Energy-conserving Scheme Research Articles

Explicit high-order conservative exponential time differencing Runge-Kutta schemes for the two-dimensional nonlinear Schrödinger equation

In this paper, we develop a class of explicit energy-preserving Runge-Kutta schemes for solving the nonlinear Schrödinger equation based on the projection technique and the exponential time differencing method. First, we reformulate the equation to an equivalent system that possesses new quadratic energy via introducing an auxiliary variable. Then, we construct a family of fully discrete exponential time differencing schemes which have better stability by using the Runge-Kutta method and the Fourier-pseudo spectral method to approximate the system in time and space, respectively. Subsequently, energy-preserving schemes are derived by combining the proposed explicit schemes and the projection technique, and the stability result is given. Finally, extensive numerical examples are presented to confirm the constructed schemes have high accuracy, energy-preserving and effectiveness in long time simulation.

A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

Eliminating finite-grid instabilities in gyrokinetic particle-in-cell simulations

Finite-grid (aliasing) instabilities are known to place severe limitations on momentum-conserving particle-in-cell (PIC) methods applied to models including charge separation effects by requiring the resolution of the Debye length. Gyrokinetic models, on the other hand, generally enforce quasi-neutrality thereby removing the Debye length analytically. Recent studies with momentum-conserving PIC applied to gyrokinetic models, however, show that a manifestation of this instability exists in certain physical parameter regimes for arbitrary spatial resolution. In this paper, we show that a simple reformulation of the discrete equations, making use of a co-located discretization of the continuity equation, eliminates this instability for all practical purposes. We perform numerical dispersion analyses for both the original and reformulated schemes, including the effects of finite mean parallel velocity and finite β. We demonstrate that the reformulated scheme is numerically stable for stationary plasmas at any spatial resolution and for plasmas in which the electron mean parallel velocity is smaller than the electron thermal velocity (generally ensured by ambipolarity in plasmas of interest). This reformulation may be particularly useful for codes with complicated meshes, where high-order shape functions or energy-conserving schemes are difficult to implement. The analysis presented here may also be useful for explaining the success of previously considered methods to remove particle instabilities, such as the split-weight scheme.

Conservative compact difference scheme for the generalized Korteweg-de Vries equation

In this paper, a linear mass and energy conservative finite difference scheme for the generalized Korteweg-de Vries (GKdV) equation is proposed and analyzed. The scheme is three-level and linear implicit and gives second- and fourth-order accuracy in time and space, respectively. It is rigorously proved by using the discrete energy method that the proposed difference scheme is uniquely solvable, stable and convergent. Numerical examples are given to confirm the stability and convergence of the numerical solution with fourth-order accuracy and the effectiveness of the present scheme for handling the single and multi-solitary waves for a long time.

Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equation

Linearly implicit and second-order energy-preserving schemes for the modified Korteweg-de Vries equation

Energy conserving discontinuous Galerkin method with scalar auxiliary variable technique for the nonlinear Dirac equation

In this paper, we propose a fully-discrete energy-conserving scheme for the nonlinear Dirac equation, by combining the scalar auxiliary variable (SAV) technique with discontinuous Galerkin (DG) discretization. We start by discussing the semi-discrete DG discretization, and show that, with suitable choices of numerical fluxes, the resulting method conserves the charge, energy exactly and preserves the multi-symplectic structure. The optimal error estimate of semi-discrete DG scheme is carried out. We combine it with the energy conserving SAV technique, and demonstrate that the fully-discrete scheme conserves the discrete global energy exactly. Both second order SAV method based on the midpoint rule and its high order extension have been studied. The proposed methods have been tested on some numerical experiments, which confirm the optimal rates of convergence and the energy conserving property. Numerical comparison with energy dissipative DG method is also provided to demonstrate that the numerical error of energy conserving method does not grow significantly in long time simulations.

Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation

In this paper, a family of high-order linearly implicit exponential integrators conservative schemes is constructed for solving the multi-dimensional nonlinear fractional Schrödinger equation. By virtue of the Lawson transformation and the generalized scalar auxiliary variable approach, the equation is first reformulated to an exponential equivalent system with a modified energy. Then, we construct a semi-discrete conservative scheme by using the Fourier pseudo-spectral method to discretize the exponential system in space direction. After that, linearly implicit energy-preserving schemes which have high accuracy are given by applying the Runge–Kutta method to approximate the semi-discrete system in temporal direction and using the extrapolation method to the nonlinear term. As expected, the constructed schemes can preserve the energy exactly and implement efficiently with a large time step. Numerical examples confirm the constructed schemes have high accuracy, energy-preserving, and effectiveness in long-time simulation.

Unconditional stability in large deformation dynamic analysis of elastic structures with arbitrary nonlinear strain measure and multi-body coupling

Stability is a fundamental feature of a time integration method in long simulations of elastic bodies in large deformations. It is well known that energy conservation is the key to achieve it. The implicit one-step conserving method of Simo and Tarnow is simple and effective for quadratic strain measures. However, the issue of unconditional stability does not yet have a definitive solution for general nonlinear strain measures and multi-body couplings. This article shows how to reach this goal by a simple modification of the Simo and Tarnow method. In practice, the mean internal force vector of the time step is evaluated using the average value of the stress at the end-points and an integral mean of the strain–displacement tangent operator over the step computed by time integration points. Compared to other proposals, the approach conserves energy for any structural model regardless of its spatial finite element discretization and does not require Lagrange multipliers, discrete derivatives, approximations of the strain–displacement law or special iterative solutions. Long term stability is proved using dynamic analyses of Reissner beams and Kirchhoff–Love shells, discretized spatially with Lagrangian and isogeometric elements respectively. Moreover, the proposed energy-conserving scheme is extended to multi-body analyses with nonlinear couplings by means of a penalty formulation, with focus on the rotational continuity for generic multi-patch Kirchhoff–Love shells.

Energy-conserving and time-stepping-varying ESAV-Hermite-Galerkin spectral scheme for nonlocal Klein-Gordon-Schrödinger system with fractional Laplacian in unbounded domains

The aim of this paper is to construct a linearized and energy-conserving numerical scheme for nonlocal-in-space Klein-Gordon-Schrödinger system in multi-dimensional unbounded domains Rd (d=1,2, and 3), where the nonlocal property of the system is described by the fractional Laplacian. Firstly, we derive the nonlocal energy conversation law of the system. Then, the Hermite-Galerkin spectral method with scaling factor is employed for the spatial approximation. In the form of the exponential scalar auxiliary variable (ESAV) approach, the Crank-Nicolson scheme with adaptive time-stepping is used for the temporal discretization. To adjust the time-stepping, we propose two kinds of time adaptive strategies depending on the time evolution of the considered system. To avoid solving nonlinear algebra system, the nonlinear terms are explicitly treated by the extrapolation technique. The main advantages of proposed numerical scheme are in three aspects: The first one is that the original nonlocal problem is directly solved in the unbounded domains to avoid the errors and singularities introduced by the domain truncation. The second one is that a linearly explicit and energy-conserving scheme is established with constant coefficients. The third one is the high efficiency without sacrificing accuracy when used in conjunction with the adaptive time-stepping strategies. Numerical experiments are given to demonstrate the accuracy, efficiency, and robustness of the proposed algorithm. As the applications of the scheme, we present numerical simulations of the interactions of 2D/3D vector solitons, which can provide a deeper understanding of nonlinear nonlocal processes in the system.

Arbitrary high-order linear structure-preserving schemes for the regularized long-wave equation

In this paper, a class of arbitrarily high-order linear momentum-preserving and energy-preserving schemes are proposed, respectively, for solving the regularized long-wave equation. For the momentum-preserving scheme, the key idea is based on the extrapolation/prediction-correction technique and the symplectic Runge-Kutta method in time, together with the standard Fourier pseudo-spectral method in space. We show that the scheme is linear, high-order, unconditionally stable and preserves the discrete momentum of the system. For the energy-preserving scheme, it is mainly based on the energy quadratization approach and the analogous linearized strategy used in the construction of the linear momentum-preserving scheme. The proposed scheme is linear, high-order and can preserve both the discrete mass and the discrete quadratic energy exactly. Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.

