基于同伦技术的Burgers方程的小波精细积分算法

以Burgers方程为例,结合区间小波精细积分方法,将同伦摄动方法的应用范围推广到多维非线性问题,给出一种求解非线性偏微分方程的新的小波精细积分方法,得到一种近似解析解的数值结果,对时间步长不敏感,更适合于求解非线性问题.     

静电磁场不规则区域问题的小波插值Galerkin算法

讨论了用小波插值Galerkin方法(WIGM)求解椭圆型偏微分方程,特别是求解区域不规则时的问题.在归纳出WIGM一般形式的基础上,推导出该方法在Sobolev空间范数下的误差界限为C2-m.提出了一种解决不规则区域中静电磁场场分析问题的数值算法,其中选用对称插值尺度函数为基函数,它的对称性及其与平均插值尺度函数的关系可以在一定程度上降低数值求解的计算量.最后通过计算实例说明该算法的有效性.     

多层快速多极子算法中的两步插值技术

多层快速多极子算法(MLFMA)在快速多极子算法(FMM)的基础上按多层聚集、层间转移和多层扩散的思路以达到优化矩阵向量积的运算的目的,其中多层聚集和多层扩散过程,随着层数递增,角谱积分采样点数逐层递增,为了快速计算角谱积分,需要采用插值技术和反插值技术以提高计算效率.应用两步插值技术替代传统的单步插值技术,大幅提高了多层快速多极子层间插值反插值操作的计算效率,对于应用普通个人计算机求解特大电大尺寸问题,具有重要意义.     

求解静电场偏微分方程的新代数多重网格法

多重网格法是求解偏微分方程大规模离散化方程最有效的方法,针对静电场偏微分方程,讨论一致线性有限元剖分下的拉格朗日有限元方程的代数多重网格法,给出了一种新的粗化算法和构造插值算子的途径。数值实验表明,新的代数多重网格法的有效性。     

插值小波尺度法探地雷达数值模拟及四阶Runge Kutta辅助微分方程吸收边界条件

应用迭代插值方法构造了插值小波尺度函数,并将该尺度函数的导数用于离散Maxwell方程组的空间微分,使用四阶Runge Kutta(four order Runge Kutta,RK4)算法计算时间导数,导出了插值小波尺度法的探地雷达(ground penetrating radar,GPR)正演公式,与常规的基于中心差分的时域有限差分算法(finite difference time domain,FDTD)相比,插值小波尺度算法提高了GPR波动方程的空间与时间离散精度.首先,采用具有解析解的层状模型,分别将FDTD算法及插值小波尺度法应用于层状模型正演,单道雷达数据与解析解拟合表明:相同的网格剖分方式,插值小波尺度法比FDTD具有更高的精度.然后,将辅助微分方程完全匹配层(auxiliary differential equation perfecting matched layer,ADE-PML)边界条件应用到插值小波尺度法GPR正演中,在均匀介质模型中对比了FDTD-CPML(坐标伸缩完全匹配层),FDTD-RK4ADE-PML、插值小波尺度RK4ADE-PML的反射误差,结果表明:插值小波尺度RK4ADE-PML吸收效果优于另外两种条件下的吸收边界.最后,应用加载UPML(各向异性完全匹配层)的FDTD和RK4ADE-PML的插值小波尺度法开展了二维GPR模型的正演,展示了RK4ADE-PML对倏逝波的良好吸收效果.     

求解双曲型守恒律方程的一类自适应多分辨方法

针对双曲型守恒律方程问题,发展一种有效的自适应多分辨分析方法.通过对嵌套网格上的数值解构造离散多分辨分析,建立小波系数与多层嵌套网格点之间的对应关系.对于小波系数较大的网格点采用高精度WENO格式计算,其余区域则直接采用多项式插值.数值试验表明,该方法在保持原规则网格方法的精度和分辨率的同时,显著地减少计算的CPU时间.     

广义Birkhoff方程的积分方法

场方法和最终乘子法是求解运动微分方程的基本方法.本文将这两种方法应用于广义Birkhoff系统,求出了场方法的基本偏微分方程和该方程的完全积分;根据Jacobi最终乘子定理求出了广义Birkhoff方程的解.并举例说明结果的应用.     

SAMR网格上扩散方程有限体格式的逼近性与两层网格算法

针对结构自适应加密网格(SAMR)上扩散方程的求解,分析几种有限体格式的逼近性,同时设计和分析一种两层网格算法.首先,讨论一种常见的守恒型有限体格式,并给出网格加密区域和细化/粗化插值算子的条件;接着,通过在粗细界面附近引入辅助三角形单元,消除粗细界面处的非协调单元,设计了一种保对称有限体元(SFVE)格式,分析表明,该格式具有更好的逼近性,且对网格加密区域和插值算子的限制更弱;最后,为SFVE格式构造一种两层网格(TL)算法,理论分析和数值实验表明该算法的一致收敛性.     

基于混合场积分方程的半空间上方金属目标电磁散射特性高效分析

研究一种可以高效求解半空间金属目标电磁散射积分方程方法,电场积分方程适用于任意结构电磁问题分析,但是生成的矩阵条件数大,迭代求解收敛性差;而磁场积分方程生成的矩阵条件数小,迭代收敛性好,但是仅能分析闭合结构问题,本文采用了混合场积分方程方法,同时具备电场积分方程的普适性与磁场积分方程的收敛性.由于混合场积分方程中涉及格林函数的梯度项,为了进一步加快计算效率,本文引入了一种针对半空间格林函数的高效四维空间插值方法,对组成半空间格林函数的索末菲积分进行列表和Lagrange插值,以实现高效的迭代求解,效率在传统混合场积分方程的基础上提高12.6倍.数值结果表明,该方法在保证精度的同时,可以显著降低求解问题的时间.     

