Invariant Energy Research Articles

An efficient linearly implicit and energy‐conservative scheme for two dimensional Klein–Gordon–Schrödinger equations

AbstractThe Klein–Gordon–Schrödinger equations describe a classical model of interaction of nucleon field with meson field in physics, how to design the energy conservative and stable schemes is an important issue. This paper aims to develop a linearized energy‐preserve, unconditionally stable and efficient scheme for Klein–Gordon–Schrödinger equations. Some auxiliary variables are utilized to circumvent the imaginary functions of Klein–Gordon–Schrödinger equations, and transform the original system into its real formulation. Based on the invariant energy quadratization approach, an equivalent system is deduced by introducing a Lagrange multiplier. Then the efficient and unconditionally stable scheme is designed to discretize the deduced equivalent system. A numerical analysis of the proposed scheme is presented to illustrate its uniquely solvability and convergence. Numerical examples are provided to validate accuracy, energy and mass conservation laws, and stability of our proposed method.

Reformulated Weak Formulation and Efficient Fully Discrete Finite Element Method for a Two-Phase Ferrohydrodynamics Shliomis Model

The two-phase ferrohydrodynamics model consisting of Cahn–Hilliard equations, Navier–Stokes equations, magnetization equations, and magnetostatic equations is a highly nonlinear, coupled, and saddle point structural multiphysics PDE system. While various works exist to develop fully decoupled, linear, second-order in time, and unconditionally energy stable methods for simpler gradient flow models, existing ideas may not be applicable to this complex model or may be only applicable to part of this model. Therefore, significant challenges remain in developing corresponding efficient fully discrete numerical algorithms with the four above-mentioned desired properties, which will be addressed in this paper by dynamically incorporating several key ideas, including a reformulated weak formulation with special test functions for overcoming two major difficulties caused by the magnetostatic equation, the decoupling technique based on the “zero-energy-contribution” property to handle the coupled nonlinear terms, the second-order projection method for the Navier–Stokes equations, and the invariant energy quadratization (IEQ) method for the time marching. Among all these ideas, the reformulated weak formulation serves as a key bridge between the existing techniques and the challenges of the target model, with all of the four desired properties kept in mind. We demonstrate the well-posedness of the proposed scheme and rigorously show that the scheme is unconditionally energy stable. Extensive numerical simulations, including accuracy/stability tests, and several 2D/3D benchmark Rosensweig instability problems for “spiking” phenomena of ferrofluids are performed to verify the effectiveness of the scheme.

Energy natural orbital characterization of nonadiabatic electron wavepackets in the densely quasi-degenerate electronic state manifold.

Dynamics and energetic structure of largely fluctuating nonadiabatic electron wavepackets are studied in terms of Energy Natural Orbitals (ENOs) [K. Takatsuka and Y. Arasaki, J. Chem. Phys. 154, 094103 (2021)]. Such huge fluctuating states are sampled from the highly excited states of clusters of 12 boron atoms (B12), which have densely quasi-degenerate electronic excited-state manifold, each adiabatic state of which gets promptly mixed with other states through the frequent and enduring nonadiabatic interactions within the manifold. Yet, the wavepacket states are expected to be of very long lifetimes. This excited-state electronic wavepacket dynamics is extremely interesting but very hard to analyze since they are usually represented in large time-dependent configuration interaction wavefunctions and/or in some other complicated forms. We have found that ENO gives an invariant energy orbital picture to characterize not only the static highly correlated electronic wavefunctions but also those time-dependent electronic wavefunctions. Hence, we first demonstrate how the ENO representation works for some general cases, choosing proton transfer in water dimer and electron-deficient multicenter chemical bonding in diborane in the ground state. We then penetrate with ENO deep into the analysis of the essential nature of nonadiabatic electron wavepacket dynamics in the excited states and show the mechanism of the coexistence of huge electronic fluctuation and rather strong chemical bonds under very random electron flows within the molecule. To quantify the intra-molecular energy flow associated with the huge electronic-state fluctuation, we define and numerically demonstrate what we call the electronic energy flux.

