Applying finite-element vertical discretization to a mass-based non-hydrostatic kernel has proved difficult due to the constraints of vertical operators. This article proposes a novel hybrid finite-element vertical discretization method for a semi-implicit mass-based nonhydrostatic kernel, which integrates a finite-differential scheme and a finite-element scheme. In the hybrid method, the finite-differential scheme which satisfies the set of constraints is applied to the linear part, while a cubic finite-element scheme with high-order accuracy is applied to the non-linear part. Furthermore, to improve the accuracy of the linear part, an enlarged set of vertical levels is applied to the differential scheme. This set of vertical levels is only used to solve semi-implicit equations, and has no impact on the grid point calculation and spectral transformations. A series of 2D idealized test cases are conducted to verify the stability and the accuracy of our new method.

A coupled method of Laplace transform and Legendre wavelets is presented to obtain exact solutions of Lane-Emden-type equations. By employing properties of Laplace transform, a new operator is first introduced and then its Legendre wavelets operational matrix is derived to convert the Lane-Emden equations into a system of algebraic equations. Block pulse functions are used to calculate the Legendre wavelets coefficient matrices of the nonlinear terms. The results show that the proposed method is very effective and easy to implement.

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The initial field has a crucial influence on numerical weather prediction (NWP). Data assimilation (DA) is a reliable method to obtain the initial field of the forecast model. At the same time, data are the carriers of information. Observational data are a concrete representation of information. DA is also the process of sorting observation data, during which entropy gradually decreases. Four-dimensional variational assimilation (4D-Var) is the most popular approach. However, due to the complexity of the physical model, the tangent linear and adjoint models, and other processes, the realization of a 4D-Var system is complicated, and the computational efficiency is expensive. Machine learning (ML) is a method of gaining simulation results by training a large amount of data. It achieves remarkable success in various applications, and operational NWP and DA are no exception. In this work, we synthesize insights and techniques from previous studies to design a pure data-driven 4D-Var implementation framework named ML-4DVAR based on the bilinear neural [d=Dong.R]network networks (BNN). The framework replaces the traditional physical model with the BNN model for prediction. Moreover, it directly makes use of [d=Dong.R]the ML model obtained from the simulation data the simulation data obtained from the learning model to implement the primary process of 4D-Var, including the realization of [d=Dong.R]the short-term forecast process the model short-term forecast and the tangent linear and adjoint models. We test a strong-constraint 4D-Var system with the Lorenz-96 model, and we compared the traditional 4D-Var system with ML-4DVAR. The experimental results demonstrate that the ML-4DVAR framework can achieve better assimilation results and significantly improve computational efficiency.

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique.