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Recently, non-local configurations have been proposed by adding beyond nearest neighbour couplings among elements in lattices to obtain roton-like dispersion relations and phase and group velocities with opposite signs. Even though the introduction of non-local elastic links in metamaterials has unlocked unprecedented possibilities, literature models and prototypes seem neither to provide criteria to compare local and non-local lattices nor to discuss any related rules governing the transition between the two configurations. A physically reasonable principle that monoatomic one-dimensional chains must obey to pass from single- to multi-connected systems is here proposed through a mass conservation law for elastic springs thereby introducing a suitable real dimensionless parameter α to tune stiffness distribution. Therefore, the dispersion relations as a function of α and of the degree of non-locality P are derived analytically, demonstrating that the proposed principle can be rather interpreted as a general mechanical consistency condition to preserve proper dynamics, involving the spring-to-bead mass ratio. Finally, after discussing qualitative results and deriving some useful inequalities, numerical simulations and two-dimensional FFTs are performed for some paradigmatic examples to highlight key dynamics features exhibited by chains with finite length as the parameters α and P vary. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels Φ satisfying Φ (x , y) = x λ 1 y λ 2 + y λ 1 x λ 2 , (x , y) ∈ (0 , ∞) 2 , with λ 1 ≤ λ 2 ≤ 1 , and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when 1 ≤ λ : = λ 1 + λ 2 ≤ 2 with λ 1 ≥ k 0 , the parameter k 0 ∈ (0 , 1) being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when 2 k 0 ≤ λ < 1 with λ 1 ≥ k 0 but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when λ 1 < k 0. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions. [ABSTRACT FROM AUTHOR]

We develop two unfitted finite element methods for the Stokes equations based on H div -conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the H div -conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary. [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]

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach's effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation. [ABSTRACT FROM AUTHOR]

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O (τ 2) in H 1 - norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

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We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the L² projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient L² projection part), positivity preserving, and mass conserving. Rigorous error estimates in the L² norm are established, which are both second order accurate in space and time. The other choice of projection, e.g., H¹ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper explores efficient numerical approximations for the anisotropic phase-field crystal model in two dimension, aiming to analyze lattice systems characterized by varying anisotropic parameters and orientations. Some linear and fully decoupled schemes are constructed for this model in accordance with the Lagrange multiplier approach, that dissipate the original energy rather than the modified energy. Unlike the scalar auxiliary variable method, the newly developed technique does not necessitate the nonlinear term of the energy to have a lower bound. This approach is highly easy-to-implement, we solve two sixth-order linear equations with constant coefficients and a nonlinear algebraic equation just requiring a few Newton-type iterations at each time step. Some numerical simulations demonstrate the accuracy, the mass conservation and the energy dissipation of the proposed schemes. [ABSTRACT FROM AUTHOR]

A variety of mathematical models have been developed to simulate the biochemical and physico-chemical aspects of the anaerobic digestion (AD) process to treat organic wastes and generate biogas. However, all these models, including the most widely accepted and implemented Anaerobic Digestion Model No.1, remain incapable of adequately representing the material balance of AD and are therefore inherently incapable of material conservation. The absence of robust mass conservation constrains reliable estimates of any kinetic parameters being estimated by regression of empirical data. To address this issue, the present work involved the development of a "framework" for a mass-conserving atomistic mathematical model which is capable of mass conservation, with a relative error in the range of machine precision value and an atom balance with a relative error of ±0.02% whilst obeying the Henry's law and electroneutrality principle. Implementing the model in an Excel spreadsheet, the study calibrated the model using the empirical data derived from batch studies. Although the model shows high fidelity as assessed via inspection, considering several constraints including the drawbacks of the model and implementation platform, the study also provides a non-exhaustive list of limitations and further scope for development. [ABSTRACT FROM AUTHOR]

We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RTk x Qk, k ≥ 0. Here Qk is the space of discontinuous polynomial functions of degree less than or equal to k and RT is the Raviart--Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on RTk X Qk, k ≥ 0. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs RT0 x Q0, RT1 x Q1, and BDM1 x Q0 (where BDM is the Brezzi--Douglas--Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh. [ABSTRACT FROM AUTHOR]

