Energy-conserving Scheme Research Articles

Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

In the article Martínez et al. (2019) [7], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.

A Novel Energy-conserving Scheme for Eight-dimensional Hamiltonian Problems

Abstract We design a novel, exact energy-conserving implicit nonsymplectic integration method for an eight-dimensional Hamiltonian system with four degrees of freedom. In our algorithm, each partial derivative of the Hamiltonian with respect to one of the phase-space variables is discretized by the average of eight Hamiltonian difference terms. Such a discretization form is a second-order approximation to the Hamiltonian gradient. It is shown numerically via simulations of a Fermi–Pasta–Ulam-β system and a post-Newtonian conservative system of compact binaries with one body spinning that the newly proposed method has extremely good energy-conserving performance, compared to the Runge–Kutta; an implicit midpoint symplectic method, and extended phase-space explicit symplectic-like integrators. The new method is advantageous over very long times and for large time steps compared to the state-of-the-art Runge–Kutta method in the accuracy of numerical solutions. Although such an energy-conserving integrator exhibits a higher computational cost than any one of the other three algorithms, the superior results justify its use for satisfying some specific purposes on the preservation of energies in numerical simulations with much longer times, e.g., obtaining a high enough accuracy of the semimajor axis in a Keplerian problem in the solar system or accurately grasping the frequency of a gravitational wave from a circular orbit in a post-Newtonian system of compact binaries. The new integrator will be potentially applied to model time-varying external electromagnetic fields or time-dependent spacetimes.

Proteomic and Metabolic Elucidation of Solar-Powered Biomanufacturing by Bio-Abiotic Hybrid System

Summary Highly efficient inorganic light absorbers combined with highly specific biocatalysts, i.e., chemolithoautotrophic microbes, have recently been proposed for solar-powered biomanufacturing to pave the way in artificial photosynthesis. In this work, we studied the global protein and metabolite changes of a Moorella thermoacetica-CdS (cadmium sulfide quantum dots) hybrid system during light harvesting. The Wood-Ljungdahl pathway (WLP) that has been considered as the main approach for CO2 fixation was found to be highly activated by CdS. Targeted metabolite quantification revealed that energy-metabolism-associated glycolysis and the tricarboxylic acid (TCA) cycle were also activated. Therefore, we propose an energy-conservation scheme to the M. thermoacetica-CdS hybrid system, i.e., glycolysis and the TCA cycle along with the oxidation of acetyl-coenzyme A (CoA) and NADH, in addition to the well-documented chemiosmotic proton gradient-driven ATP production by ATPase. We expect that the study can be helpful to guide the design of effective and biocompatible photo-biocatalytic systems in the future.

Linear-implicit and energy-preserving schemes for the Benjamin-type equations

The Benjamin-type equations are typical types of non-local partial differential equations usually describing long internal waves along the interface of two vigorously different fluid layers. In this work, we propose two kinds of novel linear-implicit and energy-preserving algorithms for the Benjamin-type equations. These algorithms are based on the invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, respectively. The IEQ and SAV are originally developed to construct energy stable schemes for the class of gradient flows. Herein, we innovate such schemes to the Benjamin-type equations and, essentially, verify them to be effective to construct energy-preserving schemes for the Hamiltonian structures. Meanwhile, numerical experiments are presented to demonstrate the efficiency of these schemes eventually.

An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation

In this paper, we propose a new approach to construct an explicit structure-preserving scheme for the nonlinear fractional wave equation. First, we reformulate the equation as a canonical Hamiltonian system. Then, we utilize the fourth-order fractional centered difference formula to discretize the equation in space direction, and obtain a conservative semi-discrete system. Subsequently, we develop an explicit fully-discrete energy-preserving scheme for the equation by using the proposed approach. Numerical examples are provided to confirm our theoretical analysis in long time computations at last.

Theoretical analysis of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

Departing from an initial-boundary-value problem governed by a Klein–Gordon–Zakarov system with fractional derivatives in the spatial variable, we provide an explicit finite-difference scheme to approximate its solutions. In agreement with the continuous system, the method proposed in this work is also capable of preserving the energy of the system, and the energy quantities are nonnegative under flexible parameter conditions. The boundedness, the consistency, the stability and the convergence of the technique are also established rigorously. The method is easy to implement, and the computer simulations confirm the main analytical and numerical properties of the new model.

