Xiaonan Wu

SOR-like Methods for Augmented Systems

Several SOR-like methods are proposed for solving augmented systems. These have many different applications in scientific computing, for example, constrained optimization and the finite element method for solving the Stokes equation. The convergence and the choice of optimal parameter for these algorithms are studied. The convergence and divergence regions for some algorithms are given, and the new algorithms are applied to solve the Stokes equations as well.

Interest Rate Modeling

As pointed out in Sect. 2.3, when the short-term interest rate is considered as a random variable, there is an unknown function λ(r, t), called the market price of risk, in the governing equation.

The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application

The exterior boundary value problems of linear elastic equations are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem on a bounded domain. Furthermore, a finite element approximation of this problem and optimal error estimates are obtained. Finally, a numerical example shows the effectiveness of this method.

Analysis and convergence of the MAC scheme. II: Navier-Stokes equations

The MAC discretization scheme for the incompressible Navier-Stokes equations is interpreted as a covolume approximation to the equations. Using some results from earlier papers dealing with covolume error estimates for div-curl equation systems, and under certain conditions on the data and the solutions of the Navier-Stokes equations, we obtain first-order error estimates for both the vorticity and the pressure.

A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations

A new mixed finite element method is formulated for the Stokes equations, in which the two components of the velocity and the pressure are defined on different meshes. First-order error estimates are obtained for both the velocity and the pressure. Also, the well-known MAC method is derived from the resulting finite element method.

Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems

We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrodinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu, H. Han, Phys. Rev. E 74 (2006) 037704] to problems in multi-dimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrodinger equations in one and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.

A Finite-Element Method for Laplace- and Helmholtz-Type Boundary Value Problems with Singularities

Laplace- and Helmholtz-type boundary value problems with singularities are considered. A sequence of approximations to the exact boundary conditions at an artificial boundary is given. Then the original problem is reduced to a boundary value problem in a domain away from the singularities. Furthermore, finite-element approximations are applied to this problem and error estimates are obtained. Finally, some numerical examples show the effectiveness of this method.

The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions

This article is concerned with the numerical solution to the time-dependent Schrodinger equation on an infinite domain. Two exact artificial boundary conditions are introduced to reduce the original problem into an initial boundary value problem with a finite computational domain. The artificial boundary conditions involve the 1/2 order fractional derivative in t. Then, a fully discrete explicit three-level difference scheme is derived. The truncation errors are analyzed in detail. The stability and convergence with the convergence order of O(h3/2+?h-1/2) are proved under the condition ?/h2<1/2 by the energy method. A numerical example is given to demonstrate the accuracy and efficiency of the proposed method. Two open problems are brought forward at the end of the article.

Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations.

An efficient method is proposed for numerical solutions of nonlinear Schrödinger equations on an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation, absorbing boundary conditions are designed to truncate the unbounded domain, which are in nonlinear form and can perfectly absorb waves outgoing from the boundaries of the truncated computational domain. The stability of the induced initial boundary value problem defined on the computational domain is examined by a normal mode analysis. Numerical examples are given to illustrate the stable and tractable advantages of the method.

Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains

This paper is concerned with the numerical solution of semilinear parabolic PDEs on unbounded spatial domains whose solutions blow up in finite time. The focus of the presentation is on the derivation of the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional computational domains and on a simple but efficient adaptive time-stepping scheme. The theoretical results are illustrated by a broad range of numerical examples, including problems with multiple blow-up points.