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In this paper, the nonlocal electron heat transport model in one and two dimensions is considered and studied. An energy stability finite element method is designed to discretize the nonlocal electron heat transport model. For the nonlinear discrete system, both Newton iteration and implicit-explicit (IMEX) schemes are employed to solve it. Then the energy stability is proved in semi-discrete and fully-discrete schemes. Numerical examples are presented to verify the energy stability of the proposed schemes as well as the optimal convergence order in L ∞ , L 2 and H 1 norm. [ABSTRACT FROM AUTHOR]

A novel in situ/operando method is introduced to measure the photon beam stability of synchrotron radiation based on orthogonal diffraction imaging of a Laue crystal/analyzer, which can decouple the energy/wavelength and Bragg angle of the photon beam using the dispersion effect in the diffraction process. The method was used to measure the energy jitter and drift of the photon beam on BL09B and BL16U at the Shanghai Synchrotron Radiation Facility. The experimental results show that this method can provide a fast way to measure the beam stability of different light sources including bending magnet and undulator with meV‐level energy resolution and ms‐level time response. [ABSTRACT FROM AUTHOR]

A novel in situ/operando method is introduced to measure the photon beam stability of synchrotron radiation based on orthogonal diffraction imaging of a Laue crystal/analyzer, which can decouple the energy/wavelength and Bragg angle of the photon beam using the dispersion effect in the diffraction process. The method was used to measure the energy jitter and drift of the photon beam on BL09B and BL16U at the Shanghai Synchrotron Radiation Facility. The experimental results show that this method can provide a fast way to measure the beam stability of different light sources including bending magnet and undulator with meV-level energy resolution and ms-level time response.

We analyze numerical approximations for axisymmetric two-phase flow in the arbitrary Lagrangian-Eulerian (ALE) framework. We consider a parametric formulation for the evolving fluid interface in terms of a one-dimensional generating curve. For the two-phase Navier-Stokes equations, we introduce both conservative and nonconservative ALE weak formulations in the 2d meridian half-plane. Piecewise linear parametric elements are employed for discretizing the moving interface, which is then coupled to a moving finite element approximation of the bulk equations. This leads to a variety of ALE methods, which enjoy either an equidistribution property or unconditional stability. Furthermore, we adapt these introduced methods with the help of suitable time-weighted discrete normals, so that the volume of the two phases is exactly preserved on the discrete level. Numerical results for rising bubbles and oscillating droplets are presented to show the efficiency and accuracy of these introduced methods. [ABSTRACT FROM AUTHOR]

Energy sustainability, especially in agriculture, has been considered due to population growth and the lack of fossil fuel resources for future generations. Extending renewable energies, including biogas, is essential for sustainable development. This qualitative research was designed and implemented to determine how to extend biogas in rural areas in Iran. The study used grounded theory to guide data collection and analysis and model presentation. The sample included experts from the Ministry of Agriculture and Renewable Energy Organization of Iran selected by purposive sampling. Data were collected by field observations and interviews and analyzed by the coding process (open, axial, and selective coding), as a result of which ten core categories were extracted. They were presented as intervening conditions, contextual conditions, and strategies and used to develop a paradigmatic model for promoting biogas use in Iran's rural areas. Finally, based on the results, suggestions are presented to accelerate the process.

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

In this paper, we consider the numerical approximations of the Cahn-Hilliard phase field model for two-phase incompressible flows with variable density. First, a temporal semi-discrete numerical scheme is proposed by combining the fractional step method (for the momentum equation) and the convex-splitting method (for the free energy). Second, we prove that the scheme is unconditionally stable in energy. Then, the $ L^2 $ convergence rates for all variables are demonstrated through a series of rigorous error estimations. Finally, by applying the finite element method for spatial discretization, some numerical simulations were performed to demonstrate the convergence rates and energy dissipations.

