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Summary: Clustering of nodes in wireless sensor networks (WSNs) plays a dominant role in gathering environmental data from the specific area of monitoring over which they are deployed for achieving a reactive decision‐making process. The design and development of an energy‐efficient clustering strategy with a potential cluster head (CH) selection process is a herculean task. This development of the CH selection scheme is referred as a non‐deterministic polynomial (NP) hard problem as it needs to optimize different parameters that influence the selection of potential sensor nodes as CH. It needs to concentrate on the process of enhancing network lifespan with energy efficiency by selecting optimal routing path during data dissemination activity. In this paper, a Hybrid Red Deer and Fireworks Optimization Algorithm (HIRDIFOA)–based energy efficient clustering technique is proposed for extending network lifespan with maximized stability in the network energy. This proposed HRDFOA integrated the exploration capability of Improved Red Deer Optimization (IRDOA) with the maximized exploitation tendency of the Modified Firework Optimization Algorithm (MFWOA) during the CH selection process. It facilitated the CH selection by evaluating the fitness functions that integrate the factors of residual energy (RE), distance between sensor and CH, distance between CH and sink, and radius of communication. It significantly adopted MWFOA for achieving sink node mobility such that data can be reliably routed from CH to sink. The outcomes of HRDFOA confirm better throughput of 19.21% with reduced energy consumption of 17.42% and reduced end‐to‐end delay of 18.52% in contrast to the competitive CH selection schemes used for investigation. [ABSTRACT FROM AUTHOR]

In this paper, a linear second-order finite difference scheme is proposed for the Allen–Cahn equation with a general positive mobility. The Crank–Nicolson scheme and the Taylor's formula are used for temporal discretization, and the central finite difference method is used for spatial approximation. The discrete maximum bound principle (MBP), the discrete energy stability and -norm error estimate are discussed, respectively. Finally, some numerical examples are presented to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In contrast to the classical Allen–Cahn equation, the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier not only satisfies the maximum bound principle (MBP) and energy dissipation law but also ensures mass conservation. Many existing schemes often fail to preserve all these properties at the discrete level or require high regularity in time on the exact solution for convergence analysis. In this paper, we construct a new class of low regularity integrators (LRIs) for time discretization of the conservative Allen–Cahn equation by repeatedly using Duhamel’s formula. The proposed first- and second-order LRI schemes are shown to conserve mass unconditionally and satisfy the MBP under some time step size constraints. Temporal error estimates for these schemes are derived under a low regularity assumption that the exact solution is only Lipschitz continuous in time, followed by a rigorous proof for energy stability of the corresponding time-discrete solutions. Various numerical experiments and comparisons in two and three dimensions are presented to verify the theoretical results and illustrate the performance of the LRI schemes, especially when the interfacial parameter approaches zero. [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]

We study the asymptotical compatibility of the Fourier spectral method in multidimensional space for the nonlocal Ohta–Kawasaka model, a more generalized version of the model proposed in our previous work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). By introducing the Fourier collocation discretization for the spatial variable, we show that the asymptotical compatibility holds in 2D and 3D over a periodic domain. For the temporal discretization, we adopt the second‐order backward differentiation formula method. We prove that for certain nonlocal kernels, the proposed time discretization schemes inherit the energy dissipation law. In the numerical experiments, we verify the asymptotical compatibility, the second‐order temporal convergence rate, and the energy stability of the proposed schemes. More importantly, we discover a novel square lattice pattern when certain nonlocal kernel are applied in the model. In addition, our numerical experiments confirm the existence of an upper bound for the optimal number of bubbles in 2D for some specific nonlocal kernels. Finally, we numerically explore the promotion/demotion effect induced by the nonlocal horizon δ$$ \delta $$, which is consistent with the theoretical studies presented in our earlier work (Y. Zhao and W. Luo, Physica D 458 (2024), 133989). [ABSTRACT FROM AUTHOR]

In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results. [ABSTRACT FROM AUTHOR]

