有限区域浅水波模式的勒让德小波谱方法研究

发布时间：2018-03-24 22:28

  本文选题：勒让德小波　切入点：方波脉冲函数　出处：《国防科学技术大学》2015年博士论文

【摘要】：当前,上至国民经济建设,下至人民的生产生活,均离不开准确的天气预报作保障。天气预报的准确率、时效性和精细化程度反映在数值模式的应用性能上,而计算方法是影响数值模式应用性能的重要因素。因此关于数值模式高效计算方法的研究变得十分重要。由于谱模式具有精度高、稳定性好和能够避免非线性不稳定问题等优点,因而成为世界各国广泛业务化的全球数值天气预报模式。然而谱模式依然面临以下两个主要问题:(1)由于基函数的全局性,当函数不光滑或在局部区域内变化较剧烈时会出现所谓的“Gibbs现象”;(2)受限于“格谱变换”的计算复杂度,谱模式计算量随着模式水平分辨率的提高而迅速增大和难于并行。上述两个问题严重制约了数值预报谱模式的发展。基函数的选取对谱方法的应用性能提升至关重要。目前国际上很多学者建议使用分段多项式作为谱方法或者有限元方法的基函数来开发新的数值方法。由于具有正则化、空间局部性和多分辨率分析等优点,小波成为谱方法理想的基函数。小波能够精确表示各种函数和算子,更重要的是基于小波的多尺度结构可以构造快速变换算法。此外,由于小波基在物理空间和频谱空间均具有良好的紧支性,因此它不仅能削弱“Gibbs现象”,提高计算精度,而且可以大大降低模式的截断波数,从而减少计算开销[1,2]。在众多小波中,勒让德小波因构造简单、权函数为1和操作矩阵具有块对角稀疏性等优点,因而受到广泛关注。分数阶微分以函数积分的形式给出的,当前时刻的微分与过去所有时刻的函数值有关,因此具有全局性和记忆性。气象中的极端天气和异常气候过程具有随机性,而分数阶微分算子的记忆性恰好能够很好的用于刻画这种随机性,因此分数阶偏微分方程在气象中具有广大的应用前景。针对目前数值预报谱模式存在的问题,本文基于谱方法和勒让德小波方法,提出了使用勒让德小波作为谱方法基函数的勒让德小波谱方法。为了使勒让德小波谱方法适用于分数阶偏微分方程的求解,本文将整数阶勒让德小波推广到任意阶。数值实验结果表明勒让德小波谱方法在保持谱收敛特性的同时能够削弱“Gibbs现象”。更重要的是,得益于勒让德小波的多尺度结构特性,该方法还具有多级并行性。论文的工作主要集中在以下六个方面:(1)系统综述了国内外数值预报谱模式的发展现状,从而指出了勒让德小波在气象的应用前景,综述了气象中的谱方法和勒让德小波求解偏微分方程的研究进展。(2)证明了二维勒让德小波向量的积分和微分定理,给出了二维勒让德小波微分操作矩阵的构造方法。分析了多尺度勒让德小波展开、积分和微分的谱收敛特性。基于勒让德小波的多尺度结构特性,提出和实现了快速勒让德小波变换算法。(3)提出了基于方波脉冲函数的勒让德小波乘积项谱系数的计算方法,并进行了相应的算法设计、分析和应用研究。(4)提出了勒让德小波谱配置法(LWSCM),分析了其稳定性和收敛性。针对多尺度LWSCM在求解边值问题时面临的边值信息传递问题,给出了分点信息交换策略,最后将LWSCM应用于有限区域浅水波模式的求解。(5)提出了勒让德小波谱Tau方法(LWSTM),对LWSCM与LWSTM的进行比较研究,系统分析了勒让德小波谱Tau方法的稳定性和收敛性。最后将LWSTM应用于有限区域浅水波模式的求解。(6)定义了分数阶勒让德小波,将整数阶勒让德小波推广到任意阶,提出了求解分数阶微分方程的变分迭代与勒让德小波混合方法(FLWVIM)和求解分数阶偏微分方程的二维分数阶勒让德小波方法(2D-FLWs)。
[Abstract]:At present, the construction of national economy, to people's production and life, all cannot do without accurate weather forecasts for protection. The weather forecast accuracy, timeliness and refinement is reflected in the application of numerical model, and the calculation method of numerical model is an important factor affecting the application performance. Therefore it becomes very important to study on the efficient calculation method of numerical model. Because the spectral model has high accuracy, good stability and can avoid the advantages of nonlinear instability, and thus become the world wide business of global numerical weather prediction models. However, spectral model still faces two main problems as follows: (1) due to the global basis functions, when function is not smooth or in the local area changed significantly when there will be a so-called "Gibbs phenomenon"; (2) the calculation is limited to "transform" the complexity of the computation model with spectral model The increase in horizontal resolution increase rapidly and difficult to parallel. These two problems seriously restrict the development of the numerical prediction model. Spectral basis function is chosen to enhance application performance of spectral method is very important. Many current international scholars have suggested that the use of piecewise polynomial basis functions as spectral method or finite element method to develop a new numerical method. Due to regularization, spatial locality and multi-resolution analysis of wavelet basis function has become the ideal method. Wavelet spectrum can accurately represent all kinds of functions and operators, more important is the wavelet multi-scale structure can be constructed based on fast transform algorithm. In addition, the wavelet has good compactness in physical space and space spectrum, so it can not only weaken the "Gibbs phenomenon", improve the calculation accuracy, but also can greatly reduce the number of truncated