A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach

In this paper, we develop a novel, linearly implicit and local energy-preserving scheme for the sine-Gordon equation. The basic idea is from the invariant energy quadratization approach to construct energy stable schemes for gradient systems, which are energy dispassion. We here take the sine-Gordon equation as an example to show that the invariant energy quadratization approach is also an efficient way to construct linearly implicit and local energy-conserving schemes for energy-conserving systems. Utilizing the invariant energy quadratization approach, the sine-Gordon equation is first reformulated into an equivalent system, which inherits a modified local energy conservation law. The new system are then discretized by the conventional finite difference method and a semi-discretized system is obtained, which can conserve the semi-discretized local energy conservation law. Subsequently, the linearly implicit structure-preserving method is applied for the resulting semi-discrete system to arrive at a fully discretized scheme. We prove that the resulting scheme can exactly preserve the discrete local energy conservation law. Moveover, with the aid of the classical energy method, an unconditional and optimal error estimate for the scheme is established in discrete $$H_h^1$$ -norm. Finally, various numerical examples are addressed to confirm our theoretical analysis and demonstrate the advantage of the new scheme over some existing local structure-preserving schemes.

Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions

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.

Symplectic simulation of dark solitons motion for nonlinear Schrödinger equation

In this paper, we study symplectic simulation of dark solitons motion of nonlinear Schrodinger equation (NLSE). The Ablowitz-Ladik model (A-L model) of NLSE can be expressed as a non-canonical Hamiltonian system. By using splitting technique, we construct explicit splitting K-symplectic methods for the A-L model. On the other hand, the A-L model can be transformed into a canonical system and standard symplectic methods can be employed to perform numerical simulation. A second order K-symplectic method and a second order symplectic method are employed to simulate one dark soliton and two dark solitons motion for the A-L model and its canonicalized system respectively. By comparing with a third-order non-symplectic Runge-Kutta method, we show the superiorities of the two symplectic methods in long-term tracking the motion of dark solitons and preserving the invariants. We also compare the CPU times of K-symplectic methods and standard symplectic methods and show that the former ones are more efficient. The energy-preserving scheme is also applied for non-canonical Hamiltonian systems. The numerical results demonstrate that the K-symplectic methods can nearly preserve the energy, the discrete invariants of A-L model and conserved quantities of NLSE, but the energy-preserving scheme can only exactly preserve the energy.

An efficient energy-preserving algorithm for the Lorentz force system

In this paper, by utilizing the invariant energy quadratization method to transform the original Hamiltonian energy functional into a quadratic form, we propose a novel energy-preserving scheme to solve the Lorentz force system. The method is a linear-implicit scheme for canonical Hamiltonian system, and can efficiently simulate the motion of charged particles in constant magnetic field. With comparison to the well-used Boris method and a similar energy-preserving BDLI method [26], numerical experiments are presented to demonstrate the energy-preserving property and computational efficiency of the method.

A linear, symmetric and energy-conservative scheme for the space-fractional Klein–Gordon–Schrödinger equations

In this paper, we propose an efficient numerical scheme for the space-fractional Klein–Gordon–Schrödinger (SFKGS) equations. Motivated by the “Invariant Energy Quadratization” (IEQ) approach, we introduce two auxiliary variables to transform the SFKGS system into a new equivalent system in which the time derivative is discretized by the Crank–Nicolson method, and the space discretization is based on the Fourier spectral method. Consequently, the numerical scheme shares two good features. The first feature is that the nonlinear terms are treated semi-explicitly and a linear symmetric system is solved at each time step. The second feature is the energy conservation at the discrete level. These two advantages are proved by the theoretical analysis and illustrated by a given numerical example.

Energy-conserving Galerkin approximations for quasigeostrophic dynamics

A method is presented for constructing energy-conserving Galerkin approximations in the vertical coordinate of the full quasigeostrophic model with active surface buoyancy. The derivation generalizes the approach of Rocha et al. (2016) [1] to allow for general bases. Details are then presented for a specific set of bases: Legendre polynomials for potential vorticity and a recombined Legendre basis from Shen (1994) [2] for the streamfunction. The method is tested in the context of linear baroclinic instability calculations, where it is compared to the standard second-order finite-difference method and to a Chebyshev collocation method. The Galerkin scheme is quite accurate even for a small number of degrees of freedom N, and growth rates converge much more quickly with increasing N for the Galerkin scheme than for the finite-difference scheme. The Galerkin scheme is at least as accurate as finite differences and can in some cases achieve the same accuracy as the finite difference scheme with ten times fewer degrees of freedom. The energy-conserving Galerkin scheme is of comparable accuracy to the Chebyshev collocation scheme in most linear stability calculations, but not in the Eady problem where the Chebyshev scheme is significantly more accurate. Finally the three methods are compared in the context of a simplified version of the nonlinear equations: the two-surface model with zero potential vorticity. The Chebyshev scheme is the most accurate, followed by the Galerkin scheme and then the finite difference scheme. All three methods conserve energy with similar accuracy, despite not having any a priori guarantee of energy conservation for the Chebyshev scheme. Further nonlinear tests with non-zero potential vorticity to assess the merits of the methods will be performed in a future work.