In this article, we develop stable and efficient numerical schemes for a binary fluid‐surfactant phase‐field model which consists of two Cahn–Hilliard type equations with respect to the free energy containing a Ginzburg–Landau double‐well potential, a logarithmic Flory–Huggins potential and a nonlinear coupling entropy. The numerical schemes, which are decoupled and linear, are established by the central difference spatial approximation in combination with the first‐ and second‐order exponential time differencing methods based on the convex splitting of the free energy. For the sake of the linearity of the schemes, the nonlinear terms, especially the logarithmic term, are approximated explicitly, which requires the bound preservation of the numerical solution to make the algorithm robust. We conduct the convergence analysis and prove the bound‐preserving property in details for both first‐ and second‐order schemes, where the high‐order consistency analysis is applied to the first‐order case. In addition, the energy stability is also obtained by the nature of the convex splitting. Numerical experiments are performed to verify the accuracy and stability of the schemes and simulate the dynamics of phase separation and surfactant adsorption. [ABSTRACT FROM AUTHOR]

This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods. [ABSTRACT FROM AUTHOR]

Image inpainting is pivotal within the realm of image processing, and many efforts have been dedicated to modeling, theory, and numerical analysis in this research area. In this paper, we propose a curvature-dependent elastic bending total variation model for the inpainting problem, in which the elastic bending energy in the phase-field framework introduces geometric information and the total variation term maintains the sharpness of the inpainting edge, referred to as elastic bending-TV model. The energy stability is theoretically proved based on the scalar auxiliary variable method. Additionally, an adaptive time-stepping algorithm is used to further improve the computational efficiency. Numerical experiments illustrate the effectiveness of the proposed model and verify the capability of our model in image inpainting. [ABSTRACT FROM AUTHOR]

In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension N = 2 , 3 , that describes the movement of a mixture of two incompressible viscous fluids. This model consists of modified Navier–Stokes equations coupled with a phase-field equation given by a convective Allen–Cahn equation, and energy transport equation for the temperature; which admits a dissipative energy inequality. We propose an energy stable numerical scheme based on the Finite Element Method, and we analyze optimal weak and strong error estimates, as well as convergence towards regular solutions. In order to construct the numerical scheme, we introduce two extra variables (given by the gradient of the temperature and the variation of the energy with respect to the phase-field function) which allows us to control the strong regularity required by the model, which is one of the main difficulties appearing from the numerical point of view. Having the equivalent model, we consider a fully discrete Finite Element approximation which is well-posed, energy stable and satisfies a set of uniform estimates which allow to analyze the convergence of the scheme. Finally, we present some numerical simulations to validate numerically our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider integrating the scalar auxiliary variable time discretization with the virtual element method spatial discretization to obtain energy-stable schemes for Allen–Cahn-type gradient flow problems. In order to optimize CPU time during calculations, we propose two step-by-step solving SAV algorithms by introducing a novel auxiliary variable to replace the original one. Then, linear, decoupled, and unconditionally energy-stable numerical schemes are constructed. However, due to truncation errors, the auxiliary variable is not equivalent to the continuous case in the original definition. Therefore, we propose a novel relaxation technique to preserve the original energy dissipation rule. It not only retains all the advantages of the above algorithms but also improves accuracy and consistency. Finally, a series of numerical experiments are conducted to demonstrate the effectiveness of our method. [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.

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]

In this paper, we introduce second-order numerical schemes for the modified phase field crystal (MPFC) model that are decoupled, linear, positivity-preserving, and unconditionally energy-stable. These schemes adopt a positivity-preserving auxiliary variable method to explicitly handle the nonlinear potential function, resulting in decoupled linear systems with constant coefficients at each time step. We rigorously demonstrate that the auxiliary variables remain positive throughout all time steps and prove the unconditionally energy stability of these schemes. The stability pertains to a discrete modified energy, rather than the original free energy or the pseudo energy of the MPFC system. Moreover, a detailed error analysis is provided. A series of numerical experiments are conducted to validate the accuracy and efficiency of our proposed schemes. [ABSTRACT FROM AUTHOR]