Energy stable schemes with second order temporal accuracy and decoupled structure for diffuse interface model of two-phase magnetohydrodynamics

In this paper, two second-order accurate in time, linear, decoupled and unconditionally energy stable schemes for the diffuse interface model of two-phase magnetohydrodynamics (MHD) are presented. The given schemes combine invariant energy quadratization (IEQ) method for the phase-field equations, second order type pressure projection method for the double saddle points MHD system, and some subtle implicit–explicit treatments for nonlinear coupled terms. The strongly coupled, highly nonlinear and double saddle points type two-phase MHD model is spilt into several smaller elliptic type problems, which can be solved efficiently. Rigorous proofs of the unconditional energy stabilities are provided in both temporal discretization and full discretization with finite element method in space approximation. Finally, various numerical results are presented to verify the accuracy and stability of the proposed schemes.

A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

<abstract><p>In this paper, we present a new method to solve the fractional nonlinear Schrödinger 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.</p></abstract>

Formulas for light deflection by a central mass, solar system relativity tests, and modification of the lens equation, for a Weyl scaling invariant dark energy

We estimate the changes in solar system tests of general relativity when "dark energy" arises from a Weyl scaling invariant action, using the modified Schwarzschild-like spherically symmetric solution obtained in this case by Adler and Ramazano\vglu. We show that the standard results of general relativity are modified by amounts that are far below current experimental errors. However, because the metric is not conformally flat when the central mass $M$ vanishes, there is a contribution to light deflection proportional to the cosmological constant $\Lambda$, with opposite sign to the one proportional to $M$. This could have implications for gravitational lensing in cosmological contexts, so we have calculated the corresponding correction to the "lens equation" for gravitational lensing. We also give formulas for the light deflection angle through order $\Lambda M$, assuming a parameterized spherically symmetric metric for the external region of the central mass.

Conservative finite difference methods for the Boussinesq paradigm equation

In this paper, we concentrate on developing and analyzing two types of finite difference schemes with energy conservation properties for solving the Boussinesq paradigm equation. Firstly, by introducing the auxiliary potential function ∂u∂t=Δv and lim|x|→∞v=0, the BPE is reformulated as an equivalent system of coupled equations. Then a class of efficient difference schemes are proposed for solving the resulting system, where one proposed scheme is a two-level nonlinear difference scheme and the other is a three-level linearized difference scheme using the invariant energy quadratization technique. Subsequently, we present the theoretical analysis of the proposed energy conservative finite difference schemes, which involves the discrete energy conservation properties, unique solvability and optimal error estimates. By using the discrete energy method, it is proven that the proposed schemes can achieve the optimal convergence rates of O(Δt2+hx2+hy2) in discrete L2-, H1- and L∞-norms. At last, numerical experiments illustrate the physical behaviors and efficiency of the proposed schemes.

On the convergence of an unconditionally stable numerical scheme for the Q-tensor flow based on the invariant quadratization method

We present a convergence analysis towards a numerical scheme designed for Q-tensor flows of nematic liquid crystals. This scheme is based on the Invariant Energy Quadratization method, which introduces an auxiliary variable to replace the original energy functional. In this work, we show that given an initial value with H2 regularity, we can obtain a uniform H2 estimate on the numerical approximations for the Q-tensor flows and then deduce the convergence to a strong solution of the parabolic-type Q-tensor equation. We also show that the limit of the auxiliary variable is equivalent to the original energy functional term in the strong sense.

Efficiently energy-dissipation-preserving ADI methods for solving two-dimensional nonlinear Allen-Cahn equation

Over the years, energy-dissipation-preserving (EDP) numerical methods (EDP-NMs) for the nonlinear system of energy dissipation have attracted considerable attention because they are good at long-term computations. More recently, the invariant energy quadratization methods (IEQMs) have been proposed by Xiaofeng Yang and his collaborators to construct linearized EDP-NMs for the nonlinear system of energy dissipation, such as gradient flows. However, the system of the linear algebraic equations corresponding to the obtained EDP-NMs involves the variable coefficient matrices in general, thus resulting in complex computations. Besides, most of efficient alternating direction implicit (ADI) methods are not able to preserve the structures of continuous problem, such as, energy dissipation, energy conservation. To overcome these shortcomings, taking two-dimensional (2D) nonlinear Allen-Cahn equation for example, two energy-dissipation-preserving ADI finite difference methods (FDMs) have been developed by using IEQMs. Numerical analyses of them, such as, the discrete energy-dissipation laws, error estimations and stabilities, have been derived in detail. Numerical results support the theoretical analysis, and confirm the efficiency and accuracy of the proposed algorithms.