A vibro-impact acoustic black hole for passive damping of flexural beam vibrations

Nonlinear flexural vibrations of slender beams holding both an Acoustic Black Hole termination and a contact non-linearity are numerically studied. The Acoustic Black Hole (ABH) effect is a passive vibration mitigation technique, which has shown attractive properties above a given cut-on frequency. In this contribution, a vibro-impact acoustic black hole (VI-ABH) is introduced, the contact nonlinearity being used as a mean to transfer energy from low to high frequencies. A numerical model of a VI-ABH is derived from a Euler-Bernoulli beam. The contact law is handled with a penalization approach, the visco-elastic layer with a Ross-Kerwin-Ungard model and the problem is solved with a modal approach combined with an energy-conserving time integration scheme. Numerical results show that the VI-ABH brings about important modifications, and changes the nature of more traditional black holes, by redistributing all the vibrational energy. It can lead to a strong decrease of the resonance magnitude at low frequencies. Under steady state noise excitation, parametric studies are realised in the cases of a single contact, a grid of contacts and bilateral contacts layouts, in order to find some optimal designs. Transient dynamics is also studied through the analysis of displacement signal envelope and energy decay time. All the numerical results constantly show that the combination of the ABH effect and an energy transfer provided by contact nonlinearity leads to very attractive mitigation template including low frequencies.

An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations

In this paper, a new explicit fourth-order scheme for solving Riesz space fractional nonlinear wave equations is developed. The scheme is designed by using a novel Riesz space fractional difference operator for spatial discretization and a multidimensional extended Runge–Kutta–Nyström method for time integration. The conservation law of the semi-discrete energy, stability and convergence of the semi-discrete system are investigated. Numerical experiments show the efficiency and energy conservation of the present scheme.

The energy conservative splitting FDTD scheme and its energy identities for metamaterial electromagnetic Lorentz system

In this paper, we develop a new energy conservative splitting FDTD scheme for solving the metamaterial electromagnetic Lorentz system. The electromagnetic Lorentz model in metamaterials is to describe the resonance of nuclei-bounded electrons in dielectrics by the damped oscillation with a restoring force. Investigating the energy properties for metamaterial electromagnetic Lorentz system is important. The new energy identities of the metamaterial electromagnetic Lorentz system are obtained, which illustrate the conservations of the global energy in Lorentz medium. An efficient energy conservative splitting FDTD scheme is proposed to solve the Lorentz dispersive system in metamaterials. The advantages of the developed scheme lie in energy conservation, unconditional stability, easy implementation, second-order convergence both in time and space as well as superconvergence, which are demonstrated rigorously in the paper. Numerical experiments show the performance of the scheme for modeling the Lorentz models.

Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM, 42(3):934–952, 2004).

Integration of Numerical Simulation and Control Scheme for Energy Conservation of Aluminum Melting Furnaces

Aluminum melting process is a semi-continuous process with high-energy consumption. In this paper, a software-based strategy which considers numerical simulation and control scheme simultaneously is employed to improve energy consumption of aluminum melting furnaces without changing the hardware. For numerical simulation, a nonlinear steady-state optimization is performed offline to obtain optimal operating points. Extensively, the optimal operating conditions include not only product exit temperature, but also ratio of combustion air flow and natural gas flow, furnace temperature, flue gas temperature and some important manipulated variables. For control scheme, a two-layer model predictive control which consists of steady state target calculation and dynamic optimization is developed to track the optimal operating conditions. In steady state target calculation layer, a priority strategy is proposed based on the different importance of controlled variables to make the steady state targets more reasonably. In dynamic optimization layer, a quadratic objective function is defined in terms of tracking both the optimal steady-state of controlled variables and manipulated variables. The method is successfully carried out in F1 aluminum melting furnace of a company in Tianjin. Compared with previous operation, the comprehensive energy consumption and the comprehensive energy consumption for unit output of product decrease 5.99% and 6.56% respectively.

A New Energy-Preserving Scheme for the Fractional Klein-Gordon-Schrödinger Equations

A New Energy-Preserving Scheme for the Fractional Klein-Gordon-Schrödinger Equations

Two Kinds of New Energy-Preserving Schemes for the Coupled Nonlinear Schrödinger Equations

Two Kinds of New Energy-Preserving Schemes for the Coupled Nonlinear Schrödinger Equations