The inclusion of a Hall term brings many more challenges in the establishment of a numerical scheme for the Hall magnetohydrodynamics (Hall-MHD) system, compared to the classical MHD equations. For the incompressible Hall-MHD system in a three-dimensional domain, we aim at constructing an efficient numerical scheme with properties of linearity, adaptive variation of time-stepping, second-order time accuracy, decoupling, and unconditional energy stability. For this purpose, the Legendre--Galerkin spectral method is applied for spatial approximation. By introducing an artificial auxiliary variable, we employ the Crank--Nicolson scheme for temporal discretization with the explicit treatment of nonlinear terms. In addition, the second-order incremental pressure-correction method is utilized in the Stokes solver to reduce the cost of computation. Based on the energy dissipation rate of the Hall-MHD system, we design an adaptive time-stepping strategy to enhance efficiency. The unconditional energy stability of the fully discrete scheme is strictly proved, where the decoupled Stokes solver needs to be analyzed in detail. We also show that the scheme is divergence-free for magnetic fields if the corresponding initial condition is divergence-free. Numerical experiments are carried out to illustrate the accuracy, efficiency, and robustness of the proposed scheme. As for applications of our scheme, numerical simulations on the O- and X-points appearing in fast magnetic reconnection and on whistler waves are presented within the Hall-MHD regime. [ABSTRACT FROM AUTHOR]

The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting, and segmentation. [ABSTRACT FROM AUTHOR]

A fully discrete semi-convex-splitting finite-element scheme with stabilization for a Cahn–Hilliard cross-diffusion system is analyzed. The system consists of parabolic fourth-order equations for the volume fraction of the fiber phase and solute concentration, modeling pre-patterning of lymphatic vessel morphology. The existence of discrete solutions is proved, and it is shown that the numerical scheme is energy stable up to stabilization, conserves the solute mass, and preserves the lower and upper bounds of the fiber phase fraction. Numerical experiments in two space dimensions using FreeFem illustrate the phase segregation and pattern formation. [ABSTRACT FROM AUTHOR]

The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal matrix-valued functions and has attracted much attention in recent years. It is well known that the MAC equation satisfies the maximum bound principle (MBP) with respect to either the matrix 2-norm or the Frobenius norm, which plays a key role in understanding the physical meaning and the wellposedness of the model. To preserve this property, we extend the explicit integrating factor Runge-Kutta (IFRK) method to the MAC equation. Moreover, we construct a new three-stage third-order and a new four-stage fourth-order IFRK schemes based on the classical Runge-Kutta schemes. Under a reasonable time-step constraint, we prove the MBP preservation of the IFRK method with respect to the matrix 2-norm, based on which, we further establish their optimal error estimates in the matrix 2-norm. Although numerical results indicate that the IFRK method preserves the MBP with respect to the Frobenius norm, a detailed analysis shows that it is hard to prove this preservation by using the same approach for the case of 2-norm. Several numerical experiments are carried out to test the convergence of the IFRK schemes and to verify the MBP preservation with respect to the matrix 2-norm and the Frobenius norm, respectively. Energy stability is also observed, which clearly indicates the orthogonality of the stationary solution of the MAC equation. In addition, we simulate the coarsening dynamics to verify the motion law of the interface for different initial conditions. [ABSTRACT FROM AUTHOR]

A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, is approximated in a semi-implicit manner. In the spatial discretization, the marker and cell difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, an efficient fully-decoupled and fully-discrete numerical scheme with second-order temporal accuracy is developed to solve the incompressible hydrodynamically coupled Cahn-Hilliard model for simulating the two-phase fluid flow system. The scheme is developed by combining the finite element method for spatial discretization and several effective time marching approaches, including the pressure-correction projection method for dealing with fluid equations and the explicit-invariant energy quadratization (explicit-IEQ) approach for dealing with coupled nonlinear terms. The obtained scheme is very efficient since it only needs to solve several decoupled, linear elliptic equations with constant coefficients at each time step. We also strictly prove the solvability and unconditional energy stability of the scheme, and verify the accuracy and stability of the scheme through plenty of numerical examples. [ABSTRACT FROM AUTHOR]