In this article, we derive error estimates for two second-order numerical schemes for solving the viscous Cahn-Hilliard equation with hyperbolic relaxation, one based on the second-order Crank-Nicolson time marching method and the other on the backward differentiation formula. In both schemes, the nonlinear potential is discretized by the Invariant Energy Quadratization (IEQ) method, which employs an auxiliary variable to produce a linear and unconditional energy-stable structure. After assuming some reasonable boundedness and continuity conditions for the nonlinear function, the optimal error estimate is rigorously derived using mathematical induction. Finally, several numerical experiments are carried out to verify the theoretical predictions of the convergence rate and energy stability of the algorithms. [ABSTRACT FROM AUTHOR]

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme. [ABSTRACT FROM AUTHOR]

This paper presents a sixth-order nonlinear parabolic problem of the square phase-field crystal model. We first demonstrate the time-discrete backward Euler scheme with mass conservation and energy stability. Then, we prove the unconditionally optimal error estimates for the time-discrete backward Euler scheme. In the end, we present 2D and 3D numerical simulations to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

A new third-order energy stable technique, which is a convex splitting scheme with the Douglas-Dupont regularization term A τ 2 (ϕ n − ϕ n − 1) , is proposed for solving the extended Fisher–Kolmogorov equation. The higher-order backward difference formula is used to deal with the time derivative term. The constructed numerical scheme is uniquely solvable and unconditionally preserves the modified discrete energy dissipative law. With the help of discrete orthogonal convolution kernels, the L 2 norm error estimate of the stabilized BDF3 scheme can be established by acting the standard inner product with the error system. Several numerical experiments are used to verify the validity of the numerical method and the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper proposes an Allen–Cahn Chan–Vese model to settle the multi-phase image segmentation. We first integrate the Allen–Cahn term and the Chan–Vese fitting energy term to establish an energy functional, whose minimum locates the segmentation contour. The subsequent minimization process can be attributed to variational calculation on fitting intensities and the solution approximation of several Allen–Cahn equations, wherein n Allen–Cahn equations are enough to partition m = 2 n segments. The derived Allen–Cahn equations are solved by efficient numerical solvers with exponential time integrations and finite difference space discretization. The discrete maximum bound principle and energy stability of the proposed numerical schemes are proved. Finally, the capability of our segmentation method is verified in various experiments for different types of images. [ABSTRACT FROM AUTHOR]

In this paper, we will develop an energy stable numerical method for computing the complex fluids in an L-shaped cavity. The complex fluids possess a number of interesting technological applications in various fields. However, numerical computations for complex fluids in an L-shaped cavity are least understood. In this paper, the Red blood cell is the main focus because of their importance and remarkable deformability presenting particular simulation challenges in the L-shaped cavity. In order to study such complex fluids possessing complicated nature and multi-scale properties, the partial differential equations have been used to formulate the mathematical model of the complex fluids. To solve the mathematical model, energy stable schemes have been first constructed. Next, the numerical results are calculated followed by discussion for different surface tensions. A nonlinear coupling system is discretized into a (1) decoupled (2) stable, and (3) linear multi-step numerical scheme by adding a stability term. Fourth-order phase field equation is transformed into two second-order elliptic problems by using some reasonable decouple techniques, which is easy to implement. Furthermore, the finite element method is used for the spatial discretization. Numerical results confirm the energy decay property and deformability of the Red blood cell in the L-shaped domain. [ABSTRACT FROM AUTHOR]

An efficient and easy-to-implement second-order Strang splitting approach is mainly applied to study the scalar Allen-Cahn (AC) equation in this paper. Base on the idea of dimensional splitting, a new time dependent function (called exponential integrating factor) is introduced for the scalar AC equation. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the explicit 2-stage strong stability preserving Runge-Kutta(SSP-RK2) method is employed for the nonlinear part. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence of the proposed scheme. Various numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the energy stability property of the correction procedure via reconstruction (CPR) method with staggered flux points based on second-order subcell limiting. The CPR method with staggered flux points uses the Gauss point as the solution point, dividing flux points based on Gauss weights, with the flux points being one more point than the solution points. For subcell limiting, a shock indicator is used to detect troubled cells where discontinuities may exist. Troubled cells are calculated by the second-order subcell compact nonuniform nonlinear weighted (CNNW2) scheme, which has the same solution points as the CPR method. The smooth cells are calculated by the CPR method. The linear energy stability of the linear CNNW2 scheme is proven theoretically. Through various numerical experiments, we demonstrate that the CNNW2 scheme and CPR method based on subcell linear CNNW2 limiting are energy-stable and that the CPR method based on subcell nonlinear CNNW2 limiting is nonlinearly stable. [ABSTRACT FROM AUTHOR]

