An efficient method to compute the permeability of disordered fibrous arrays is proposed. A stabilized mixed finite element method is used with an immersed domain approach to represent the porous material at its microscopic scale. Therefore, the Stokes equations are solved in the whole domain (including solid part) using a penalization method. The accuracy is controlled by refining the mesh around the fluid-solid interface defined by a level-set function. Using homogenization techniques, the permeability of an RVE is obtained. Furthermore, a new method to generate disordered fibers in function of the porosity, $Φ$, and other microstructural parameters is proposed and a study of the effect of inter-fiber spacing on $\mathcal{K}$, the permeability tensor, is performed. This task was achieved using parallel computation and over 460 simulations were carried out in two-dimensional RVEs consisting of over 555 fibers.

This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations (PDEs) with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rannacher time-stepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.  

The paper deals with a numerical method for solving nonlinear integro-parabolic problems of Fredholm type. A monotone iterative method, based on the method of upper and lower solutions, is constructed. This iterative method yields two sequences which converge monotonically from above and below, respectively, to a solution of a nonlinear difference scheme. This monotone convergence leads to an existence-uniqueness theorem. An analysis of convergence rates of the monotone iterative method is given. Some basic techniques for construction of initial upper and lower solutions are given, and numerical experiments with two test problems are presented.

A new numerical algorithm for telegraph equations with homogeneous boundary conditions is proposed. Due to the damping terms in telegraph equations, there is no royal conservation law according to Noether's theorem. The algorithm originates from the discovery of a transform applied to a telegraph equation, which transforms the telegraph equation to a Klein-Gordon equation. The Symplectic method is then brought in this algorithm to solve the Klein-Gordon equation, which is based on the fact that the Klein-Gordon equation with the homogeneous boundary condition is a perfect Hamiltonian system and the symplectic method works very well for Hamiltonian systems. The transformation itself and the inverse transformation theoretically bring no error to the numerical computation. Therefore the error only comes from the symplectic scheme chosen. The telegraph equation is finally explicitly computed when an explicit symplectic scheme is utilized. A relatively long time result can be expected due to the application of the symplectic method. Meanwhile, we present order analysis for both one-dimensional and multi-dimensional cases in the paper. The efficiency of this approach is demonstrated with numerical examples.

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

A parameter-self-adjusting Levenberg-Marquardt method (PSA-LMM) is proposed for solving a nonlinear system of equations $F(x) = 0$, where $F : \mathbb{R}^n$ →$\mathbb{R}^n$ is a semismooth mapping. At each iteration, the LM parameter $μ_k$ is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA-LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.