In order to reveal the influence of thermal effect on lubrication performances of compound textured plain bearings， a solving model for lubrication performances of the bearings is established considering thermal effect of lubricant. The effects of texture position, rotational speed and length to diameter ratio on lubrication performances of the bearings are analyzed， and compared with results of single-layer textured bearings and non-textured bearings. The results show that：1） whether the thermal effect is considered or not, the compound textured bearings have the highest load capacity and the smallest friction coefficient when the texture is located in bearing pressure rising zone； with the increase of rotational speed， the load capacity and friction force of the bearings increase， the compound textured bearings have the highest load capacity and the smallest friction force； with the increase of length to diameter ratio, the load capacity increases but the friction coefficient decreases, the compound textured bearings have the highest load capacity and the smallest friction coefficient. 2） After considering thermal effect， the load capacity， friction force and friction coefficient are lower than those without considering thermal effect under same working parameters， but the load capacity of compound textured bearings is always higher than that of single-layer textured bearings and non-textured bearings, the friction force and friction coefficient are the smallest. [ABSTRACT FROM AUTHOR]

The global existence of mass‐conserving weak solutions to the Safronov–Dubovskiǐ coagulation equation is shown for the coagulation kernels satisfying at most linear growth for large sizes. In contrast to previous works, the proof mainly relies on the de la Vallée–Poussin theorem, which only requires the finiteness of the first moment of the initial condition. By showing the necessary regularity of solutions, it is shown that the weak solutions constructed herein are indeed classical solutions. Under additional restrictions on the initial data, the uniqueness of solutions is also shown. Finally, the continuous dependence on the initial data and the large‐time behavior of solutions are also addressed. [ABSTRACT FROM AUTHOR]

Recently, an efficient and multidimensional central scheme has been proposed by applying the limiting process diagonally (Yan et al. (2023) [25]). The aforementioned approach is applied to solve the two-dimensional shallow water equations with non-flat bottom topography in this study. The nonnegativity of the updated water depth is preserved under a reasonable CFL condition. The well-balancedness is achieved by simply applying central difference discretization to the source term using the surface gradient method. The conservation of water mass is obtained by changing the scheme into conservation form. All the key features of the present scheme have been rigorously proven. Several numerical experiments are provided to illustrate the robustness of the proposed scheme, including simulations of shallow water equations with Manning's bottom friction and Coriolis forces. [ABSTRACT FROM AUTHOR]

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]

A class of high-order mass- and energy-conserving methods is proposed for the nonlinear Schrödinger equation based on Gauss collocation in time and finite element discretization in space, by introducing a mass- and energy-correction post-process at every time level. The existence, uniqueness, and high-order convergence of the numerical solutions are proved. In particular, the error of the numerical solution is O(τk+1 + hp) in the L∞ (0,T;H¹) norm after incorporating the accumulation errors arising from the post-processing correction procedure at all time levels, where and denote the degrees of finite elements in time and space, respectively, which can be arbitrarily large. Several numerical examples are provided to illustrate the performance of the proposed new method, including the conservation of mass and energy and the high-order convergence in simulating solitons and bi-solitons. [ABSTRACT FROM AUTHOR]

Cells realize various life activities such as migrations and the ingestion of extracellular substances by deforming their cytoplasm and cell membrane. Many reaction-diffusion systems with mass conservation have been used for describing these cell activities. Among them, a mass-conserved three-component reaction-diffusion system was proposed to describe the dynamics of wavelike actin polymerization in macropinocytosis (Yochelis et al. in Phys. Rev. E 101:022213, 2020). This system numerically exhibits dynamical patterns such as annihilation, crossover, and nucleation of pulses in a relatively large interval. In this article, to investigate the dynamics of wavelike actin polymerization in macropinocytosis, we first establish the condition for the diffusion driven instability in the system. We then rigorously prove the existence of spiky stationary solutions to the system in a bounded interval with the Neumann condition. These solutions play a crucial role in the nucleation of pulses although it is only numerically demonstrated. By reducing the stationary problem to a scalar second order nonlinear equation with a nonlocal term, we construct the desired solution by converting the equation into an integral equation. [ABSTRACT FROM AUTHOR]