The photoinjector laser system of Shenzhen Superconducting Soft X-Ray Free Electron Laser (S3FEL) is reported in this paper. This laser system operates at up to 1 MHz and produces more than 50 μJ infrared (IR) laser pulses. With a customized fourth harmonic generation (FHG) module, more than 2 μJ ultraviolet (UV) laser pulses were obtained. The power standard deviations of the IR laser and the UV laser are 0.093% and 0.395% respectively. While the pulse energy standard deviations are 1.087% and 1.746% correspondingly. We implemented the pulse stacking scheme to generate flat-top pulses. With four birefringent uniaxial crystals, the Gaussian pulses were converted to flat-top shape, featuring 10 ps pulse width and 0.5 ps rising and falling edges. A cut-Gaussian transverse profile with very sharp rising and falling edges can be produced after the spatial pulse shaper.

The energy security of each country is one of the main factors of its proper functioning. Currently, in the era of problems related to energy security resulting from, among other things, the war in Ukraine, this topic is particularly important. This article presents issues related to Poland's energy security, understood as the financial and economic stability of enterprises operating in the energy industry. This stability is considered in two aspects: macroeconomic, where the focus is mainly on the aspect of state intervention in market processes; and microeconomic, where factors determining the financial security of energy enterprises were identified, including internal and external factors affecting the functioning of these entities. In order to achieve the assumed research goals, the analysis of the indicated problems was based on non-reactive research, consisting in the assessment of the available information. It included studies of normative acts, official statistical data, industry reports and analyses, as well as data obtained in the form of a public information request. Two basic research methods were used in the work—dogmatic–legal and comparative analyses. The identification of factors affecting the security of companies in the sector was carried out on the basis of data on the entire energy sector in Poland for the years 2015–2021 on a semi-annual basis. Vector-autoregressive models were used for the analysis. As a result of the analyses, it was established that market failures and public safety are the premises justifying the public financing of enterprises in the electricity generation, transmission, distribution and trade sectors. At the same time, the conducted research showed that the level of financial security of energy enterprises in Poland was affected by the ratio of the value of goods and materials sold to net sales revenue, as well as the level of EBIT (earnings before deducting interest and taxes) margin, and among external factors, the level of GDP (gross domestic product), CPI (consumer price index) and Crude Oil were important. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new space-fractional modified phase field crystal equation which has some similar properties including the mass conservation and energy dissipation. Then, we propose a second-order scheme based on a new Lagrange multiplier method that conserves the mass and dissipates the energy. For the new method, there are only two decoupled linear equations with constant coefficients and one nonlinear algebraic system to be solved at each time step which makes it efficient. Finally, we give some numerical experiments to verify the accuracy and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete scheme is proposed to solve the nonlinear Schrödinger-Possion equations. The scheme is developed by the scalar auxiliary variable (SAV) approach, the Crank-Nicolson temproal discretization and the Galerkin-Legendre spectral spatial discretization. The fully discrete scheme is proved to be mass- and energy- conserved. Moreover, unconditional energy stability and convergence of the scheme are obtained by the use of the Gagliardo-Nirenberg and some Sobolev inequalities. Numerical results are presented to confirm our theoretical findings.

