Legendre Wavelets Spectral Methods For Shallow Water Model In Limited-Area

In recent years, it puts forward higher requirements for the accuracy, timeliness, the degree of precision and service level of prediction. In other words, it gives more requirements on the performance of the numerical weather predication(NWP) model. Therefore,the research on the efficient numerical method for NWP model is very important and urgent. Because the spectral model has many advantages, such as high precision, good stability and can avoid nonlinear instability problem, most of the centres employ it for the short term and medium weather forecasting. However, the spectral model still confronts with two major issues:(1) it has the problem of so-called ”Gibbs phenomenon”in representation of discontinuous physical quantities,(2) its computational time increases rapidly with the horizontal resolution and it’s difficult to be parallel problems. These severely restrict the development of NWP spectral model.The choice of the basis functions is crucial to the performance of spectral method.Due to the properties of wavelet such as orthogonality, spatial locality and multi-resolution analysis, wavelet methods have been widely applied in many different fields of science and engineering. Various functions and operators can be represented accurately in terms of wavelets. Moreover, it can establish a connection with the fast numerical algorithms. Wavelet basis have compact or finite support, and as well as exhibit good locality in both the physical space and wavenumber space. Therefore it can weaken the ”Gibbs phenomenon”, localise the error, greatly reduce the truncation wavenumber of model and improve the accuracy. The operational matrix of the Legendre wavelets is sparse, has lower dimension, and most importantly, is equal on every subinterval. These features can decrease the storage and computational complexity. As a result, the Legendre wavelets gain special attentions.The fractional derivative is given in the form of integration, and is dependent on all of the values of the function in the past, so it was non-locality and has the virtue of memory. The fractional derivative operator can describe the stochastic appear in the extreme weather and abnormal climate preferably, so it can look forward to the fractional partial equations(FPDEs) will play an important role in describing the processes of meteorological.To address the issues of spectral model in NWP, we propose an new spectral method using Legendre wavelets as basis functions, and then present an Legendre wavelets spectral model for shallow water equations. Finally, we generalize the Legendre wavelets to arbitrary order for the numerical solution of the FPDEs. The numerical results show that the presented methods still have the exponential convergence and can weaken the ”Gibbs phenomenon”. What’s more, it can save the computation time by multi-level parallelism inherited from the hierarchical scale structure of Legendre wavelets. The main contributions are listed as follows:(1) We provide a comprehensive overview of the spectral models in NWP, the application prospects of Legendre wavelets and the shallow water equations. Then, we discuss the research progress on the spectral method in NWP and Legendre wavelets method for the numerical solution of Partial differential equation(PDE).(2) We present an rigorous proof for two-dimensional(2D) Legendre wavelets operational matrices of integral and derivative, and give an general procedure for constructing the 2D Legendre wavelets operational matrices of derivative. In addition, we analyze the spectral convergence of Legendre wavelets. Finally, we develop a fast Legendre wavelets transform(FLWT) algorithm based on the multi-scale structure of Legendre wavelets.(3) We propose an algorithm based on the block pulse functions(BPFs) to calculate the Legendre wavelets expansion coefficients of nonlinear term and study its computation complexity and accuracy. Finally, we employ two applications to illustrate presented algorithm.(4) We propose an Legendre wavelets spectral collocation method(LWSCM), and then study the stability and convergence of it. What’s more, we present a method for exchanging the information of the boundaries between two sub-domains for multi-scale LWSCM. Finally, we apply the LWSCM to solve the limited-area shallow water equations.(5) We propose an Legendre wavelets spectral tau method(LWSTM) and present a comparative study on LWSCM and LWSTM, and then apply LWSTM to solve the limitedarea shallow water equations.(6) We give the definition of the fractional Legendre wavelets(FLWs) and then present a coupled of variational iteration method(FVIM) and fractional Legendre wavelets method(FLWM) for the numerical solutions of fractional differential equations. What’s more, wepresent a two-dimensional FLWM to obtain the approximate solutions of the fractional partial differential equations(FPDEs).