Multiple fluid simulation has raised wide research interest in recent years. Despite the impressive successes of current works, simulation of scenes containing mixing or unmixing of high-density-ratio phases using particle-based discretizations still remains a challenging task. In this paper, we propose a peridynamic mixture-model theory that stably handles high-density-ratio multi-fluid simulations. With assistance of novel scalar-valued volume flow states, a particle based discretization scheme is proposed to calculate all the terms in the multi-phase Navier-Stokes equations in an integral form, We also design a novel mass updating strategy for enhancing phase mass conservation and reducing particle volume variations under high density ratio settings in multi-fluid simulations. As a result, we achieve significantly stabler simulations in mixture-model multi-fluid simulations involving mixing and unmixing of high density ratio phases. Various experiments and comparisons demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

The active thickness of the translational landslides plays a pivotal role in evaluating its hazards and simulating its instability. Existing techniques have difficulties in estimating the accurate active thickness due to the limitations of observation conditions and imaging geometry, leading to deviations in failure simulations. To overcome these challenges, this study proposes an enhanced method that utilizes multi-orbit Interferometric Synthetic Aperture Radar (InSAR) observations to estimate the active thickness of the translational landslides and subsequently conduct more accurate instability simulations. The method involves integrating multi-orbit InSAR imaging parameters with the spatial geometry of the landslide to establish a slope coordinate system. This system enables the projection of one-dimensional InSAR Line Of Sight (LOS) displacements onto the three-dimensional displacements of the landslide. Subsequently, the active thickness is estimated by combining InSAR three-dimensional displacements with the mass conservation method. Finally, the estimated thickness is incorporated into the geological model construction to simulate the dynamic movement of the landslide. The method was applied to the Xiongba translational landslide in Gongga County, Tibet Autonomous Region, China. The results show that the deformation is mainly concentrated at its forefront, with maximum three-dimensional deformation rates of 4.7 m/a, 2.3 m/a, and 1.24 m/a. The landslide encompasses an estimated area of around 5.33 square kilometers, and its active thickness varies from 0 to 106.59 m. The maximum displacement distance reaches 1469.76 m, with a peak velocity of 60.37 m/s. The proposed method provides scientific support for assessing, analyzing, and preventing landslide disasters.

Artificial Neural Networks (ANNs) have emerged as a powerful tool in combustion simulations to replace memory-intensive tabulation of integrated chemical kinetics. Complex reaction mechanisms, however, present a challenge for standard ANN approaches as modeling multiple species typically suffers from inaccurate predictions on minor species. This paper presents a novel ANN approach which can be applied on complex reaction mechanisms in tabular data form, and only involves training a single ANN for a complete reaction mechanism. The approach incorporates a network architecture that automatically conserves mass and employs a particular loss weighting based on species depletion. Both modifications are used to improve the overall ANN performance and individual prediction accuracies, especially for minor species mass fractions. To validate its effectiveness, the approach is compared to standard ANNs in terms of performance and ANN complexity. Four distinct reaction mechanisms (H2, C7H16, C12H26, OME34) are used as a test cases, and results demonstrate that considerable improvements can be achieved by applying both modifications.

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.

A variety of mathematical models have been developed to simulate the biochemical and physico-chemical aspects of the anaerobic digestion (AD) process to treat organic wastes and generate biogas. However, all these models, including the most widely accepted and implemented Anaerobic Digestion Model No.1, remain incapable of adequately representing the material balance of AD and are therefore inherently incapable of material conservation. The absence of robust mass conservation constrains reliable estimates of any kinetic parameters being estimated by regression of empirical data. To address this issue, the present work involved the development of a “framework” for a mass-conserving atomistic mathematical model which is capable of mass conservation, with a relative error in the range of machine precision value and an atom balance with a relative error of ±0.02% whilst obeying the Henry’s law and electroneutrality principle. Implementing the model in an Excel spreadsheet, the study calibrated the model using the empirical data derived from batch studies. Although the model shows high fidelity as assessed via inspection, considering several constraints including the drawbacks of the model and implementation platform, the study also provides a non-exhaustive list of limitations and further scope for development.