The Cahn–Hilliard–Navier–Stokes model is extensively used for simulating two-phase incompressible fluid flows. With the absence of exterior force, this model satisfies the energy dissipation law. The present work focuses on developing a linear, decoupled, and energy dissipation-preserving time-marching scheme for the hydrodynamics coupled Cahn–Hilliard model. An efficient time-dependent auxiliary variable approach is first introduced to design equivalent equations. Based on equivalent forms, a BDF2-type linear scheme is constructed. In each time step, the unique solvability and the energy dissipation law can be analytically estimated. To enhance the energy stability and the consistency, we correct the modified energy by a practical relaxation technique. Using the finite difference method in space, the fully discrete scheme is described, and the numerical solutions can be separately implemented. Numerical results indicate that the proposed scheme has desired accuracy, consistency, and energy stability. Moreover, the flow-coupled phase separation, the falling droplet, and the dripping droplet are well simulated. [ABSTRACT FROM AUTHOR]

In this work, we present first- and second-order energy-stable linear schemes for the Swift-Hohenberg equation based on first-order backward Euler and Crank-Nicolson schemes, respectively. We prove rigorously that the schemes satisfy the energy dissipation property. We also derive the error analysis for our schemes. Moreover, we adopt a spectral-Galerkin approximation for the spatial variables and establish error estimates for the fully discrete second-order Crank-Nicolson scheme. Numerical results are presented to validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel fully discrete semi-implicit stabilized finite element method for the Cahn-Hilliard-Navier-Stokes phase-field model by using the lowest equal-order (P 1 / P 1 / P 1 / P 1) finite element pair, which consists of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the first order semi-implicit scheme combined with convex splitting approximation for the temporal discretization. We prove that the fully discrete scheme is unconditional energy stable and mass conservative. We also carry out optimal error estimates both in time and space for the phase function, chemical potential and velocity in the appropriate norms. Finally, several numerical experiments are presented to confirm the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper we provide a detailed analysis of the preconditioned steepest descent (PSD) iteration solver for a convex splitting numerical scheme to the Cahn-Hilliard equation with variable mobility function. In more details, the convex-concave decomposition is applied to the energy functional, which in turn leads to an implicit treatment for the nonlinear term and the surface diffusion term, combined with an explicit update for the expansive concave term. In addition, the mobility function, which is solution-dependent, is explicitly computed, which ensures the elliptic property of the operator associated with the temporal derivative. The unique solvability of the numerical scheme is derived following the standard convexity analysis, and the energy stability analysis could also be carefully established. On the other hand, an efficient implementation of the numerical scheme turns out to be challenging, due to the coupling of the nonlinear term, the surface diffusion part, and a variable-dependent mobility elliptic operator. Since the implicit parts of the numerical scheme are associated with a strictly convex energy, we propose a preconditioned steepest descent iteration solver for the numerical implementation. Such an iteration solver consists of a computation of the search direction (involved with a Poissonlike equation), and a one-parameter optimization over the search direction, in which the Newton’s iteration becomes very powerful. In addition, a theoretical analysis is applied to the PSD iteration solver, and a geometric convergence rate is proved for the iteration. A few numerical examples are presented to demonstrate the robustness and efficiency of the PSD solver. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L 2 -gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L 2 -gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H − 1 -gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework. [ABSTRACT FROM AUTHOR]

• An energy stable finite element method (FEM) for droplet dynamics is derived. • New FEM scheme ensures no spurious energy is introduced into the discrete system. • The FEM scheme is derived within arbitrary Lagrangian Eulerian framework. • The moving contact line problem is incorporated into the model. • Supporting (solid) surface may exhibit non-homogeneous properties. An energy stable finite element scheme within arbitrary Lagrangian Eulerian (ALE) framework is derived for simulating the dynamics of millimetric droplets in contact with solid surfaces. Supporting surfaces considered may exhibit non–homogeneous properties which are incorporated into the governing system through generalized Navier boundary conditions (GNBC). Numerical scheme is constructed such that the counterpart of (continuous) energy balance holds on the discrete level. This ensures that no spurious energy is introduced into the discrete system, i.e. the discrete formulation is stable in the energy norm. The newly proposed scheme is numerically validated to confirm the theoretical predictions. Of a particular interest is the case of droplet on a non–homogeneous inclined surface. This case shows the capabilities of the scheme to capture the complex droplet dynamics (sliding and rolling) while maintaining stability during the long time simulation. [ABSTRACT FROM AUTHOR]

