Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email: [email protected] or [email protected]

Splitting and composition methods in the numerical integration of differential equations

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table

Numerical methods for large-scale Lyapunov equations with symmetric banded data

The numerical solution of large-scale Lyapunov matrix equations with symmetric banded data has so far received little attention in the rich literature on Lyapunov equations. We aim to contribute to this open problem by introducing two efficient solution methods, which respectively address the cases of well conditioned and ill conditioned coefficient matrices. The proposed approaches conveniently exploit the possibly hidden structure of the solution matrix so as to deliver memory and computation saving approximate solutions. Numerical experiments are reported to illustrate the potential of the described methods

A Provably Stable Discontinuous Galerkin Spectral Element Approximation for Moving Hexahedral Meshes

We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian formulation to map the governing equations to a fixed reference domain. The approximation is made stable by a discretization of a skew-symmetric formulation of the problem. We prove that the discrete approximation is stable, conservative and, for constant coefficient problems, maintains the free-stream preservation property. We also provide details on how to add the new skew-symmetric ALE approximation to an existing discontinuous Galerkin spectral element code. Lastly, we provide numerical support of the theoretical results