In this study, we propose positivity preserving conservative characteristic methods with temporal second-order and spatial fourth-order accuracy for solving convection dominated diffusion problems. The method of characteristics is utilized to avoid strict restrictions on time step sizes, providing greater flexibility in computation. To preserve mass, conservative piecewise parabolic interpolation is used to obtain values at tracking points. Additionally, we leverage the finite difference implementation of the continuous finite element method and construct various fourth-order approximation operators for the Laplace operator, which are then applied to develop conservative numerical schemes with positivity preserving property. The proposed methods are theoretically proven to preserve the positivity property of solutions and ensure mass conservation. Numerical examples are conducted to validate the performance of developed schemes, demonstrating their spatial and temporal convergence orders, as well as conservation property. [ABSTRACT FROM AUTHOR]

In the conventional finite element method, only a few types of elements can directly use the weight functions to generate a proper, diagonally lumped mass matrix (LMM), which satisfies mass conservation and positive definiteness. This study proposes a new set of weight functions that can be efficiently constructed and used for generating a proper LMM based on classical 6‐, 9‐, and 10‐node triangular elements. Furthermore, we prove that these new elements are all complete and consistent, which ensures the convergence of the numerical solution. Finally, some modal and forced/free vibration analyses are conducted to demonstrate the computational accuracy and validation of the proposed LMM. [ABSTRACT FROM AUTHOR]

In this work, we introduce a weighted finite volume scheme for multiple fragmentation problems and report a convergence criterion of the scheme. It is observed that the finite volume method mentioned in Kumar and Kumar (Appl Math Comput 219(10):5140–5151, 2013) has not estimated the physical moments of clusters with satisfactory precision. Therefore, to control this deficiency, a weight function, and a correction factor are introduced in the numerical flux to approximate the conservative formulation of the multiple fragmentation equation. The proposed scheme preserves the first two physical moments with high accuracy in the cell overlapping case for newly born clusters. It is shown that the new formulation converges weakly under certain growth restrictions on the kernels. Finally, simulation results and numerical validations are presented. [ABSTRACT FROM AUTHOR]

The nonlocal Cahn‐Hilliard equation has attracted much attention these years. Despite the advantage of describing more practical phenomena for modeling phase transitions of microstructures in materials, the nonlocal operator in the equation brings a lot of extra computational costs compared with the local Cahn‐Hilliard equation. Thus high order time integration schemes are needed in numerical simulations. In this paper, we propose two classes of exponential time differencing (ETD) schemes for solving the nonlocal Cahn‐Hilliard equation. We first use the Fourier collocation method to discretize the spatial domain, and then the ETD‐based multistep and Runge‐Kutta schemes are adopted for the time integration. In particular, some specific multistep and Runge‐Kutta schemes up to fourth order are constructed. We rigorously establish the energy stabilities of the multistep schemes up to fourth order and the second order Runge‐Kutta scheme, which show that the first order ETD and the second order Runge‐Kutta schemes unconditionally decrease the original energy. We also theoretically prove the mass conservations of the proposed schemes. Several numerical experiments in two and three dimensions are carried out to test the temporal convergence rates of the schemes and to verify their mass conservations and energy stabilities. The long time simulations of coarsening dynamics are also performed to verify the power law for the energy decay. [ABSTRACT FROM AUTHOR]

The predictor-corrector-type (P-C) numerical solution to the 1D Richards equation only requires one matrix inversion operation per time step, making it attractive in terms of computational cost. However, the mass conservation could be violated at the saturated-unsaturated interface. A new post-allocation procedure is designed for the P-C method, which redistributes moisture after the corrector step to achieve strict mass balance. A novel adaptive time-stepping strategy is proposed to further improve model efficiency and robustness. It adjusts time step size based on both moisture difference and the Courant number. The proposed solution method and time control strategies are tested and compared with an analytical solution, the previous P-C solution and other existing iterative solutions. The new method shows good conservation property and good agreements to the existing solutions. Compared to the iterative methods that occasionally experience convergence issues, the proposed P-C method is more robust. The new time-control strategy improves computational efficiency compared to the original P-C method, but it remains less efficient than iterative methods for most of the tested scenarios because of its explicit treatment of the corrector step.

