In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct effi...
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In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear *** experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.
For antisymmetric tensors, the paper examines a low-rank approximation that is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least-squares structure-preserving a...
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For antisymmetric tensors, the paper examines a low-rank approximation that is represented via only three vectors. We describe a suitable low-rank format and propose an alternating least-squares structure-preserving algorithm for finding such an approximation. Moreover, we show that this approximation problem is equivalent to the problem of finding the best multilinear low-rank antisymmetric approximation and, consequently, equivalent to the problem of finding the best unstructured rank-1 approximation. The case of partial antisymmetry is also discussed. The algorithms are implemented in the Julia programming language and their numerical performance is discussed.
In this paper, we present a new method to solve the fractional nonlinear Schrodinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting ...
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In this paper, we present a new method to solve the fractional nonlinear Schrodinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.
The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Sc...
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The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger ***,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar ***,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space *** that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete *** expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time ***,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.
This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms a...
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This paper presents two kinds of strategies to construct structure-preserving algorithms with homogeneous Neumann boundary conditions for the sine-Gordon equation, while most existing structure-preserving algorithms are only valid for zero or periodic boundary conditions. The first strategy is based on the conventional second-order central difference quotient but with a cell-centered grid, while the other is established on the regular grid but incorporated with summation by parts (SBP) operators. Both the methodologies can provide conservative semi-discretizations with different forms of Hamiltonian structures and the discrete energy. However, utilizing the existing SBP formulas, schemes obtained by the second strategy can directly achieve higher-order accuracy while it is not obvious for schemes based on the cell-centered grid to make accuracy improved easily. Further combining the implicit midpoint method and the scalar auxiliary variable (SAV) approach, we construct symplectic integrators and linearly implicit energy-preserving schemes for the two-dimensional sine-Gordon equation, respectively. Extensive numerical experiments demonstrate their effectiveness with the homogeneous Neumann boundary conditions. (C) 2019 Elsevier Inc. All rights reserved.
This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrodinger equation with wave operator. A fourth-order energy-conserving difference scheme is devel...
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This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrodinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.& COPY;2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
In this paper,based on discrete gradient,a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is *** solution of this system is a damped nonlinear ***,lots of nonl...
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In this paper,based on discrete gradient,a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is *** solution of this system is a damped nonlinear ***,lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this *** new integrator gives a discrete analogue of the dissipation property of the original ***,since the integrator is based on the variation-of-constants formula for oscillatory systems,it preserves the oscillatory structure of the *** properties of the new integrator are *** convergence is analyzed for the implicit iterations based on the discrete gradient integrator,and it turns out that the convergence of the implicit iterations based on the new integrator is independent of k Mk,where M governs the main oscillation of the system and usually k Mk≫*** significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian *** experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature。
In this paper, we discuss the Cholesky decomposition of the Hermitian positive definite quaternion matrix. For the first time, the structure-preserving Gauss transformation is defined, and then a novel structure-prese...
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In this paper, we discuss the Cholesky decomposition of the Hermitian positive definite quaternion matrix. For the first time, the structure-preserving Gauss transformation is defined, and then a novel structure-preserving algorithm, which is applied to its real representation matrix, is proposed. Our algorithm needs only real number operations, does not depend on the quaternion toolbox for matlab (QTFM) and has more portability. Although the flops of our algorithm are theoretically about the same as those based on quaternion arithmetic operations or QTFM, numerical experiments show that our algorithm runs faster. (C) 2013 Elsevier Inc. All rights reserved.
The N -coupled nonlinear Schr & ouml;dinger equations are reformulated into an expanded form by using a scalar auxiliary variable method, and then a scheme preserving exactly original mass and energy conservation ...
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The N -coupled nonlinear Schr & ouml;dinger equations are reformulated into an expanded form by using a scalar auxiliary variable method, and then a scheme preserving exactly original mass and energy conservation laws is proposed based on discretizing the expanded form. The scheme is efficient as it consists of decoupled linear systems with constant coefficients, along with a nonlinear algebraic equation that can be solved with negligible computational cost. Some numerical experiments are carried out to demonstrate the behavior of wave solutions, the accuracy of solution and the preservation of physical invariants.
We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principl...
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We introduce a novel computational framework designed to explore the dynamic interactions between fluid and solid particles or structures immersed in a viscous fluid medium adhering to the generalized Onsager principle. This innovative framework harnesses the power of the phase -field -embedding method, in which each solid component, whether rigid or elastic, is characterized by a volume -preserving phase field. The unified velocity within the fluid -solid ensemble governs the movement of both solid particles and the surrounding fluid, specifically for passive particles. Active particles, however, are not only influenced by this unified velocity but are also driven by their self-propelling velocities. To capture exclusive volume interactions among particles and between particles and boundaries, we employ repulsive potential forces at a coarser scale. These forces effectively model repulsion and collision effects. Rigid particles maintain structural integrity by enforcing a zero velocity gradient tensor within their spatial domains, necessitating the introduction of a constraining stress tensor. In contrast, elastic particles are governed by a quasi -linear constitutive equation describing the elastic stress within their domains, allowing for accurate modeling of their deformations. The motion of solid particles is tracked by monitoring the dynamics of their centers of mass. This approach facilitates the development of a hybrid, thermodynamically consistent hydrodynamic model applicable to both rigid and elastic particles. To numerically solve this thermodynamically consistent model for elastic particles, we present a structure -preserving numerical algorithm. Notably, in the limit of an infinite elastic modulus, this algorithm converges to the one employed for modeling rigid particles. Finally, we substantiate the effectiveness, accuracy, and stability of our proposed scheme through a series of numerical experiments. These experiments not only validate the computati
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