We propose and analyze an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations in this paper. The proposed method is a first-order velocity correction projection algorithms in time marching, including standard velocity correction and rotation velocity correction, which can completely decouple all variables in the model. Meanwhile, the schemes are not only linear and only need to solve a series of linear partial differential equations with constant coefficients at each time step, but also the standard velocity correction algorithm can produce the Neumann boundary condition for the pressure field, but the rotational velocity correction algorithm can produce the consistent boundary which improve the accuracy of the pressure field. Thus, improving our computational efficiency. Then, we give the energy stability of the algorithms and give a detailed proofs. The key idea to establish the stability results of the rotation velocity correction algorithm is to transform the rotation term into a telescopic symmetric form by means of the Gauge–Uzawa formula. Finally, numerical experiments show that the rotation velocity correction projection algorithm is efficient to solve the thermally coupled magneto-hydrodynamic equations. [ABSTRACT FROM AUTHOR]

In this work, we present three linear numerical schemes to model nematic liquid crystals using the Landau-de Gennes Q -tensor theory. The first scheme is based on using a truncation procedure of the energy, which allows for an unconditionally energy stable first order accurate decoupled scheme. The second scheme uses a modified second order accurate optimal dissipation algorithm, which gives a second order accurate coupled scheme. Finally, the third scheme uses a new idea to decouple the unknowns from the second scheme which allows us to obtain accurate dynamics while improving computational efficiency. We present several numerical experiments to offer a comparative study of the accuracy, efficiency and the ability of the numerical schemes to represent realistic dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical modeling of gas flow in porous media with compressible gas and rock. To develop an effective numerical method for simulating this problem, we propose an alternative equation by introducing the poroelasticity equation and the porosity variation equation to account for the influence of rock deformation on porosity. In addition, we introduce a free energy for the skeleton of rocks and take into account rock compressibility as well. Instead of the pressure gradient, we utilize the chemical potential gradient as the primary driving force. This formulation is proved to satisfy an energy dissipation law. By applying the improved energy factorization method to handle the Helmholtz free energy density, we propose a semi-implicit time discretization scheme. The discrete pressure is carefully calculated by the discrete chemical potential and Helmholtz free energy so as to inherit the energy dissipation law. A fully discrete scheme is constructed based on the discontinuous Galerkin and mixed finite element methods with the upwind strategy. We prove that the fully discrete scheme still satisfies the energy dissipation law as well as possesses the feature of mass conservation. Additionally, we can prove the boundedness of density under reasonable assumptions on porosity. Numerical results are provided to show the performance of the proposed scheme and validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

Novel time domain finite element methods are proposed to numerically solve the system of Maxwell's equations with a cubic nonlinearity in the spatial 3D case. The effects of linear and nonlinear electric polarization are precisely modeled in this approach. In order to achieve an energy stable discretization at the semi-discrete and the fully discrete levels, a novel technique is developed to handle the discrete nonlinearity, with spatial discretization either using edge and face elements (Nédélec-Raviart-Thomas) or discontinuous spaces and edge elements (Lee-Madsen). In particular, the proposed time discretization scheme is unconditionally stable with respect to the electromagnetic energy and is free of any Courant-Friedrichs-Lewy-type condition. Optimal error estimates are presented at semi-discrete and fully discrete levels for the nonlinear problem. The methods are robust and allow for discretization of complicated geometries and nonlinearities of spatially 3D problems that can be directly derived from the full system of nonlinear Maxwell's equations. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete scheme is proposed to solve the nonlinear Schrödinger-Possion equations. The scheme is developed by the scalar auxiliary variable (SAV) approach, the Crank-Nicolson temproal discretization and the Galerkin-Legendre spectral spatial discretization. The fully discrete scheme is proved to be mass- and energy- conserved. Moreover, unconditional energy stability and convergence of the scheme are obtained by the use of the Gagliardo-Nirenberg and some Sobolev inequalities. Numerical results are presented to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We present a variant of second order accurate in time backward differentiation formula schemes for the Cahn-Hilliard equation, with a Fourier collocation spectral approximation in space. A three-point stencil is applied in the temporal discretization, and the concave term diffusion term is treated explicitly. An additional Douglas-Dupont regularization term is introduced, which ensures the energy stability with a mild requirement. Various numerical simulations including the verification of accuracy, coarsening process and energy decay rate are presented to demonstrate the efficiency and the robustness of proposed schemes. [ABSTRACT FROM AUTHOR]

