This paper presents the results of a dissipation and dispersion analysis of a linearized version of a novel finite-difference scheme for the fully nonlinear weakly dispersive shallow water equations. It is a modification of the Rusanov scheme, which is a third-order approximation of the gas dynamics equations. We derive stability conditions, phase change and phase error formulas, and investigate the behavior of the harmonic attenuation coefficient. The study uncovers some previously unknown properties of the Rusanov scheme. Notably, both the original and the modified schemes exhibit high-order smallness in the phase error for long waves. Furthermore, we identify the optimal parameter values that ensure monotonicity of the harmonic attenuation coefficient, strong suppression of short-wave harmonics, and small change in their phase.
Using the idea of Ogita and Aishima [1] and Ulm’s method [2], we designed an inexact Ulm-type method based on matrix multiplication for inverse singular value problems, which avoids shortcomings in solving a system of linear equations in each iteration of the algorithm in [3]. Thus, it looks more stable and requires less computations. Under the condition that the given singular values are distinct, the algorithm has the quadratic convergence in the sense of root convergence. Moreover, the numerical experiments described in the last section show that the new method is better than some known algorithms.
In this paper, a new fast compressive Fourier collocation algorithm for high-dimensional diffusion equations is constructed. The new algorithm is based on the high-dimensional sparse and fast Fourier transform and the compressive sampling matching pursuit algorithm. The computational complexity of the proposed algorithm is proportional to d^2 , where d is the dimension of a PDE domain. The error of an approximate solution is estimated and some numerical experiments are given.
In this paper, we study a numerical method to approximate a solution with weak singularity at the initial time for nonlinear fractional integro-differential equations. For this, the fractional order of Legendre polynomials is used as a basis and the given problem is reduced to a finite-dimensional one by applying a spectral collocation method. We theoretically prove the spectral convergence rate of a function approximation based on the fractional-order Legendre polynomials and, on this basis, present an error analysis of the proposed method. Some numerical examples are provided to show the convergence rate and efficiency of our method.
With the swift advancement of artificial intelligence technologies, automatically solving plane geometry problems has emerged as a popular area of research. To improve the automatic solution effect of mathematical plane geometry problems, an efficient deterministic prediction model is constructed by combining the powerful feature extraction ability of Transformer, the time series processing ability of long short-term memory network, and the generation ability based on the bidirectional encoder representation of Transformer. The outcomes of the ablation experiment show that the proposed model performs the best in regard to accuracy and F1 score, with 92.5