Numerical modeling of immiscible two‐phase flow in deformable porous media has become increasingly significant due to its applications in oil reservoir engineering, geotechnical engineering and many others. The coupling between two‐phase flow and geomechanics gives rise to a major challenge to the development of physically consistent mathematical models and effective numerical methods. In this article, based on the concept of free energies and guided by the second law of thermodynamics, we derive a thermodynamically consistent mathematical model for immiscible two‐phase flow in poro‐viscoelastic media. The model uses the fluid and solid free energies to characterize the fluid capillarity and solid skeleton elasticity, so that it rigorously follows an energy dissipation law. The thermodynamically consistent formulation of the pore fluid pressure is naturally derived for the solid mechanical equilibrium equation. Additionally, the model ensures the mass conservation law for both fluids and solids. For numerical approximation of the model, we propose an energy stable and mass conservative numerical method. The method herein inherits the energy dissipation law through appropriate energy approaches and subtle treatments for the coupling between two phase saturations, the effective pore pressure and porosity. Using the locally conservative cell‐centered finite difference methods on staggered grids with the upwind strategies for saturations and porosity, we construct the fully discrete scheme, which has the ability to conserve the masses of both fluids and solids as well as preserve the energy dissipation law at the fully discrete level. In particular, the proposed method is an unbiased algorithm, that is, treating the wetting phase, the non‐wetting phase and the solid phase in the same way. Numerical results are also given to validate and verify the features of the proposed model and numerical method. [ABSTRACT FROM AUTHOR]

We present a structure-preserving Eulerian algorithm for solving L²-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system’s free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes. [ABSTRACT FROM AUTHOR]

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard L² norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the L² norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field. [ABSTRACT FROM AUTHOR]

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable. [ABSTRACT FROM AUTHOR]

In this work, we construct a fully discrete scheme with finite difference method based on the staggered grids for the viscous Cahn–Hilliard equation. The constructed scheme can satisfy the unconditional dissipation law with original energy. We carry out a rigorous error analysis with superconvergence by introducing an auxiliary function depending on the chemical potential. We obtain second order accuracy in both space and time with l ∞ (0 , T ; H 1 (Ω)) norm for the phase function and l 2 (0 , T ; l 2 (Ω)) for the chemical potential on non-uniform grids. Numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel second-order generalised rotational discrete gradient scheme for numerically approximating the orthonormal frame gradient flow of biaxial nematic liquid crystals. This scheme relies on reformulating the original gradient flow system into an equivalent generalised “rotational” form. A second-order discrete gradient approximation of the energy variation is then devised such that it satisfies an energy difference relation. The proposed numerical scheme has two remarkable properties: (i) it strictly obeys the orthonormal property of the tensor field and (ii) it satisfies the energy dissipation law at the discrete level, regardless of the time step sizes. We provide ample numerical results to validate the accuracy, efficiency, unconditional stability and SO(3)-preserving property of this scheme. In addition, comparisons of the simulation results between the biaxial orthonormal frame gradient flow model and uniaxial Oseen–Frank gradient flow are made to demonstrate the ability of the former to characterize non-axisymmetric local anisotropy. [ABSTRACT FROM AUTHOR]

In this paper, we propose SGEM, stochastic gradient with energy and momentum, to solve a class of general non-convex stochastic optimization problems, based on the AEGD method introduced in AEGD (adaptive gradient descent with energy) Liu and Tian (Numerical Algebra, Control and Optimization, 2023). SGEM incorporates both energy and momentum so as to inherit their dual advantages. We show that SGEM features an unconditional energy stability property and provide a positive lower threshold for the energy variable. We further derive energy-dependent convergence rates in the general non-convex stochastic setting, as well as a regret bound in the online convex setting. Our experimental results show that SGEM converges faster than AEGD and generalizes better or at least as well as SGDM in training some deep neural networks. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we analyze the numerical approximation of two‐phase magnetohydrodynamic flows. Firstly, an equivalent new system is designed by introducing two scalar auxiliary variables. One of variables is used to linearize the phase field function and the other is used to deal with the highly coupled and nonlinear terms. Secondly, by combining with a novel decoupling technique based on the "zero‐energy‐contribution" feature and the pressure correction method, the linearized second order BDF numerical scheme, which has the advantage of fully decoupled structure, is constructed. Furthermore, we strictly prove the unconditional energy stability and error analysis of the scheme, and give a detailed implementation procedure that only requires to calculate several linear elliptic equations with constant coefficients. Finally, the results of numerical simulations are presented to validate the rates of convergence and energy stability. [ABSTRACT FROM AUTHOR]

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.

