The gas-kinetic theory based flux splitting method has been successfully proposed for solving one- and two-dimensional ideal magnetohydrodynamics by Xu et al. [J. Comput. Phys., 1999; 2000], respectively. This paper extends the kinetic method to solve three-dimensional ideal magnetohydrodynamics equations, where an adaptive parameter $\eta$ is used to control the numerical dissipation in the flux splitting method. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy and resolve strong discontinuous waves in three dimensional ideal MHD problems.

In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging, scientific computing, reverse engineering and geometric modelling. The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation (PDE)-based diffusion model derived by a minimal volume-like variational formulation. The evolution is driven both by the distance from the data set and by the curvature analytically computed by it. The distance function is computed by implicit local interpolants defined in terms of radial basis functions. Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale. The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction. Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions. Moreover, we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.

The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented. Existence and uniqueness of optimal solutions is proved. A collective Gauss-Seidel scheme and a multigrid scheme are discussed. Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments.

The optimal preconditioner and the superoptimal preconditioner were proposed in 1988 and 1992 respectively. They have been studied widely since then. Recently, Chen and Jin [6] extend the superoptimal preconditioner to a more general case by using the Moore-Penrose inverse. In this paper, we further study some useful properties of the optimal and the generalized superoptimal preconditioners. Several existing results are extended and new properties are developed.

Image restoration is often solved by minimizing an energy function consisting of a data-fidelity term and a regularization term. A regularized convex term can usually preserve the image edges well in the restored image. In this paper, we consider a class of convex and edge-preserving regularization functions, i.e., multiplicative half-quadratic regularizations, and we use the Newton method to solve the correspondingly reduced systems of nonlinear equations. At each Newton iteration, the preconditioned conjugate gradient method, incorporated with a constraint preconditioner, is employed to solve the structured Newton equation that has a symmetric positive definite coefficient matrix. The eigenvalue bounds of the preconditioned matrix are deliberately derived, which can be used to estimate the convergence speed of the preconditioned conjugate gradient method. We use experimental results to demonstrate that this new approach is efficient, and the effect of image restoration is reasonably good.

We consider a finite difference scheme for a nonlinear wave equation, whose solutions may lose their smoothness in finite time, i.e., blow up in finite time. In order to numerically reproduce blow-up solutions, we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A  numerical blow-up time is defined and is proved to converge, under a certain hypothesis, to the real blow-up time as the grid size tends to zero.

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The $L^2$ norm and $H^1$ semi-norm error estimates are obtained for the first scheme and the second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.