奇异摄动积分微分方程的高精度方法研究

The singularly perturbed problem is always closely concerned in scientific computing filed and is urgently needed to be solved. A typical singularly perturbed problem is the singularly perturbed Volterra integro-differential equation. This kind equation is widely used in physics and biology fields, such as diffusion-dissipation processes, synchronous control systems, renewal processes and filament stretching and so on. Research on numerical computing methods, theoretical analysis and applications of this kind equation has been one of the most popular topics in the field of scientific computing. This project will study high order numerical computing methods for solving this kind equation. Previous related results focused on the “asymptotic analysis”, and few results on problems of “high order numerical scheme”, “super-convergence” and “uniform convergence”, so this project is a great challenge. We will design some stabilized and highly effective numerical computing schemes to solve the singularly perturbed Volterra integro-differential equation, such as the discontinuous Galerkin method and DG-CFEM method and so on. At the same time, we will study the regularity of this kind equation, and analyze feasibility, super-convergence and uniform convergence of these high order methods.

奇异摄动问题一直是科学计算领域密切关注且迫切需要解决的问题。奇异摄动Volterra积分微分方程是一类典型的奇异摄动问题。这类方程广泛应用于物理、生物等领域中，如扩散耗散过程、同期控制系统、更新过程和拉伸纤维等。这类方程的数值计算方法、理论分析及应用的研究一直是科学计算领域同行研究的热点。本项目将研究求解这类方程的高精度数值计算方法。以前相关的结果主要集中在“渐进分析”问题上，关于“高精度的数值计算格式”、“超收敛性”和“一致收敛性”这些问题的结果却很少，所以本项目的研究具有极大的挑战。我们将设计一些稳定、高效的数值计算格式来求解奇异摄动Volterra积分微分方程，比如间断有限元方法、DG-CFEM方法等。同时，我们将研究这类方程的正则性，并分析这些高精度方法的可行性、超收敛性以及一致收敛性。

奇异摄动积分微分方程广泛应用于物理、生物等领域，其高精度数值计算方法一直是科学计算界同行研究的热点问题之一。本项目的研究成果主要内容为：对于奇异摄动Volterra积分微分方程，运用局部间断有限元方法（包括h-version LDG方法和p-version LDG方法）以及耦合方法（CFEM-LDG方法）求解该问题具有高精度性质。本项目得到如下三个重要结论：第一，在局部加密网格下，h-version LDG解的数值通量在节点处收敛阶能达到2p+1阶，而且具有与小参数无关的一致收敛性；第二，将区间粗略剖分为边界层和外部区域，p-version LDG解的数值通量在节点处达到指数收敛，而且具有与小参数无关的一致收敛性；第三，在局部加密网格下，CFEM-LDG解的数值通量在节点处收敛阶能达到2p阶，而且具有与小参数无关的一致收敛性。