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A study on CFL conditions for the DG solution of conservation laws on adaptive moving meshes

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arXiv 2021年

作者： Zhang, Min Huang, Weizhang Qiu, Jianxian School of Mathematical Sciences Peking University Beijing100871 China Department of Mathematics University of Kansas LawrenceKS66045 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin (DG) solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping. A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy (CFL) conditions established for fixed and uniform meshes. In this work, we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations. A stability analysis is presented for a moving mesh DG method for linear scalar conservation laws. Based on the analysis, a new selection strategy of the time step is proposed, which takes into consideration the coupling of the α-function (that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity) and the heights of the mesh elements. The analysis also suggests several stable combinations of the choices of the α-function in the numerical scheme and in the time step selection. Numerical results obtained with a moving mesh DG method for Burgers’ and Euler equations are presented. For comparison purpose, numerical results obtained with an error-based time step-size selection strategy are also *** Codes 65M50, 65M60 Copyright © 2021, The Authors. All rights reserved.

关键词： Jacobian matrices

New efficient time-stepping schemes for the anisotropic phase-field dendritic crystal growth model

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arXiv 2021年

作者： Li, Minghui Azaiez, Mejdi Xu, Chuanju School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Per-formance Scientific Computing Xiamen University Xiamen361005 China Institut Polytechnique de Bordeaux Laboratoire I2M CNRS UMR5295 France

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model, and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes are emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility *** Codes 74A50 (Primary) 65M12, 65M70, 65Z05 (Secondary) Copyright © 2021, The Authors. All rights reserved.

关键词： Anisotropy

High order well-balanced asymptotic preserving finite difference Weno schemes for the shallow water equations in all froude numbers

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arXiv 2021年

作者： Huang, Guanlan Xing, Yulong Xiong, Tao School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Department of Mathematics The Ohio State University ColumbusOH43210 United States School of Mathematical Sciences Xiamen University Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Fujian Xiamen361005 China

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-at bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced ow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stifiy accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions. © 2021, CC BY.

关键词： Equations of motion

High order finite difference Hermite WENO fixed-point fast sweeping method for static Hamilton-Jacobi equations

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arXiv 2021年

作者： Ren, Yupeng Xing, Yulong Qiu, Jianxian School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Department of Mathematics The Ohio State University ColumbusOH43210 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we combine the nonlinear HWENO reconstruction in [41] and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in [21]. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function φ. They are now computed from the current value of φ and the previous spatial derivatives of φ. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter Ε used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods. Copyright © 2021, The Authors. All rights reserved.

关键词： Hamiltonians

Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

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arXiv 2022年

作者： Jin, Diancong Hong, Jialin Sheng, Derui School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the Γ -convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations. Copyright © 2022, The Authors. All rights reserved.

关键词： Finite difference method

Stability and error analysis of imex sav schemes for the magneto-hydrodynamic equations

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arXiv 2021年

作者： Li, Xiaoli Wang, Weilong Shen, Jie School of Mathematics Shandong University JinanShandong250100 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University XiamenFujian361005 China Department of Mathematics Purdue University West LafayetteIN47907 United States

We construct and analyze first- and second-order implicit-explicit (IMEX) schemes based on the scalar auxiliary variable (SAV) approach for the magneto-hydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable. We derive rigorous error estimates for the velocity, pressure and magnetic field of the first-order scheme in the two dimensional case without any condition on the time step. Numerical examples are presented to validate the proposed schemes. Copyright © 2021, The Authors. All rights reserved.

关键词： Errors

Spiking hierarchy in an adaptive exponential integrate-and-fire network synchronization

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arXiv 2021年

作者： Wu, Xiaoyue Lin, Congping Zhang, Yiwei School of Mathematics and Statistics Huazhong University of Sciences and Technology Wuhan China Center for Mathematical Sciences Huazhong University of Science and Technology Wuhan China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China

Neuronal network synchronization has received wide interests. Network connection structure is known to play a key role in its synchronization. In the present manuscript, we study the influence of initial membrane potentials together with network topology on bursting synchronization, in particular the sequential spiking order in stabilized inter bursts. We find a hierarchical phenomenon on spiking order. We grade neurons into different layers: Primary neurons are those initiate the spike and lower-layer neurons are then determined via network connections successively. This constructs a directed graph to indicate spiking propagation among different layers of neurons. Neurons in upper layers spike earlier than those in lower layers. More interestingly, we find that among the same layer, neuron spiking order is mainly associated with stimuli they receive from upper layer via network connections;more stimuli leads to earlier spiking. Furthermore, we find that with effectively the same stimuli from upper layer, neurons with less connections spike earlier. © 2021, CC BY.

关键词： Neurons

High order residual distribution conservative finite difference HWENO scheme for steady state problems

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arXiv 2021年

作者： Lin, Jianfang Ren, Yupeng Abgrall, Rémi Qiu, Jianxian School of Mathematical Sciences Zhejiang University Zhejiang Hangzhou310058 China School of Mathematical Sciences Xiamen University Fujian Xiamen361005 China Institute of Mathematics University of Zurich Zurich8057 Switzerland School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we develop a high order residual distribution (RD) method for solving steady state conservation laws in a novel Hermite weighted essentially non-oscillatory (HWENO) framework recently developed in [24]. In particular, we design a high order HWENO integration for the integrals of source term and fluxes based on the point value of the solution and its spatial derivatives, and the principles of residual distribution schemes are adapted to obtain steady state solutions. Two advantages of the novel HWENO framework have been shown in [24]: first, compared with the traditional HWENO framework, the proposed method does not need to introduce additional auxiliary equations to update the derivatives of the unknown variable, and just compute them from the current point value of the solution and its old spatial derivatives, which saves the computational storage and CPU time, and thereby improve the computational efficiency of the traditional HWENO framework. Second, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller at the same grid. Thus, it is also a compact scheme when we design the higher order accuracy, compared with that in [11] Chou and Shu proposed. Extensive numerical experiments for one- and two-dimensional scalar and systems problems confirm the high order accuracy and good quality of our scheme. Copyright © 2021, The Authors. All rights reserved.

关键词： Physical properties

High order semi-implicit WENO schemes for all mach full euler system of gas dynamics

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arXiv 2021年

作者： Boscarino, Sebastiano Qiu, Jing-Mei Russo, Giovanni Xiong, Tao Department of Mathematics and Computer Science University of Catania Catania95127 Italy Department of Mathematics University of Delaware NewarkDE19716 United States School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Fujian Xiamen361005 China

In this paper, we propose a new high order semi-implicit scheme for the all Mach full Euler equations of gas dynamics. Material waves are treated explicitly, while acoustic waves are treated implicitly, thus avoiding severe CFL restrictions for low Mach flows. High order accuracy in time is obtained by semi-implicit temporal integrator based on the IMEX Runge-Kutta (IMEX-RK) framework. High order in space is achieved by finite difference WENO schemes with characteristic-wise reconstructions adapted to the semi-implicit IMEX-RK time discretization. Type A IMEX schemes are constructed to handle not well-prepared initial conditions. Besides, these schemes are proven to be asymptotic preserving and asymptotically accurate as the Mach number vanishes for well-prepared initial conditions. Divergence-free property of the time-discrete schemes is proved. The proposed scheme can also well capture discontinuous solutions in the compressible regime, especially for two dimensional Riemann problems. Numerical tests in one and two space dimensions will illustrate the effectiveness of the proposed schemes. © 2021, CC BY.

关键词： Mach number