On a numerical artifact of solving shallow water equations with a discontinuous bottom: Analysis and a nontransonic fix

In this paper, we study a numerical artifact of solving the nonlinear shallow water equations with a discontinuous bottom topography. For various first-order schemes, the numerical solution of the momentum will form a spurious spike at the discontinuous points of the bottom, which should not exist in the exact solution. The height of the spike cannot be reduced even after the mesh is refined. For subsonic problems, this numerical artifact may cause the wrong convergence to a function far away from the exact solution. To explain the formation of the spurious spike, we perform a convergence analysis by proving a Lax--Wendroff type theorem. It is shown that the spurious spike is caused by the numerical viscosity in the computation of the water height at the discontinuous bottom. The height of the spike is proportional to the magnitude of the viscosity constant in the Lax--Friedrichs flux. Motivated by this conclusion, we propose a modified scheme by adopting the central flux at the bottom discontinuity in the equation of mass conservation, and show that this numerical artifact can be removed in many cases. For various numerical tests with nontransonic Riemann solutions, we observe that the modified scheme is able to retrieve the correct convergence.Comment: 37 page

The Runge--Kutta discontinuous Galerkin method with stage-dependent polynomial spaces for hyperbolic conservation laws

In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent polynomial spaces for the spatial discretization operators. To be more specific, two different DG operators, associated with $\mathcal{P}^k$ and $\mathcal{P}^{k-1}$ piecewise polynomial spaces, are used at different RK stages. The resulting method is referred to as the sdRKDG method. It features fewer floating-point operations and may achieve larger time step sizes. For problems without sonic points, we observe optimal convergence for all the sdRKDG schemes; and for problems with sonic points, we observe that a subset of the sdRKDG schemes remains optimal. We have also conducted von Neumann analysis for the stability and error of the sdRKDG schemes for the linear advection equation in one dimension. Numerical tests, for problems including two-dimensional Euler equations for gas dynamics, are provided to demonstrate the performance of the new method.Comment: 24 page

Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise

One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results

EViT: An Eagle Vision Transformer with Bi-Fovea Self-Attention

Thanks to the advancement of deep learning technology, vision transformer has demonstrated competitive performance in various computer vision tasks. Unfortunately, vision transformer still faces some challenges such as high computational complexity and absence of desirable inductive bias. To alleviate these problems, a novel Bi-Fovea Self-Attention (BFSA) is proposed, inspired by the physiological structure and characteristics of bi-fovea vision in eagle eyes. This BFSA can simulate the shallow fovea and deep fovea functions of eagle vision, enable the network to extract feature representations of targets from coarse to fine, facilitate the interaction of multi-scale feature representations. Additionally, a Bionic Eagle Vision (BEV) block based on BFSA is designed in this study. It combines the advantages of CNNs and Vision Transformers to enhance the ability of global and local feature representations of networks. Furthermore, a unified and efficient general pyramid backbone network family is developed by stacking the BEV blocks in this study, called Eagle Vision Transformers (EViTs). Experimental results on various computer vision tasks including image classification, object detection, instance segmentation and other transfer learning tasks show that the proposed EViTs perform effectively by comparing with the baselines under same model size and exhibit higher speed on graphics processing unit than other models. Code is available at https://github.com/nkusyl/EViT.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl