Energy-conserving Scheme Research Articles

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Mathematical analysis and numerical comparison of energy-conservative schemes for the Zakharov equations

AbstractFurihata and Matsuo proposed in 2010 an energy-conserving scheme for the Zakharov equations, as an application of the discrete variational derivative method (DVDM). This scheme is distinguished from conventional methods (in particular the one devised by Glassey (Math Comput 58(197):83–102, 1992)) in that the invariants are consistent with respect to time, but it has not been sufficiently studied both theoretically and numerically. In this study, we theoretically prove the solvability under the loosest possible assumptions. We also prove the convergence of this DVDM scheme by improving the argument by Glassey. Furthermore, we perform intensive numerical experiments for comparing the above two schemes. It is found that the DVDM scheme is superior in terms of accuracy, but since it is fully-implicit, the linearly-implicit Glassey scheme is better for practical efficiency. In addition, we proposed a way to choose a solution for the first step that would allow Glassey’s scheme to work more efficiently.

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

The non-uniform grids in the axisymmetric coordinate system pose a significant challenge for electrostatic particle-in-cell/Monte Carlo collision (PIC/MCC) simulations because they require numerous macroparticles to manage numerical heating around the mid-axis. To address this, we have developed a stochastic weighted particle control method that selectively samples small-weight particles, effectively controlling the particle number without inducing numerical heating. This method is based on a rejection-acceptance probability merging scheme, which is easy to implement and has a low time complexity. We have also made essential modifications, including a corrected density deposition scheme, an energy conservation scheme, and the introduction of target weights. By applying this particle control method, the number of macroparticles in the simulation can be reduced by more than one order of magnitude, significantly reducing the required computing time and storage. Furthermore, appropriately setting target weights also enables enhanced resolution of dilute regions with an acceptable increase in computational cost.

Energy-conserving event-triggered control for multiagent systems with limited resources

An energy-conserving event-driven control method is developed for multiagent systems with unknown disturbances to achieve low-energy formation control. Firstly, an energy-conserving event-driven scheme is developed to save energy by introducing a second trigger function, which is utilized to switch the control input to zero when the consensus error is small. Then, a dynamic energy-conserving event-driven scheme is designed to further decrease the frequency of communication and control by establishing a positive dynamic variable to increase the trigger threshold. Under the designed energy-conserving event-based schemes, the energy consumption during the system stability is significantly reduced, the communication burden is decreased, and the system is Zeno-free and bounded stable. Finally, a four-agent formation example is illustrated to show the effectiveness of the proposed energy-conserving schemes.

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method

In this paper, by taking the NLS-type equations as examples, we propose a novel high-order explicit energy-preserving Lie-group method for Hamiltonian PDEs. The main idea is to combine recently developed generalized SAV (G-SAV) approach (Cheng et al., 2021) and explicit Lie group method, in which the fourth-order or higher-order Lie group method only composes two exponentials in each stage, and this is the key point that the proposed methods can preserve energy of Hamiltonian PDEs. This is our first attempt to construct high-order explicit energy-preserving methods without using projection technique (Hairer et al., 2006 [14]), and some stability conclusions of our proposed methods for the NLS-type equations are also presented. Finally, we present 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Energy-conserving finite difference scheme based on velocity interpolation applicable to unsteady flows using collocated grids

The collocation method uses the Rhie–Chow scheme to find the cell interface velocity by pressure-weighted interpolation. The accuracy of this interpolation method in unsteady flows has not been fully clarified. This study constructs a finite difference scheme for incompressible fluids using a collocated grid in a general curvilinear coordinate system. The velocity at the cell interface is determined by weighted interpolation based on the pressure difference to prevent pressure oscillations. The Poisson equation for the pressure correction value is solved with the cross-derivative term omitted to improve calculation efficiency. In addition, simultaneous relaxation of velocity and pressure is applied to improve convergence. Even without the cross-derivative term, calculations can be stably performed, and convergent solutions are obtained. In unsteady inviscid flow, the conservation of kinetic energy is excellent even in a non-orthogonal grid, and the calculation result has second-order accuracy to time. In viscous analysis at a high Reynolds number, the error decreases compared with that of the Rhie–Chow interpolation method. The present numerical scheme improves calculation accuracy in unsteady flows. The possibility of applying this computational method to high Reynolds number flows is demonstrated through several analyses.

Parallel and energy conservative/dissipative schemes for sine–Gordon and Allen–Cahn equations

The sine–Gordon and Allen–Cahn equations are two typical models in the fields of conservative Hamiltonian systems and dissipative gradient flows, respectively. As the demand for numerical methods that respect intrinsic energy conservation/dissipation laws turns into a fundamental principle, the subsequent computational efficiency is getting more and more desired. Linearly implicit methods, which only require solutions of linear algebraic systems at each time step, have become a popular way to design efficient schemes. However, the overall computational cost of these methods depends heavily on the speed at which the linear system can be solved. In this paper, we propose an alternative approach for developing highly efficient energy-conservative or dissipative schemes for the sine–Gordon and Allen–Cahn equations. Our approach is based on spatial finite difference approximations and is fully implicit at first glance, but actually it is completely decoupled point by point, allowing for the implementation of a scalar nonlinear equation successively with a lower complexity that is comparable only to the degrees of freedom. We further establish the connections between our approach and the classic Itoh–Abe discrete gradient and the partitioned averaged vector field methods, and then propose the generalized Itoh–Abe discrete gradient method, which offers great flexibility in the ordering of updating each unknown point. One immediate benefit is that we can specifically choose an ordering to achieve a naturally parallel computation of the resulting scheme, significantly improving the computational efficiency. Various numerical experiments are presented to illustrate the performance of the proposed schemes.

A class of linearly implicit energy-preserving schemes for conservative systems

We consider a kind of differential equations y˙(t)=R(y(t))y(t)+f(y(t)) with energy conservation. Such conservative models appear for instance in quantum physics, engineering and molecular dynamics. A new class of energy-preserving schemes is constructed by the ideas of scalar auxiliary variable (SAV) and splitting, from which the nonlinearly implicit schemes have been improved to be linearly implicit. The energy conservation and error estimates are rigorously derived. Based on these results, it is shown that the new proposed schemes have unconditionally energy stability and can be implemented with a cost of solving a linearly implicit system. Numerical experiments are done to confirm these good features of the new schemes.

Fully decoupled and high-order linearly implicit energy-preserving RK-GSAV methods for the coupled nonlinear wave equation

This paper is concerned with high-order accurate, linearly implicit and energy-preserving schemes for the coupled nonlinear wave equation. To this end, a novel auxiliary variable approach proposed in a recent paper (Ju et al., 2022) is applied to reformulate the governing equation in an equivalent system, which possesses a modified energy that consists of primary functional and quadratic functional. Subsequently, we utilize the extrapolation strategy/prediction–correction technique for treating the nonlinear terms of the equivalent system to obtain a linearized energy-preserving system. Then, a series of fully decoupled high-order linear energy-preserving schemes are constructed by using the symplectic Runge–Kutta (RK) methods. We show that the modified energy functional can be precisely conserved under certain circumstances for the coefficients of the symplectic RK methods, and the unique solvability of the proposed time-stepping scheme is analyzed. In numerical implementations, a class of linear algebraic equations with constant-coefficient matrices need to be solved at each time step and only less computational costs are required. Extensive numerical experiments of the coupled nonlinear wave equation with local/nonlocal diffusion operators are provided to illustrate the energy conservation law and the computational efficiency of the schemes in long-time simulations.

A variant of the discrete gradient method for the solution of the semilinear wave equation under different boundary conditions

In this paper, energy-preserving schemes based upon the discrete gradient are used to numerically solve the semilinear wave equation under periodic boundary conditions, Dirichlet boundary conditions and Neumann boundary conditions. Both the integral form and the evolutionary behaviour of the energy depend on the associated boundary conditions. The wave equations are semi-discretized in different manners for different boundary conditions such that the Hamiltonian functions of the resulting semi-discrete Hamiltonian systems are the discrete analogue of the original energy. Furthermore, the Hamiltonian functions of the resulting semi-discrete systems have similar evolutionary behaviours as the original energy. It is noted that the semi-discrete system exhibits an oscillatory structure. Subsequently, the semi-discrete approximation of the energy evolution as well as the oscillatory structure is passed to the fully-discrete problem by applying extended discrete gradient formula in the temporal direction. Numerical experiments are carried out to show the effectiveness of the new schemes.

A Novel Energy Conservation Scheme for IoT-Based Wireless Networks: A Use Case of E-Commerce Systems for Consumer Electronics

A Novel Energy Conservation Scheme for IoT-Based Wireless Networks: A Use Case of E-Commerce Systems for Consumer Electronics

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of O(h4) for a small time stepsize h.

Energy-preserving schemes for conservative PDEs based on periodic quasi-interpolation methods

In this paper, we present a novel energy-preserving scheme for solving time-dependent partial differential equations with periodic solutions on non-uniform grids. The proposed scheme combines the periodic quasi-interpolation approaches with the discrete gradient method. One of the main advantages of our method is its ability to ensure energy conservation without relying on the requirement of an anti-symmetric differentiation matrix. We rigorously verify the energy conservation properties of the full-discrete scheme, as well as establish the convergence results. Furthermore, we extend the capabilities of our method by incorporating an adaptive strategy, allowing for the design of an adaptive energy-preserving scheme. This adaptive feature enhances the accuracy and efficiency of the numerical solution. To demonstrate the effectiveness and energy-preserving ability of the proposed scheme, we present several numerical examples. These examples showcase the superior accuracy and robustness of our method.