This paper studies the energy stability property of the correction procedure via reconstruction (CPR) method with staggered flux points based on second-order subcell limiting. The CPR method with staggered flux points uses the Gauss point as the solution point, dividing flux points based on Gauss weights, with the flux points being one more point than the solution points. For subcell limiting, a shock indicator is used to detect troubled cells where discontinuities may exist. Troubled cells are calculated by the second-order subcell compact nonuniform nonlinear weighted (CNNW2) scheme, which has the same solution points as the CPR method. The smooth cells are calculated by the CPR method. The linear energy stability of the linear CNNW2 scheme is proven theoretically. Through various numerical experiments, we demonstrate that the CNNW2 scheme and CPR method based on subcell linear CNNW2 limiting are energy-stable and that the CPR method based on subcell nonlinear CNNW2 limiting is nonlinearly stable.

The energy security of each country is one of the main factors of its proper functioning. Currently, in the era of problems related to energy security resulting from, among other things, the war in Ukraine, this topic is particularly important. This article presents issues related to Poland’s energy security, understood as the financial and economic stability of enterprises operating in the energy industry. This stability is considered in two aspects: macroeconomic, where the focus is mainly on the aspect of state intervention in market processes; and microeconomic, where factors determining the financial security of energy enterprises were identified, including internal and external factors affecting the functioning of these entities. In order to achieve the assumed research goals, the analysis of the indicated problems was based on non-reactive research, consisting in the assessment of the available information. It included studies of normative acts, official statistical data, industry reports and analyses, as well as data obtained in the form of a public information request. Two basic research methods were used in the work—dogmatic–legal and comparative analyses. The identification of factors affecting the security of companies in the sector was carried out on the basis of data on the entire energy sector in Poland for the years 2015–2021 on a semi-annual basis. Vector-autoregressive models were used for the analysis. As a result of the analyses, it was established that market failures and public safety are the premises justifying the public financing of enterprises in the electricity generation, transmission, distribution and trade sectors. At the same time, the conducted research showed that the level of financial security of energy enterprises in Poland was affected by the ratio of the value of goods and materials sold to net sales revenue, as well as the level of EBIT (earnings before deducting interest and taxes) margin, and among external factors, the level of GDP (gross domestic product), CPI (consumer price index) and Crude Oil were important.

The Cahn–Hilliard–Navier–Stokes model is extensively used for simulating two-phase incompressible fluid flows. With the absence of exterior force, this model satisfies the energy dissipation law. The present work focuses on developing a linear, decoupled, and energy dissipation-preserving time-marching scheme for the hydrodynamics coupled Cahn–Hilliard model. An efficient time-dependent auxiliary variable approach is first introduced to design equivalent equations. Based on equivalent forms, a BDF2-type linear scheme is constructed. In each time step, the unique solvability and the energy dissipation law can be analytically estimated. To enhance the energy stability and the consistency, we correct the modified energy by a practical relaxation technique. Using the finite difference method in space, the fully discrete scheme is described, and the numerical solutions can be separately implemented. Numerical results indicate that the proposed scheme has desired accuracy, consistency, and energy stability. Moreover, the flow-coupled phase separation, the falling droplet, and the dripping droplet are well simulated.

