AI 中文总结
本文提出并分析了一类用于一维线性对流扩散方程的Hermite型混合变量方法，证明其具有超收敛性及半离散稳定性，并通过数值实验验证。
AI 中文摘要
我们针对一维周期域上的线性对流扩散方程，开发并分析了一类Hermite型方法。这些方法同时演化节点值和单元平均值，因此被称为混合变量（HV）方法。利用Hermite插值理论，我们构造了具有任意阶精度的HV方法，并证明了该方法在空间精度阶数大于局部截断误差的意义下是超收敛的。在论文的第二部分，我们利用正三角级数理论证明了，对于线性扩散方程和线性对流扩散方程，所有中心HV方法在半离散层面都是稳定的。超收敛性质和中心HV格式的稳定性均通过大量数值算例得到了验证。
英文摘要
We develop and analyze a class of Hermite-type methods for linear advection-diffusioon equations on one-dimensional periodic domains. The methods evolve both nodal values and cell averages and are therefore referred to as hybrid-variable (HV) methods. Using Hermite interpolation theory, we construct the HV methods to arbitrary order of accuracy, and prove that the method is supraconvergent in the sense that the spatial order of accuracy of the method is larger than the local truncation error. In the second part of the paper, we prove using the theory of positive trigonometric series that all central HV methods for linear diffusion equations and linear advection-diffusion equations are stable at the semi-discretized level. Both the supraconvergence property and the stability of central HV schemes are verified by extensive numerical examples.
Comments38 pages, 12 figures