Computational methods for thermal radiative transfer problems exhibit high computational costs and a prohibitive memory footprint when the spatial and directional domains are finely resolved. A strategy to reduce such computational costs is dynamical low-rank approximation (DLRA), which represents and evolves the solution on a low-rank manifold, thereby significantly decreasing computational and memory requirements. Efficient discretizations for the DLRA evolution equations need to be carefully constructed to guarantee stability while enabling mass conservation. In this work, we focus on the Su--Olson closure leading to a linearized internal energy model and derive a stable discretization through an implicit coupling of internal energy and particle density. Moreover, we propose a rank-adaptive strategy to preserve local mass conservation. Numerical results are presented which showcase the accuracy and efficiency of the proposed low-rank method compared to the solution of the full system. [ABSTRACT FROM AUTHOR]

This article presents a mass-conserved Allen-Cahn type two-phase ferrofluid flow model and establishes its corresponding energy law. The model is a highly coupled, nonlinear saddle point system consisting of the mass-conserved Allen-Cahn equation, the Navier-Stokes equation, the magnetostatic equation, and the magnetization equation. We develop a unified framework of the scalar auxiliary variable (SAV) method and the zero energy contribution (ZEC) approach, which constructs a mass-conserved, fully decoupled, second-order accurate in time, and unconditionally energy-stable linear scheme. We incorporate several distinct numerical techniques, including refor- mulations of the equations to remove the linear couplings and implicit nonlocal integration, the projection method to decouple the velocity and pressure, a symmetric implicit-explicit format for symmetric positive definite nonlinearity, and the continuous finite element method discretization. We also analyze the mass-conserved property, unconditional energy stability, and well-posedness of the scheme. To demonstrate the effectiveness, stability, and accuracy of the developed model and numerical algorithm, we implemented several numerical examples, involving a ferrofluid hedgehog in 2D and a ferromagnetic droplet in 3D. It is worth mentioning that the proposed unified framework of the SAV-ZEC method is also applicable to designing efficient schemes for other coupled-type fluid flow phase-field systems. [ABSTRACT FROM AUTHOR]

Two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility (between stable values 0 and 1) are presented, by using two different non-centered approximation of the mobility. We prove that both schemes are energy stable and preserve the maximum principle approximately, i.e. the amount of the solution being outside of the interval [ 0 , 1 ] goes to zero in terms of a truncation parameter. Additionally, we present several numerical results in order to show the accuracy and the well behavior of the proposed schemes, comparing both schemes and the corresponding centered scheme. [ABSTRACT FROM AUTHOR]

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]

The phase-field surfactant model is popular to study the dynamics of surfactant-laden phase separation in a binary mixture. In this work, we numerically investigate the H - 1 -gradient flow based phase-field surfactant mathematical model using an energy dissipation-preserving numerical method. The proposed method adapts a Lagrange multiplier method. The present method not only preserves the unconditional stability, but also satisfies the original energy dissipation law, which is different from the modified energy dissipation laws estimated by the scalar auxiliary variable and invariant energy quadratization methods. An effective scheme is introduced to solve the weakly coupled discrete equations. In one time cycle, we only need to calculate four linear, fully decoupled discrete equations with constant coefficients and compute two nonlinear algebraic equations using Newton's iteration. The computational experiments indicate that the proposed method is accurate and satisfies the original energy stability. Moreover, the long-time behaviors of surfactant-laden phase separation can also be well simulated. [ABSTRACT FROM AUTHOR]