We introduce a new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation. An appropriate memory variable treatment of stress-strain convolution terms, result into a symmetric system of first order linear hyperbolic partial differential equations, which we discretize using a high-order discontinuous Galerkin finite element method. The accuracy of the resulting numerical scheme is verified using convergence studies against analytical plane wave solutions and analytical solutions of the viscoelastic wave equation. Computational experiments are shown for various combinations of homogeneous and heterogeneous viscoelastic media in two and three dimensions. [ABSTRACT FROM AUTHOR]

We develop a trigonometric scalar auxiliary variable (TSAV) approach for constructing linear, totally decoupled, and energy-stable numerical methods for gradient flows. An auxiliary variable r based on the trigonometric form of the nonlinear potential functional removes the bounded-from-below restriction. By adding a positive constant greater than 1, the positivity preserving property of r can be satisfied. Furthermore, the phase-field variables and auxiliary variable r can be treated in a totally decoupled manner, which simplifies the algorithm. A practical stabilization method is employed to suppress the effect of an explicit nonlinear term. Using our proposed approach, temporally first-order and second-order methods are easily constructed. We prove analytically the discrete energy dissipation laws of the first- and second-order schemes. Furthermore, we propose a multiple TSAV approach for complex systems with multiple components. A comparison of stabilized-SAV (S-SAV) and stabilized-TSAV (S-TSAV) approaches is performed to show their efficiency. Two-dimensional numerical experiments demonstrated the desired accuracy and energy stability. [ABSTRACT FROM AUTHOR]

In this paper, we construct a second-order accurate, energy stable and maximum bound principle-preserving scheme for the Allen–Cahn equation with a general mobility based on the stabilized exponential scalar auxiliary variable (SESAV) approach. Some extra stabilizing terms are added to the discretized scheme for the purpose of improving numerical stability. We first proved the maximum bound principle (MBP) under reasonable constraints on time step size and the stabilization parameter. Then, we found that the proposed scheme is unconditionally energy-stable. Finally, a numerical example is carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, time-domain finite element methods for the full system of Maxwell's equations with cubic nonlinearities in 3D are presented, including a selection of computational experiments. The new capabilities of these methods are to efficiently model linear and nonlinear effects of the electrical polarization. The novel strategy has been developed to bring under control the discrete nonlinearity model in space and time. It results in energy stable discretizations both at the semi-discrete and the fully discrete levels, with spatial discretization using edge and face elements (Nédeléc-Raviart-Thomas formulation). In particular, the proposed time discretization schemes are unconditionally stable with respect to a specially defined nonlinear electromagnetic energy, which is an upper bound of the electromagnetic energy commonly used. The approaches presented prove to be robust and allow the modeling of 3D optical problems that can be directly derived from the full system of Maxwell's nonlinear equations, and allow the treatment of complex nonlinearities and geometries of various physical systems coupled with electromagnetic fields.

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L2-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L2-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H−1-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework.

We propose and analyze an effective decoupling algorithm for unsteady thermally coupled magneto-hydrodynamic equations in this paper. The proposed method is a first-order velocity correction projection algorithms in time marching, including standard velocity correction and rotation velocity correction, which can completely decouple all variables in the model. Meanwhile, the schemes are not only linear and only need to solve a series of linear partial differential equations with constant coefficients at each time step, but also the standard velocity correction algorithm can produce the Neumann boundary condition for the pressure field, but the rotational velocity correction algorithm can produce the consistent boundary which improve the accuracy of the pressure field. Thus, improving our computational efficiency. Then, we give the energy stability of the algorithms and give a detailed proofs. The key idea to establish the stability results of the rotation velocity correction algorithm is to transform the rotation term into a telescopic symmetric form by means of the Gauge–Uzawa formula. Finally, numerical experiments show that the rotation velocity correction projection algorithm is efficient to solve the thermally coupled magneto-hydrodynamic equations.