Arbitrary high-order linearly implicit energy-preserving algorithms for Hamiltonian PDEs

In this paper, we present a novel strategy to systematically construct linearly implicit energy-preserving schemes with arbitrary order of accuracy for Hamiltonian PDEs. Such a novel strategy is based on the newly developed exponential scalar auxiliary variable (ESAV) approach that can remove the bounded-from-below restriction of nonlinear terms in the Hamiltonian functional and provides a totally explicit discretization of the auxiliary variable without computing extra inner products. So it is more effective and applicable than the traditional scalar auxiliary variable (SAV) approach. To achieve arbitrary high-order accuracy and energy preservation, we utilize symplectic Runge-Kutta methods for both solution variables and the auxiliary variable, where the values of the internal stages in nonlinear terms are explicitly derived via an extrapolation from numerical solutions already obtained in the preceding calculation. A prediction-correction strategy is proposed to further improve the accuracy. Fourier pseudo-spectral method is then employed to obtain fully discrete schemes. Compared with the SAV schemes, the solution variables and the auxiliary variable in these ESAV schemes are now decoupled. Moreover, when the linear terms have constant coefficients, the solution variables can be explicitly solved by using the fast Fourier transform. Numerical experiments are carried out for three Hamiltonian PDEs to demonstrate the efficiency and conservation of the ESAV schemes.

Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation

<p style='text-indent:20px;'>A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.</p>

Linearly Implicit and High-Order Energy-Preserving Schemes for Highly Oscillatory Hamiltonian Systems

Linearly Implicit and High-Order Energy-Preserving Schemes for Highly Oscillatory Hamiltonian Systems

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

A mass and energy conservative fourth-order compact difference scheme for the Klein-Gordon-Dirac equations$$^{\star }$$

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

The well-known energy dissipation anomaly in the inviscid limit, related to velocity singularities according to Onsager, still needs to be demonstrated by numerical experiments. The present work contributes to this topic through high-resolution numerical simulations of the inviscid three-dimensional Taylor–Green vortex problem using a novel high-order discontinuous Galerkin discretisation approach for the incompressible Euler equations. The main methodological ingredient is the use of a discretisation scheme with inbuilt dissipation mechanisms, as opposed to discretely energy-conserving schemes, which – by construction – rule out the occurrence of anomalous dissipation. We investigate effective spatial resolution up to $8192^3$ (defined based on the $2{\rm \pi}$ -periodic box) and make the interesting phenomenological observation that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but that the sequence of discrete solutions seemingly converges to a solution with non-zero kinetic energy dissipation rate. Taking the fine-resolution simulation as a reference, we measure grid-convergence with a relative $L^2$ -error of $0.27\,\%$ for the temporal evolution of the kinetic energy and $3.52\,\%$ for the kinetic energy dissipation rate against the dissipative fine-resolution simulation. The present work raises the question of whether such results can be seen as a numerical confirmation of the famous energy dissipation anomaly. Due to the relation between anomalous energy dissipation and the occurrence of singularities for the incompressible Euler equations according to Onsager's conjecture, we elaborate on an indirect approach for the identification of finite-time singularities that relies on energy arguments.

Exponential integrator preserving mass boundedness and energy conservation for nonlinear Schrödinger equation

In this paper, a novel linearly implicit conservative scheme is developed for the two-dimensional nonlinear Schrödinger equation. The scheme is based on a new linearly implicit exponential time differencing method for the temporal discretization and the Fourier pseudo-spectral approximation for the spatial derivative. By rigorous analysis, we prove that the proposed scheme can preserve the energy conservation. Furthermore, although in fact the proposed scheme does not preserve mass conservation, we can prove that it can preserve the discrete L2 norm boundedness of the numerical solution, which may be helpful for the analysis of unconditional convergence of the energy conservative scheme. Then an optimal error estimate of the proposed scheme is established without any restriction on the grid ration. Numerical experiments show that the proposed scheme is remarkable efficiency comparing with some other existing structure-preserving schemes.