考虑密度与温度耦合关系的非结构网格SIMPLEC算法

提出一种新的密度压力速度和温度耦合的算法,使用静焓方程构造温度修正量与压力修正量之间的关系,类似Rhie-Chow方法用动量方程插值,网格面上的温度压力修正量插值采用能量方程的形式.CGNS API(CFD General Notation System Application Programming Interface)作为非结构化网格求解器的前处理和后处理,基于网格的FVM(Finite Volume Method)作为偏微分方程求解方法.燃烧模拟采用Arrinius-Eddy Dissipation模型.最后给出两个算例作为说明.     

Stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system

A stochastic collocation method is proposed to investigate the secondary bifurcation of a two-dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces, and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle. The accuracy of the stochastic collocation method is improved by incorporating higher order schemes, such as piecewise cubic interpolation and piecewise cubic spline interpolation, instead of a piecewise linear interpolation formula. For an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables, an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy. Numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems.     

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.     

Multi-Symplectic Wavelet Collocation Method for Maxwell's Equations

In this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell's equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.     

FGMs稳态热传导分析的重心Lagrange插值配点法

提出一种求解二维功能梯度材料（FGMs）稳态热传导问题的重心Lagrange插值配点法.基于Chebyshev节点构造二维重心Lagrange插值函数及其偏导数,然后基于配点法将其直接代入FGMs热传导问题的控制方程和边界条件,得到系统离散方程.重心Lagrange插值配点法是一种真正的无网格方法,很好地融合了重心Lagrange插值和配点格式的优势,具有高效、稳定、高精度和易于数值实现的优点.采用重心Lagrange插值配点法分别对指数型、二次型和三角型FGMs热传导问题进行数值模拟.结果表明：该方法具有较高的计算效率和计算精度,对材料梯度参数的变化不敏感.可以进一步拓展到FGMs瞬态问题和FGMs的热力耦合分析.     

Viscous detonation in H2‒O2‒Ar using intrinsic low-dimensional manifolds and wavelet adaptive multilevel representation

A standard ignition delay problem for a mixture of hydrogen-oxygen-argon in a shock tube is extended to the viscous regime and solved using the method of intrinsic low-dimensional manifolds (ILDM) coupled with a wavelet adaptive multilevel representation (WAMR) spatial discretization technique. An operator-splitting method is used to describe the reactions as a system of ordinary differential equations at each spatial point. The ILDM method is used to eliminate the stiffness associated with the chemistry by decoupling processes which evolve on fast and slow time scales. The fast time scale processes are systematically equilibrated, thereby reducing the dimension of the phase space required to describe the reactive system. The WAMR technique captures the detailed spatial structures automatically with a small number of basis functions thereby further reducing the number of variables required to describe the system. A maximum of only 300 collocation points and 15 scale levels yields results with striking resolution of fine-scale viscous and induction zones. Additionally, the resolution of physical diffusion processes minimizes the effects of potentially reaction-inducing artificial entropy layers associated with numerical diffusion.     

A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates

In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence.     

On Convergence of a Least-Squares Kansa's Method for the Modified Helmholtz Equations

We analyze a least-squares asymmetric radial basis function collocation method for solving the modified Helmholtz equations. In the theoretical part, we proved the convergence of the proposed method providing that the collocation points are sufficiently dense. For numerical verification, direct solver and a subspace selection process for the trial space (the so-called adaptive greedy algorithm) is employed, respectively, for small and large scale problems.     

Weighted-Residual Methods for the Solution of Two-Particle Lippmann–Schwinger Equation Without Partial-Wave Decomposition

Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves. The aim of the present work is to show that the weighted-residual approach in combination with local basis functions give rise to convenient computational schemes for the solution of the multi-variable integral equations without the partial wave expansion. The weighted-residual approach provides a unifying framework for various variational and degenerate-kernel methods for integral equations of scattering theory. Using a direct-product basis of localized quadratic interpolation polynomials, Galerkin, collocation and Schwinger variational realizations of the weighted-residual approach have been implemented for a model potential. It is demonstrated that, for a given expansion basis, Schwinger variational method exhibits better convergence with basis size than Galerkin and collocation methods. A novel hybrid-collocation method is implemented with promising results as well.     

随机扰动对拟小波方法求解对流扩散方程的影响

引进拟小波方法数值求解对流扩散方程，研究结果表明，计算带宽W有一个极值，当计算带宽W取该极值时，该方程的拟小波解的精度最高，且好于迎风格式。当边界发生随机不等幅扰动时，对于积分时间较长的情况，拟小波格式的效果要稍逊于迎风格式；当边界发生随机等幅扰动时，若计算带宽W取大于等于20的整数时，方程拟小波解的精度与迎风格式相同；当参数受到随机扰动时，W取10时的拟小波解的均方根误差要小于迎风格式；在初值发生随机扰动且计算带宽W取10时，方程的拟小波解的精度最高，好于迎风格式。     

Symplectic wavelet collocation method for Hamiltonian wave equations

This paper introduces a novel symplectic wavelet collocation method for solving nonlinear Hamiltonian wave equations. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, collocation method is conducted for the spatial discretization, which leads to a finite-dimensional Hamiltonian system. Then, appropriate symplectic scheme is employed for the integration of the Hamiltonian system. Under the hypothesis of periodicity, the properties of the resulted space differentiation matrix are analyzed in detail. Conservation of energy and momentum is also investigated. Various numerical experiments show the effectiveness of the proposed method.