Two efficient numerical schemes based on L 1 formula and Finite-Difference Time-Domain (FDTD) method are constructed for Maxwell's equations in a Cole-Cole dispersive medium. The temporal discretizations are built upon the leap-frog method and Crank-Nicolson method, respectively. We carry out the energy stability and error analysis rigorously by the energy method. Both schemes have been proved convergence with order O ((Δ t) 2 − α + (Δ x) 2 + (Δ y) 2) , where Δ t , Δ x , Δ y are respectively the step sizes in time, space in x- and y-direction. The parameter α is a measure of the dispersion broadening. Numerical experiments are performed to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this study, we develop linear and energy stable numerical schemes for the Swift–Hohenberg equation with quadratic-cubic nonlinearity. A modified scalar auxiliary variable (SAV) approach is used to construct the temporally first- and second-order accurate discretizations. Different from the classical SAV approach, the proposed schemes permit us to solve the governing equations in a step-by-step manner, i.e., the calculation of inner product is not needed. We analytically prove the energy stability. We solve the resulting system of discrete equations using the linear multigrid method. We perform various numerical examples to show the accuracy and energy stability of the proposed method. The pattern formations in two- and three-dimensional spaces are also simulated. [ABSTRACT FROM AUTHOR]

In this work, we investigate (energy) stability of global radial basis function (RBF) methods for linear advection problems. Classically, boundary conditions (BC) are enforced strongly in RBF methods. By now it is well-known that this can lead to stability problems, however. Here, we follow a different path and propose two novel RBF approaches which are based on a weak enforcement of BCs. By using the concept of flux reconstruction and simultaneous approximation terms (SATs), respectively, we are able to prove that both new RBF schemes are strongly (energy) stable. Numerical results in one and two spatial dimensions for both scalar equations and systems are presented, supporting our theoretical analysis. [ABSTRACT FROM AUTHOR]

• Linear and totally decoupled numerical schemes are proposed. • Our method inherits all advantages of classical SAV approach and further simplifies the algorithm. • The temporally first-order and second-order schemes can be easily constructed by using our method. • The desired accuracy and energy dissipation law are confirmed by the numerical experiments. • The coarsening dynamics of two-phase surfactant system can be well simulated by our method. In this work, we develop a new linear, decoupled numerical scheme for the typical phase-field surfactant model. An improved scalar auxiliary variable (SAV) approach is used to discretize the governing equations in time. Different from the classical SAV approach, this improved form can calculate the phase field function ϕ , surfactant function ψ , and auxiliary variables in a step-by-step manner, i.e., the auxiliary variables are treated totally explicitly, thus we can directly calculate ϕ and ψ instead of computing the inner products. At each time step, the surfactant ψ can be directly obtained by an explicit way, then ϕ is updated by solving a linear system with constant coefficient. Therefore, the implementation of this improved SAV approach is easier than the classical SAV approach. The energy stability in first-order case can be analytically proved by using the our method. The numerical experiments show that our proposed method not only achieves desired first- and second-order accuracy but also satisfies the desired discrete energy dissipation law even if some larger time steps are used. Furthermore, the coarsening dynamics with different average concentrations can be well simulated by using our method. The co-continuous and drop patterns are generated in the even compositions case and uneven compositions case, respectively. [ABSTRACT FROM AUTHOR]

• Fully discrete energy stable scheme for phase-field moving contact line model. • Rigorous proof that the fully discrete scheme is unconditionally energy stable. • 2D and 3D results demonstrate accuracy and energy stability of scheme. In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone. [ABSTRACT FROM AUTHOR]

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt2 + N−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, time-domain finite element methods for the full system of Maxwell's equations with cubic nonlinearities in 3D are presented, including a selection of computational experiments. The new capabilities of these methods are to efficiently model linear and nonlinear effects of the electrical polarization. The novel strategy has been developed to bring under control the discrete nonlinearity model in space and time. It results in energy stable discretizations both at the semi-discrete and the fully discrete levels, with spatial discretization using edge and face elements (Nédeléc-Raviart-Thomas formulation). In particular, the proposed time discretization schemes are unconditionally stable with respect to a specially defined nonlinear electromagnetic energy, which is an upper bound of the electromagnetic energy commonly used. The approaches presented prove to be robust and allow the modeling of 3D optical problems that can be directly derived from the full system of Maxwell's nonlinear equations, and allow the treatment of complex nonlinearities and geometries of various physical systems coupled with electromagnetic fields. [ABSTRACT FROM AUTHOR]