modes, thereby reducing the calculation The overhead of [1,2]. in many wavelet, Legendre small wave has the advantages of simple structure, the weight function is 1 and operation has the advantages of sparse block diagonal matrix, which has attracted extensive attention. The fractional differential is presented in terms of function integral, differential current and past all the time about the numerical function, so it has global and memory in the extreme weather. And abnormal climate process is random, and the memory of the fractional differential operator that can be very good for describing this kind of randomness, so the fractional partial differential equation has broad application prospect in meteorology. Aiming at the problems of numerical prediction of spectral model, the spectral method and Legendre wavelet based on the method proposed by Legendre Legendre wavelet as wavelet spectral method for spectral basis functions. In order to make the Legendre spectral method is suitable for the fractional partial differential. The solution, the integer order Legendre wavelet is extended to arbitrary order. Numerical results table miner Jean de spectrum method to weaken the "Gibbs phenomenon" while maintaining spectral convergence characteristics. More importantly, thanks to Jules multiscale structure characteristics of Wavelet De, the method also has the multi-level parallelism. This dissertation focuses on the following six aspects: (1) summarizes the development status of domestic and foreign numerical prediction spectral model, which points out the Legendre wavelet application in meteorology, summarized the research progress of spectral method in meteorology and Legendre wavelet to solve partial differential equations. (2) proved that the two wheeler let integral and differential theorem of De Xiaobo vector, gives two wheeler method to construct wavelet de let differential operation matrix. Analysis of multi-scale Legendre wavelet expansion, spectral convergence characteristics based on Le Jean de Pepo integral and differential. Multi scale structure character, proposes and realizes the fast wavelet transform algorithm. Legendre (3) proposed the Fang Bo pulse function of Legendre wavelet product spectrum coefficient calculation method based on the algorithm and the corresponding design, analysis and application research. (4) proposed the Legendre spectral collocation method (LWSCM). Analysis of the stability and convergence. The problem of the information transfer value in the face of multi-scale LWSCM in solving the boundary value problem of edge point information exchange strategy is given, finally, the LWSCM was applied to solve the finite area of shallow water wave model. (5) proposed the Legendre spectrum Tau method (LWSTM), a comparative study was made on LWSCM and LWSTM, analyzed the stability and convergence of the Legendre spectrum Tau method. Finally, the LWSTM was applied to solve the finite area of shallow water wave model. (6) the definition of fractional order Legendre wave, the integer order of Legendre Wavelet is extended to any order, and the variational fractional iteration and Legendre wavelet hybrid method (FLWVIM) for solving fractional differential equations is proposed, and the two-dimensional fractional Legendre wavelet method (2D-FLWs) for solving fractional partial differential equations is proposed.

【学位授予单位】：国防科学技术大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：P456.7;O241.8
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本文编号：1660311