一种基于Associated Hermite正交函数求解对流扩散方程的算法

提出了一种基于AH(Associated Hermite)正交基函数求解对流扩散方程的无条件稳定算法。该算法将方程的时间项通过Hermite多项式作为正交基函数进行展开,利用Galerkin方法消除时间变量项,从而导出有限维AH域隐式差分方程,突破了传统显式差分格式稳定性条件的限制,最后通过对AH域展开系数的求解得到该对流扩散方程的数值解。在数值算例中,将该算法与传统显示差分法和交替方向隐式差分法进行对比分析,数值计算结果表明,算法无条件稳定且其计算精度与时间步长无关,对于具有精细结构的对流换热问题,该算法具有明显的效率优势,且保持了较高的精度。     

Unified compact ADI methods for solving nonlinear viscous and nonviscous wave equations

In this paper, two unified alternating direction implicit (ADI) methods, based on the combination of fourth-order compact difference for the approximations of the second spatial derivatives with approximation factorization of difference operators, are presented for solving a two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations, respectively. By the discrete energy method, it is shown that their solutions converge to exact solutions with an order of two in time and four in space in L²- and H¹-norms. Finally, numerical findings testify the computational efficiency of the algorithms.     

High order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients

In this paper, we develop implicit difference schemes of O(k⁴ + k²h² + h⁴), where k > 0, h > 0 are grid sizes in time and space coordinates, respectively, for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions. The proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable ADI method of O(k⁴ + k²h² + h⁴). Some physical nonlinear problems are provided to demonstrate the accuracy of the implementation.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

三维泊松方程基于Richardson外推法的高阶紧致差分方法

基于Richardson外推法提出了数值求解三维泊松方程的高阶紧致差分方法.方法通过利用四阶和六阶紧致差分格式,分别在细网格和粗网格上求解,然后利用Richardson外推技术和算子插值方法,得到三维泊松方程在细网格上的六阶和八阶精度的数值解.数值实验结果验证了该方法的精确性和有效性.     

A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations

This paper is concerned with the numerical solution of symmetric large‐scale Lyapunov equations with low‐rank right‐hand sides and coefficient matrices depending on a parameter. Specifically, we consider the situation when the parameter dependence is sufficiently smooth, and the aim is to compute solutions for many different parameter samples. On the basis of existing results for Lyapunov equations and parameter‐dependent linear systems, we prove that the tensor containing all solution samples typically allows for an excellent low multilinear rank approximation. Stacking all sampled equations into one huge linear system, this fact can be exploited by combining the preconditioned CG method with low‐rank truncation. Our approach is flexible enough to allow for a variety of preconditioners based, for example, on the sign function iteration or the alternating direction implicit method. Copyright © 2013 John Wiley & Sons, Ltd.     

An inverse‐free ADI algorithm for computing Lagrangian invariant subspaces

The numerical computation of Lagrangian invariant subspaces of large‐scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low‐rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse‐free representation of the subspace and avoids inverses of ill‐conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low‐rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

解三维抛物型方程的一个ADI格式

构造了一个解三维抛物型方程的高精度ADI格式,格式绝对稳定,截断误差为O（△t^2＋△x^4）;然后应用Richerdson外推法,外推一次得到了具有O（△t^3＋△x^6）阶精度的近似解.     

A class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.