摘要
本文讨论带光滑核的第二类Volterra积分微分方程以及奇异摄动Volterra积分微分方程的高阶数值格式。方程中积分项的出现,表明该问题具有记忆性质。如何快速高效地求解这类方程是计算数学领域关心的重要问题之一。近年来,由于意识到谱收敛的优越性,人们逐渐开始用谱方法来求解这类方程。本文采用谱Jacobi-Petrov Galerkin方法和伪谱Jacobi-Petrov-Galerkin方法求解第二类Volterra积分微分方程。而后者的主要思想是采用离散带权内积近似逼近前者中的带权内积。本文从理论上证明了这些方法在Lω2和L∞意义下具有指数收敛性质。数值结果表明这些方法不仅具有运算速度快的特点,而且的确能达到谱收敛精度。受此启发,本文证明了当核函数和源函数满足一定的条件时,方程的解满足所谓的M条件。在此基础上,进一步证明了两类重要的谱方法,即谱和伪谱Legendre-Petrov-Galerkin方法以及谱和伪谱Chebyshev-Petrov-Galerkin方法的超几何收敛性。
对于奇异摄动Volterra积分微分方程,本文首先证明了解的正则性。对于这类问题,当小参数ε趋于零时,解在边界层变化非常快。本文通过选取局部加密网格策略和高阶数值格式相结合的方法来求解此类问题。首先,采用间断有限元方法求解,并证明了该方法的稳定性。当核函数正定时,本文进一步证明了在Shishkin网格下DG方法具有一致收敛性质。在边界层区域,DG解在L2意义下具有(?)(lnN/N)p阶收敛；节点处DG解的数值通量具有(lnN/N)2p+1阶的超收敛性。数值算例不仅验证了该方法稳定,而且DG解在L2范数下能达到p+1阶最佳收敛；节点处DG解的数值通量具有2p+1阶的超收敛性。随后,本文采用耦合方法求解该方程,即在边界层区域,运用连续有限元求解；在外部区域,运用间断有限元求解。对于正定核情形,本文给出了解的存在唯一性证明。数值算例表明,在Shishkin网格下,该方法不仅稳定,而且解在L2范数下能达到p+1阶最佳收敛性；在节点处的数值解具有2p阶的超收敛性。耦合方法在Shishkin网格下同样具有一致收敛性质。
The high-order numerical schemes for Volterra integro-differential equa-tions of the second kind and singularly perturbed Volterra integro-differential equations with the smooth kernel are discussed in this paper. The integral item in these equations leads to the memory property for the problem. How to solve these equations more efficiently and rapidly is one of the most important issues in the field of computational mathematics. In recent years, for realizing the advantage of spectral convergence, people begin to solve these equations by the spectral method. In this paper, spectral Jacobi-Petrov-Galerkin method and pseudo-spectral Jacobi-Petrov-Galerkin method are used to solve Volterra integro-differential equations of the second kind. The key idea of the latter is that the weighted inner product of the former can be approximated by the weighted discrete inner product. This paper has proved the exponential convergence property of these methods in the sense of Lw2and L∞norms theo-retically. Numerical results show that these methods not only are rapid in the computation, but also achieve the spectral convergence indeed. Inspired by the numerical experiment, that the analytic solution of the equation satisfies the so-called M condition is proved in the paper when the kernel function and the source function satisfy certain conditions. Furthermore, the supergeomet-ric convergence property for two special spectral methods. i.e., spectral and pseudo-spectral Legendre-Petrov-Galerkin methods and spectral and pseudo-spectral Chebyshev-Petrov-Galerkin methods, is proved rigorously
For singularly perturbed Volterra integro-differential equations, this pa-per firstly has proved the regularity property of the solution. For this kind problem, when the parameter∈limits to zero, the solution undergoes a rapid transition in the layer region. In the paper, we choose the strategy of the local grid refinement and seek the high-order numerical scheme to solve the problem. First, this paper uses the discontinuous Galerkin(DG) method to solve such problem and proves the stability property of the method. If the kernel function is positive definite, this paper has further proved that the DG method has the uniform superconvergence property under the Shishkin mesh. In the layer region, the DG solution has the convergence rate (?)(ln N/N)p in L2norm, and the DG solution for the one-side flux at nodes achieves the super-convergence rate (ln N/N)2p+1. Numerical experiment validates that not only is the DG method stable, but also the DG solution has the optimal convergence rate p+1in L2norm and the DG solution for the one-side flux at nodes has the superconvergence rate2p+1. In the following part, we use the coupled method to solve the equation, i.e., the finite element method is used in the layer region and the DG method is applied out of the layer region. For the case of positive definite kernel, the existence and uniqueness of the solution by the coupled method is proved in the paper. Numerical experiment shows that not only is this method stable, but also the solution achieves the optimal convergence rate p+1in L2norm, and the numerical solution at nodes has the convergence rate2p. The coupled method has the uniform convergence property under the Shishkin mesh.
引文
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