The exponential invariant energy quadratization approach for general multi-symplectic Hamiltonian PDEs

Many conservative systems in physical sciences can be described by multi-symplectic Hamiltonian PDEs (MS-HPDEs) admitting three important inherent properties named as multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we develop a novel strategy to systematically derive linearly implicit local energy-preserving schemes for general MS-HPDEs, which is named the exponential invariant energy quadratization approach (EIEQ). Such novel strategy is based on the exponential form of non-quadratic terms of the state function that can remove the bounded-from-below restriction, so it is more applicable than the traditional IEQ approach widely employed to construct linearly implicit schemes. Moreover, we provide a completely explicit discretization of the auxiliary variable combined with the nonlinear term, which obtains linearly implicit schemes of the constant coefficient, making their more effective than the IEQ schemes for rapid simulations. We also present the multiple EIEQ approach to improve the applicability of the EIEQ approach for MS-HPDEs with more non-quadratic terms. In addition, when periodic or homogeneous boundary conditions are considered, the proposed schemes are global energy-preserving and can be explicitly solved by using the fast Fourier transform. Finally, ample numerical tests are carried out to demonstrate the computational efficiency, conservation and accuracy of EIEQ schemes.

Construction of a Second-order Six-dimensional Hamiltonian-conserving Scheme

Abstract Research has analytically shown that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen, and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of this, a new second-order energy-conserving implicit scheme is proposed. Numerical simulations of a galactic model hosting a BL Lacertae object and magnetized rotating black hole background support these analytical results. The new method with appropriate time steps is used to explore the effects of varying the parameters on the presence of chaos in the two physical models. Chaos easily occurs in the galactic model as the mass of the nucleus, the internal perturbation parameter, and the anisotropy of the potential of the elliptical galaxy increase. The dynamics of charged particles around the magnetized Kerr spacetime is easily chaotic for larger energies of the particles, smaller initial angular momenta of the particles, and stronger magnetic fields. The chaotic properties are not necessarily weakened when the black-hole spin increases. The new method can be used for any six-dimensional Hamiltonian problems, including globally hyperbolic spacetimes with readily available (3 + 1) split coordinates.

Analysis of a Galerkin finite element method for the Maxwell–Schrödinger system under temporal gauge

Abstract This paper is concerned with the numerical solution of the Maxwell–Schrödinger system under the temporal gauge, which describes light–matter interactions. We first propose a semidiscrete finite element scheme for the system and establish stability estimates for the finite element solution. Due to the lack of control over its divergence we cannot get $\textbf{H}^{1}$$a \;priori$ estimates for the vector potential, making it difficult to obtain error estimates by usual techniques. We apply an exhaustion argument to overcome this difficulty and derive error estimates for the finite element approximation. An energy-conserving time-stepping scheme is proposed to solve the semidiscrete system.

High-order explicit conservative exponential integrator schemes for fractional Hamiltonian PDEs

The paper aims to develop a class of high-order explicit exponential time differencing energy-preserving schemes for some conservative fractional PDEs based on the general Hamiltonian form of these equations. The equation is first reformulated into an equivalent system that possesses a new quadratic energy by introducing an auxiliary variable. Then, explicit schemes are derived via applying the exponential time differencing Runge-Kutta method approximations for time integration and Fourier pseudo-spectral discretization in space, which can maintain the accuracy and improve the stability to the system with stiffness. Subsequently, highly efficient energy-preserving schemes are given by combining the proposed explicit schemes and the projection technique. Finally, the advantages of the proposed schemes are illustrated by the numerical results of the fractional nonlinear Schrödinger equation.