本文選題：勒讓德小波　切入點(diǎn)：方波脈沖函數(shù)　出處：《國防科學(xué)技術(shù)大學(xué)》2015年博士論文

【摘要】：當(dāng)前,上至國民經(jīng)濟(jì)建設(shè),下至人民的生產(chǎn)生活,均離不開準(zhǔn)確的天氣預(yù)報(bào)作保障。天氣預(yù)報(bào)的準(zhǔn)確率、時(shí)效性和精細(xì)化程度反映在數(shù)值模式的應(yīng)用性能上,而計(jì)算方法是影響數(shù)值模式應(yīng)用性能的重要因素。因此關(guān)于數(shù)值模式高效計(jì)算方法的研究變得十分重要。由于譜模式具有精度高、穩(wěn)定性好和能夠避免非線性不穩(wěn)定問題等優(yōu)點(diǎn),因而成為世界各國廣泛業(yè)務(wù)化的全球數(shù)值天氣預(yù)報(bào)模式。然而譜模式依然面臨以下兩個(gè)主要問題:(1)由于基函數(shù)的全局性,當(dāng)函數(shù)不光滑或在局部區(qū)域內(nèi)變化較劇烈時(shí)會(huì)出現(xiàn)所謂的“Gibbs現(xiàn)象”;(2)受限于“格譜變換”的計(jì)算復(fù)雜度,譜模式計(jì)算量隨著模式水平分辨率的提高而迅速增大和難于并行。上述兩個(gè)問題嚴(yán)重制約了數(shù)值預(yù)報(bào)譜模式的發(fā)展�；瘮�(shù)的選取對(duì)譜方法的應(yīng)用性能提升至關(guān)重要。目前國際上很多學(xué)者建議使用分段多項(xiàng)式作為譜方法或者有限元方法的基函數(shù)來開發(fā)新的數(shù)值方法。由于具有正則化、空間局部性和多分辨率分析等優(yōu)點(diǎn),小波成為譜方法理想的基函數(shù)。小波能夠精確表示各種函數(shù)和算子,更重要的是基于小波的多尺度結(jié)構(gòu)可以構(gòu)造快速變換算法。此外,由于小波基在物理空間和頻譜空間均具有良好的緊支性,因此它不僅能削弱“Gibbs現(xiàn)象”,提高計(jì)算精度,而且可以大大降低模式的截?cái)嗖〝?shù),從而減少計(jì)算開銷[1,2]。在眾多小波中,勒讓德小波因構(gòu)造簡單、權(quán)函數(shù)為1和操作矩陣具有塊對(duì)角稀疏性等優(yōu)點(diǎn),因而受到廣泛關(guān)注。分?jǐn)?shù)階微分以函數(shù)積分的形式給出的,當(dāng)前時(shí)刻的微分與過去所有時(shí)刻的函數(shù)值有關(guān),因此具有全局性和記憶性。氣象中的極端天氣和異常氣候過程具有隨機(jī)性,而分?jǐn)?shù)階微分算子的記憶性恰好能夠很好的用于刻畫這種隨機(jī)性,因此分?jǐn)?shù)階偏微分方程在氣象中具有廣大的應(yīng)用前景。針對(duì)目前數(shù)值預(yù)報(bào)譜模式存在的問題,本文基于譜方法和勒讓德小波方法,提出了使用勒讓德小波作為譜方法基函數(shù)的勒讓德小波譜方法。為了使勒讓德小波譜方法適用于分?jǐn)?shù)階偏微分方程的求解,本文將整數(shù)階勒讓德小波推廣到任意階。數(shù)值實(shí)驗(yàn)結(jié)果表明勒讓德小波譜方法在保持譜收斂特性的同時(shí)能夠削弱“Gibbs現(xiàn)象”。更重要的是,得益于勒讓德小波的多尺度結(jié)構(gòu)特性,該方法還具有多級(jí)并行性。論文的工作主要集中在以下六個(gè)方面:(1)系統(tǒng)綜述了國內(nèi)外數(shù)值預(yù)報(bào)譜模式的發(fā)展現(xiàn)狀,從而指出了勒讓德小波在氣象的應(yīng)用前景,綜述了氣象中的譜方法和勒讓德小波求解偏微分方程的研究進(jìn)展。(2)證明了二維勒讓德小波向量的積分和微分定理,給出了二維勒讓德小波微分操作矩陣的構(gòu)造方法。分析了多尺度勒讓德小波展開、積分和微分的譜收斂特性�；诶兆尩滦〔ǖ亩喑叨冉Y(jié)構(gòu)特性,提出和實(shí)現(xiàn)了快速勒讓德小波變換算法。(3)提出了基于方波脈沖函數(shù)的勒讓德小波乘積項(xiàng)譜系數(shù)的計(jì)算方法,并進(jìn)行了相應(yīng)的算法設(shè)計(jì)、分析和應(yīng)用研究。(4)提出了勒讓德小波譜配置法(LWSCM),分析了其穩(wěn)定性和收斂性。針對(duì)多尺度LWSCM在求解邊值問題時(shí)面臨的邊值信息傳遞問題,給出了分點(diǎn)信息交換策略,最后將LWSCM應(yīng)用于有限區(qū)域淺水波模式的求解。(5)提出了勒讓德小波譜Tau方法(LWSTM),對(duì)LWSCM與LWSTM的進(jìn)行比較研究,系統(tǒng)分析了勒讓德小波譜Tau方法的穩(wěn)定性和收斂性。最后將LWSTM應(yīng)用于有限區(qū)域淺水波模式的求解。(6)定義了分?jǐn)?shù)階勒讓德小波,將整數(shù)階勒讓德小波推廣到任意階,提出了求解分?jǐn)?shù)階微分方程的變分迭代與勒讓德小波混合方法(FLWVIM)和求解分?jǐn)?shù)階偏微分方程的二維分?jǐn)?shù)階勒讓德小波方法(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.

【學(xué)位授予單位】：國防科學(xué)技術(shù)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：P456.7;O241.8
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本文編號(hào)：1660311