论文标题
抛物线透明度差异方程的可变时步IMEX BDF2方案的急剧误差估计值，金融中产生的初始奇异性
Sharp error estimate of variable time-step IMEX BDF2 scheme for parabolic integro-differential equations with initial singularity arising in finance
论文作者
论文摘要
最近开发的DOC内核技术在BDF2方案的稳定性和收敛分析方面取得了巨大成功。但是，这种分析技术似乎并不直接适用于初始奇点的问题。在对具有初始奇异性的解决方案的数值模拟中，始终采用诸如分级网格之类的可变时步方案以实现最佳收敛，其第一个相邻的时步比率可能会变得很大，因此不满足所获得的限制。在本文中，我们重新访问[W. Wang，Y。Chen和H. Fang，\ Emph {Siam J. Numer。肛门。}，57（2019），pp。1289-1317]，以计算具有初始奇异性的部分截面方程式（pides）。我们在相邻时步比率的轻度限制条件下获得急剧误差估计，$ r_ {k}：=τ_{k}/τ_{k-1} \; （k \ geq 3）<r _ {\ max} = 4.8645 $，对第一个比率（即$ r_2> 0 $）的要求非常温和。这导致了我们对可变时间步长IMEX BDF2方案的分析验证，当通过简单的策略（即等级的网格网格$ t_k = t（k/n）^γ$处理初始奇异性时）。在这种情况下，使用$ n $和$ n $和$α$分别代表总网格点并指示精确解决方案的规律性，分别代表了$ \ MATHCAL {o}（n^{ - \ min \ {2，γα\}}）$的收敛。这就是，通过服用$γ_ {\ text {opt}} = 2/α$来实现光学收敛。提供了数值示例以证明我们的理论分析。
The recently developed technique of DOC kernels has been a great success in the stability and convergence analysis for BDF2 scheme with variable time steps. However, such an analysis technique seems not directly applicable to problems with initial singularity. In the numerical simulations of solutions with initial singularity, variable time-steps schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme presented in [W. Wang, Y. Chen and H. Fang, \emph{SIAM J. Numer. Anal.}, 57 (2019), pp. 1289-1317] to compute the partial integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios $r_{k}: =τ_{k}/τ_{k-1} \; (k\geq 3) < r_{\max} = 4.8645 $ and a much mild requirement on the first ratio, i.e., $r_2>0$. This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e., the graded mesh $t_k=T(k/N)^γ$. In this situation, the convergence of order $\mathcal{O}(N^{-\min\{2,γα\}})$ is achieved with $N$ and $α$ respectively representing the total mesh points and indicating the regularity of the exact solution. This is, the optical convergence will be achieved by taking $γ_{\text{opt}}=2/α$. Numerical examples are provided to demonstrate our theoretical analysis.
