High Order Methods For Solving Volterra Integro-differential Equations

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 rigorouslyFor 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.