基于修正的块广义Adams卷积求积公式,实现了Riemann-Liouville分数阶物质导数的高精度离散.借助Laplace变换,论证了时间离散格式的收敛性.理论分析与数值实验表明,修正的卷积求积公式可结合空间谱配置方法高效求解向后分数阶Feynman-Kac方程.最后,研究变阶时间分数阶积分偏微分方程的数值逼近.时间方向采用基于块广义Adams方法的卷积求积公式进行离散化处理.利用扇形算子的预解估计与卷积求积公式的理论分析,严格证明了时间离散格式的最优误差估计.数值实验验证了所提出的算法在变阶动态系统中的计算稳定性.更多还原

【Abstract】 Convolution integrals have been widely applied in modeling various phenomena,in-cluding anomalous diffusion,endemic infectious diseases,communication systems and viscoelasticity,where the study of their numerical computation plays a crucial role in advancing these fields.Convolution quadrature,a numerical algorithm for efficiently computing convolution integrals,is particularly effective when the Laplace transform of the kernel function is known,while inheriting the convergence of underlying discrete method.This approach exhibits superior advantages such as good stability,high precision and analytical tractability.In this thesis,we construct a family of convolution quadra-tures based on the block generalized Adams method,and subsequently apply them to temporal discretization for solving three specific fractional diffusion equations.The main contents are as follows:Firstly,a family of high-performance convolution quadratures is designed for ef-ficient evaluation of convolution integrals.By the block generalized Adams method to discretize the underlying initial value problem on a uniform grid,departing from the well-established approaches that rely on linear multistep formulas or Runge-Kutta methods.The convergence order of the proposed convolution quadrature can be controlled without requiring grid point changes,enhancing flexibility.Through strategic selection of the lo-cal interpolation polynomial and block size,the method achieves high-order convergence for calculation of convolution integrals with hyperbolic kernels.A convergence analysis is established for the proposed convolution quadrature,and the theoretical findings are numerically validated.Secondly,the numerical solution for time-fractional subdiffusion equations is investi-gated.The temporal approximation is achieved using convolution quadrature,developed via the block generalized Adams method.By incorporating a correction term,the opti-mal convergence order is derived for the semi-discrete scheme in time.Additionally,the stability of the convolution quadrature is analyzed.The spectral method is employed for spatial approximation to derive the fully discrete scheme.Numerical experiments validate that the proposed scheme achieves high-order convergence even with a uniform grid for temporal discretization.Furthermore,a time stepping method for solving the fractional Feynman-Kac equa-tion is investigated.The effective discretization of Riemann-Liouville fractional substan-tial derivative is achieved by formulating a modified block generalized Adams convolution quadrature.The convergence rate for the time-discrete solutions is derived with the help of the Laplace transform.Both theoretical analysis and numerical experiments demon-strate that the modified convolution quadrature can be combined with the spatial spectral collocation method to efficiently solve the backward fractional Feynman-Kac equation.Finally,the numerical approximation of variable-order time fractional integro-partial differential equations is investigated.The temporal direction is efficiently discretized us-ing the block generalized Adams convolution quadrature.By employing the resolvent estimates of sectorial operator and leveraging the theoretical analysis of the convolution quadrature,the optimal error estimate for the temporal discretization scheme is rigor-ously established.Numerical experiments validate the computational stability of the proposed algorithm in variable-order dynamical systems.更多还原