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This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L 1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability. [ABSTRACT FROM AUTHOR]

In this paper, we propose a failure-informed multi-stage training algorithm for approximating the double-valley dark soliton solutions of the three-component nonlinear Schrödinger equation (NLSE). Although the physics-informed neural network (PINN) method has achieved remarkable results in many equations, it is ineffective when directly applied to the considered equation due to the non-vanishing background of dark solitons and the complexity of the multi-valley structure. To improve the approximation ability of the PINN, the algorithm we proposed generates effective samples adaptively based on Gaussian mixture distribution by combining the time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. Furthermore, the algorithm is extended to learn the solution operator of three-component NLSE within a specialized solution space. The results of numerical simulation demonstrate that the improved algorithm has better approximation ability and faster convergence rate in approximating the solution and solution operator of the considered equation while using a smaller training set. • We explore a deep learning method for solving three-component nonlinear Schrödinger equation. • We proposed an improved PINNS by time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. • The data-driven double-valley dark soliton solutions are derived by the proposed method. • The proposed method can be easily extended to the physics-informed DeepONets. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]