Dynamical gravastars

We combine the ideas of a Weyl scaling invariant dark energy action, which eliminates black hole horizons, with the ``gravastar'' idea of a jump in the hole interior from a normal matter equation of state to an equation of state where pressure plus density approximately sum to zero. Using the Tolman-Oppenheimer-Volkoff equation, which requires continuous pressure, we present Mathematica notebooks in which the structure of the gravastar is entirely governed by the action and the equation of state, with the radii where structural changes occur emerging from the dynamics, rather than being specified in advance. The notebooks work even with zero cosmological constant, but when the cosmological constant is nonzero, there is a very small black hole ``wind'' that we calculate by a relativistic extension of standard pressure driven isothermal stellar wind theory.

A second-order unconditionally stable method for the anisotropic dendritic crystal growth model with an orientation-field

In this article, we develop a linear, unconditionally energy stable computational scheme for solving the dendritic crystal growth model with the orientational field. We apply the phase field model to describe the evolution of crystal with rotation. The model, which couples the heat equation and anisotropic Allen–Cahn type equation, is a complicated nonlinear system. The time integration is based on the second-order Crank–Nicolson method. The anisotropic coefficient is treated by using the invariant energy quadratization. We mathematically prove that the proposed method is unconditionally energy stable. The second-order spatial and temporal accuracy will be preserved for the numerical approximation. Various computational tests are performed to show the accuracy, stability, and efficiency of the proposed scheme.

Explicit exactly energy-conserving methods for Hamiltonian systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.

Arbitrarily high-order energy stable s-stage RK-IEQ scheme for the nonlocal Cahn–Hilliard equation

In this paper, for solving the nonlocal-type Cahn–Hilliard equation, we propose an arbitrary high-order energy stable s-stage RK-IEQ scheme, which is based on the combination of the Invariant Energy Quantization (IEQ) approach and the s-stage Runge–Kutta method. We prove that the proposed scheme is unconditionally energy stable and carry out several numerical examples in both 2D and 3D to demonstrate the accuracy and stability of the scheme.

Triangle singularity in the production of Tcc+(3875) and a soft pion

The double-charm tetraquark meson $T_{cc}^+(3875)$ can be produced in high-energy proton-proton collisions by the creation of the charm mesons $D^{*+} D^0$ at short distances followed by their binding into $T_{cc}^+$. The $T_{cc}^+$ can also be produced by the creation of $D^{*+} D^{*+}$ at short distances followed by their rescattering into $T_{cc}^+ \pi^+$. A charm-meson triangle singularity produces a narrow peak in the $T_{cc}^+ \pi^+$ invariant mass distribution 6.1 MeV above the threshold with a width of about 1 MeV. Well beyond the peak, the differential cross section decreases with the invariant kinetic energy $E$ of $T_{cc}^+ \pi^+$ as $E^{-1/2}$. The fraction of $T_{cc}^+$ that are accompanied by $\pi^+$ with $E< m_\pi$ is estimated to be roughly 3%. The fraction of $T_{cc}^+$ events with $T_{cc}^+ \pi^+$ in the narrow peak from the triangle singularity could be comparable.

Fully discrete Spectral-Galerkin scheme for a ternary Allen–Cahn type mass-conserved Nakazawa–Ohta phase-field model for triblock copolymers

In this paper, we first use the Allen–Cahn relaxation dynamics to simulate the phase-field model of triblock copolymers, where the mass-conserving property is achieved by adding two additional nonlocal Lagrange multipliers. The numerical approximations of the obtained model are further considered, and an efficient fully-discrete numerical scheme based on the Spectral-Galerkin approach for spatial discretization and the so-called explicit-IEQ (invariant energy quadratization) method for time marching is developed. Two auxiliary variables are introduced to reformulate the governing system into an equivalent form, which facilitates the design of numerical algorithms by simply discretizing nonlinear terms explicitly. The resulting scheme is not only second-order accurate in time and spectrally accurate in space, but also easy to implement, i.e., the scheme can be carried out by only solving multiple independently decoupled, linear, and constant-coefficient elliptic equations at each time step. We also prove the unconditional energy stability of the developed scheme and demonstrate its effectiveness by implementing several benchmarking numerical examples in 2D and 3D.