When a fluid fills an infinite layer between two rigid plates in relative motion, and it is simultaneously subject to a gradient of pressure not parallel to the motion, the base flow is a combination of Couette–Poiseuille in the direction along the boundaries' relative motion, but it also possess a Poiseuille component in the transverse direction. For this reason the linearised equations include all variables x, y, z, and not only explicitly two variables x, z as it typically happens in the literature. For convenience, we indicate as streamwise the direction of the relative motions of the plates, and spanwise the orthogonal direction. We use Chebyshev collocation method to investigate the monotonic behaviour of the energy along perturbations of general streamwise Couette–Poiseuille plus spanwise Poiseuille base flow, thus obtaining energy-critical Reynolds numbers depending on two parameters. We finally compute the spectrum of the linearisation at such base flows, and hence determine spectrum-critical Reynolds numbers depending on the two parameters. The choice of convex combinations of Couette and Poiseuille flows along the streamwise direction, and spanwise Poiseuille flow, affects the value of the energy-critical Reynolds and wave numbers in interesting ways. Also the spectrum-critical Reynolds and wave numbers depend on the type of base flow in peculiar ways. These dependencies are not described in the literature. [ABSTRACT FROM AUTHOR]

A model is developed to describe thermal convection in a mixture of fluids in a porous medium, where the layer is heated from below while simultaneously the fluid density at the base of the porous layer is greater than that higher up. In addition to buoyancy forces which are essentially due to gravity the fluid mixture is subject to Korteweg stresses which arise because of density gradients in the mixture. A complete stability analysis is provided and the critical Rayleigh number for convective motion is derived for both stationary and oscillatory convection and this is complemented with a global energy stability analysis. The analogous problem in a bidisperse porous medium is also briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper, we present a second-order accurate and linear numerical scheme for the phase field crystal equation and prove its convergence in the discrete L\infty sense. The key ingredient of the error analysis is to justify the boundedness of the numerical solution, so that the nonlinear term, treated explicitly in the scheme, can be bounded appropriately. Benefiting from the existence of the sixth-order dissipation term in the model, we first estimate the discrete H2 norm of the numerical error. The error estimate in the supremum norm is then obtained by the Sobolev embedding, so that the uniform bound of the numerical solution is available. We also show the mass conservation and the energy stability in the discrete setting. The proposed scheme is linear with constant coefficients so that it can be solved efficiently via some fast algorithms. Numerical experiments are conducted to verify the theoretical results, and long-time simulations in two- and three-dimensional spaces demonstrate the satisfactory and high effectiveness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

A unified discrete gradient structure of the second order nonuniform integral averaged approximations for the Caputo fractional derivative and the Riemann--Liouville fractional integral is established in this paper. The required constraint of the step-size ratio is weaker than that found in the literature. With the proposed discrete gradient structure, the energy stability of the variable step Crank--Nicolson type numerical schemes is derived immediately, which is essential to the longtime simulations of the time fractional gradient flows and the nonlinear integro-differential models. The discrete energy dissipation laws fit seamlessly into their classical counterparts as the fractional indexes tend to one. In particular, we provide a framework for the stability analysis of variable step numerical schemes based on the scalar auxiliary variable type approaches. The time fractional Swift--Hohenberg model and the time fractional sine-Gordon model are taken as two examples to elucidate the theoretical results at great length. Extensive numerical experiments using the adaptive time-stepping strategy are provided to verify the theoretical results in the time multiscale simulations. [ABSTRACT FROM AUTHOR]

Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully constructed to guarantee stability while preserving crucial properties of the original problem. Important physical effects that one likes to preserve with DLRA include capturing the diffusion limit in the high-scattering regimes as well as dissipating energy. In this work we propose and analyze a dynamical low-rank method based on the the "unconventional" basis update & Galerkin step integrator. We show that this method is asymptotic preserving, i.e., it captures the diffusion limit, and energy stable under a CFL condition. The derived CFL condition captures the transition from the hyperbolic to the parabolic regime when approaching the diffusion limit. [ABSTRACT FROM AUTHOR]

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept. [ABSTRACT FROM AUTHOR]