New Conservative Schemes for Zakharov Equation

New first-order and second-order energy preserving schemes are proposed for the Zakharov system. The methods are fully implicit and semi-explicit. It has been found that the first order method is also massconserving. Concrete schemes have been applied to simulate the soliton evolution of the Zakharov system. Numerical results show that the proposed methods capture the remarkable features of the Zakharov equation. We have obtained that the semi-explicit methods are more efficient than the fully implicit methods. Numerical results also demonstrate that the new energy-preserving schemes accurately simulate the soliton evolution of the Zakharov system.

Mass-, and Energy Preserving Schemes with Arbitrarily High Order for the Klein–Gordon–Schrödinger Equations

Mass-, and Energy Preserving Schemes with Arbitrarily High Order for the Klein–Gordon–Schrödinger Equations

An energy-conserving Fourier particle-in-cell method with asymptotic-preserving preconditioner for Vlasov-Ampère system with exact curl-free constraint

We present an efficient and accurate energy-conserving implicit particle-in-cell (PIC) algorithm for the electrostatic Vlasov system, with particular emphasis on its high robustness for simulating complex plasma systems with multiple physical scales. This method consists of several indispensable elements: (i) the reformulation of the original Vlasov-Poisson system into an equivalent Vlasov-Ampère system with divergence-/curl-free constraints; (ii) a novel structure-preserving Fourier spatial discretization, which exactly preserves these constraints at the discrete level; (iii) a preconditioned Anderson-acceleration algorithm for the solution of the highly nonlinear system; and (iv) a linearized and uniform approximation of the implicit Crank-Nicolson scheme for various Debye lengths, based on the generalized Ohm's law, which serves as an asymptotic-preserving preconditioner for the proposed method. Numerical experiments are conducted, and comparisons are made among the proposed energy-conserving scheme, the classical leapfrog scheme, and a Strang operator-splitting scheme to demonstrate the superiority of the proposed method, especially for plasma systems crossing physical scales.

Unconditional superconvergence analysis of an energy conservation scheme with Galerkin FEM for nonlinear Benjamin–Bona–Mahony equation

In this paper, a general energy conservation Crank–Nicolson (CN) fully-discrete finite element method (FEM) scheme is developed for solving the nonlinear Benjamin–Bona–Mahony (BBM) equation, and the unconditional supercloseness and superconvergence behavior are discussed with the nonconforming EQ1rot element. Different from the time–space splitting technique, the boundedness of numerical solution in the broken H1-norm is obtained directly based on the above energy conservation property, and the well-posedness of numerical solution is proved rigorously through the Brouwer fixed point theorem. Then, with the help of the special characters of this element and the interpolation post-processing technique, the unconditional superclose and superconvergence results are deduced strictly without any restriction between mesh size h and time step τ. Further, our analysis is extended to some other popular conforming and nonconforming finite elements. Finally, two numerical experiments are implemented to confirm the theoretical analysis. It is worth mentioning that this paper not only improve the previous results, but also can be regard as a framework for solving the BBM equation and some other PDEs with Galerkin FEMs.

Explicit high accuracy energy-preserving Lie-group sine pseudo-spectral methods for the coupled nonlinear Schrödinger equation

The newly developed generalized scalar auxiliary variable (GSAV) approaches are very popular methods that can be applied to construct high-accuracy energy-preserving schemes for conservative systems. However, these proposed high-order schemes are fully-implicit and have the disadvantages of huge calculating quantities and long computing time for solving nonlinear systems. Therefore, we develop in this paper explicit energy-preserving schemes to deal with the coupled nonlinear Schrödinger equation by combining the Lie-group method and GSAV approaches. The given schemes are very efficient, have high accuracy, and can inherit the modified energy of the system. Ample numerical results are presented to confirm the developed schemes' efficiency, accuracy, and conservation.

High-order Lagrange multiplier method for the coupled Klein-Gordon-Schrödinger system

In this work, a novel class of high-order energy-preserving algorithms are developed for simulating the coupled Klein-Gordon-Schrödinger equations. We introduce a Lagrange multiplier approach and derive a new system which warrants exactly the original total energy, instead of the modified quadratic energy in the previous energy quadratization approach. The Lagrange system is then reformulated equivalently into an optimization problem subject to PDE constraints. We discretize the optimization system by applying the Gauss collocation method together with the prediction-correction technique in time and the sine pseudo-spectral method in space, which leads to a family of fully discrete high-order energy-preserving schemes. In the numerical experiments, we compare this method with other previous methods and verify the high accuracy as well as its ability to maintain the original energy at each time layer. The feasibility and validity of new method are also witnessed for the KGS model in the nonrelativistic limit regime. Additionally, enlightened by this seminal idea, another type of high-order numerical algorithms conserving two invariants of mass and energy are also presented and relevant experimental results are listed to verify the power of the suggested method.