Efficient linear, fully-decoupled and energy stable numerical scheme for a variable density and viscosity, volume-conserved, hydrodynamically coupled phase-field elastic bending energy model of lipid vesicles

We first establish a variable density and viscosity, volume-conserved, hydrodynamically coupled phase-field variable elastic bending energy model for lipid vesicles, and then construct an efficient time-discrete scheme for solving it. The numerical scheme combines the penalty method for solving the Navier–Stokes equation, the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, and the operator-splitting method. Hence it is not only fully decoupled but also owns some desired properties of linearity and unconditional energy stability. The feature of full decoupling is achieved by introducing some auxiliary variables and designing additional ordinary differential equations which are used for discretizing the coupled and nonlinear terms. The solvability and the unconditional energy stability of the numerical scheme have been further rigorously and numerically proven. Several numerical examples are carried out on the sedimentation process of the vesicle cell to show the effectiveness of the model and scheme.

Cosmological constant corrections to the photon sphere and black hole shadow radii

We review the equations determining the photon sphere radius and the black hole shadow radius, and calculate the cosmological constant corrections arising when the dark energy action has the usual form, and when dark energy arises from a Weyl scaling invariant dark energy action. For black hole targets of the Event Horizon Telescope, the corrections are very small.

Numerical approximations of flow coupled binary phase field crystal system: Fully discrete finite element scheme with second-order temporal accuracy and decoupling structure

In this article, we first establish a new flow-coupled binary phase-field crystal model and prove its energy law. Then by using some newly introduced variables, we reformulate this three-phase model into an equivalent form, which makes it possible to construct a fully discrete linearized decoupling scheme with unconditional energy stability and second-order time accuracy to solve this model for the first time. The energy law of the reformulated model is also proved. Then we incorporate the explicit-IEQ (invariant energy quadratization) method for the nonlinear potentials, the projection method for the Navier-Stokes equations, the Crank-Nicolson method for time marching, and the finite element method for spatial discretization together to develop the fully discrete scheme for the reformulated and equivalent system. By using the nonlocal splitting technique, at each time step, only a few decoupled constant-coefficient elliptic equations are required to be solved, even though the original and reformulated models are much more complicated in the form. The developed algorithm is further proved to be unconditionally energy stable, and a detailed implementation process is also provided. Various numerical experiments in 2D and 3D are carried out to verify the effectiveness of the developed scheme, including the binary crystal growth under the action of shear flow and the sedimentation process of many binary particles.

Bounds on some energy-like invariants of corona and edge corona of graphs

The Laplacian-energy-like invariant of a finite simple graph is the sum of square roots of all its Laplacian eigenvalues and the incidence energy is the sum of square roots of all its signless Laplacian eigenvalues. In this paper, we give the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona of two graphs. We also observe that the bounds on the Laplacian-energy-like invariant and incidence energy of the corona and edge corona are sharp when the graph is the corona or edge corona of two complete graphs.

Are astrophysical “black” holes leaky?

We continue a study by Adler and Ramazanoǧlu (AR) of “black” holes as modified by a scale invariant dark energy action. For the spherically symmetric Schwarzschild-like case, AR found that there is no event horizon; hence spacetime is not divided by the “black” hole into causally disconnected regions. We review the formalism for locating trapped surfaces and apparent horizons, and show that the modified “black” hole has no trapped surfaces. Thus, one suspects that it is “leaky,” and that there will be a “black hole wind” of particles streaming out from the location of the nominal horizon. This will have astrophysical consequences, for example, the wind may feed and stabilize star formation in the vicinity of the “black” hole. We initiate a study of the stationary axially-symmetric rotating “black” hole as modified by a scale invariant dark energy action, i.e. one that is Kerr-like. To set-up the axial case, we note that the conserving completion of the dark energy stress-energy tensor can be calculated algebraically by solving a [Formula: see text] matrix equation. Using Mathematica we calculate and simplify the modified system of Einstein equations for the axial case in quasi-isotropic coordinates, and give appropriate boundary conditions. Solution of these equations, which may require developing a special purpose numerical program, is rendered tricky by a residual general coordinate invariance of quasi-isotropic coordinates. This leads to the interesting mathematical question of representing a positive planar function as the gradient squared of a harmonic function.