The predictor-corrector-type (P-C) numerical solution to the 1D Richards equation only requires one matrix inversion operation per time step, making it attractive in terms of computational cost. However, the mass conservation could be violated at the saturated-unsaturated interface. A new post-allocation procedure is designed for the P-C method, which redistributes moisture after the corrector step to achieve strict mass balance. A novel adaptive time-stepping strategy is proposed to further improve model efficiency and robustness. It adjusts time step size based on both moisture difference and the Courant number. The proposed solution method and time control strategies are tested and compared with an analytical solution, the previous P-C solution and other existing iterative solutions. The new method shows good conservation property and good agreements to the existing solutions. Compared to the iterative methods that occasionally experience convergence issues, the proposed P-C method is more robust. The new time-control strategy improves computational efficiency compared to the original P-C method, but it remains less efficient than iterative methods for most of the tested scenarios because of its explicit treatment of the corrector step.

In this paper, we consider a decoupled and mass-conservative numerical scheme for the incompressible miscible displacement problem in porous media by employing multipoint flux mixed finite element method for velocity-pressure equation and mass-conservation characteristic finite element method for concentration equation. This feature is designed to avoid handling saddle-point system and keep mass globally. Convergence and L2 error estimates of full discrete scheme are obtained. Numerical experiments are also presented to verify the correctness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we develop a generalized moment method with a continuous weight function for the Smoluchowski coagulation equation in its continuous form to study the mass conservation property of this equation. We first establish some basic inequalities for the generalized moment and prove the mass conservation property under a sufficient condition on the kernel and an initial condition, utilizing these inequalities. Additionally, we provide some concrete examples of coagulation kernels that exhibit mass conservation properties and show that these kernels exhibit either polynomial or exponential growth along specific particular curves. [ABSTRACT FROM AUTHOR]

The two-fluid model is widely used to describe two-phase flows in complex systems such as nuclear reactors. Although the two-phase flow was successfully simulated, the standard two-fluid model suffers from an ill-posed nature. There are several remedies for the ill-posedness of the one-dimensional (1D) two-fluid model; among those, artificial viscosity is the focus of this study. Some previous works added artificial diffusion terms to both mass and momentum equations to render the two-fluid model well-posed and demonstrated that this method provided a numerically converging model. However, they did not consider mass conservation, which is crucial for analyzing a closed reactor system. In fact, the total mass is not conserved in the previous models. This study improves the artificial viscosity model such that the 1D incompressible two-fluid model is well-posed, and the total mass is conserved. The water faucet and Kelvin-Helmholtz instability flows were simulated to test the effect of the proposed artificial viscosity model. The results indicate that the proposed artificial viscosity model effectively remedies the ill-posedness of the two-fluid model while maintaining a negligible total mass error.

The anode recirculation mode is increasingly being adopted in today's fuel cell systems. The recycling of hydrogen gas can effectively improve fuel utilization and the wider economy. However, using the purge strategy for the recirculation exhaust has a significant impact on the operational performance and economic efficiency of fuel cell systems.Experiments have shown that, when the purge interval increases from 6 s to 10 s, the recirculation pump power increases by about 20%, the nitrogen content in the exhaust gas increases, and the stack voltage shows a 10 V attenuation. The accumulation of nitrogen permeation in the anode circuit leads to the degradation of the fuel cell performance. Therefore, it is necessary to discharge the accumulated nitrogen through the purge valve in a timely manner. However, opening the exhaust valve with excessively high frequency can result in the unreacted hydrogen being discharged, which reduces the economic efficiency of the fuel cell. This paper is based on the principle of mass conservation and models each subsystem of the anode circuit in the recirculation pump mode of the fuel cell separately, including the proportional valve model, the hydrogen consumption model of the fuel cell, the nitrogen permeation model of the fuel cell, the neural network model of the circulating pump, and the purge valve model. These submodels are integrated to construct a nitrogen content observer for the hydrogen circuit, which can estimate the nitrogen content. The accuracy of the model is validated through experimental data. The estimation error is less than 5.5%. The nitrogen content in the anode circuit can be effectively estimated, providing a model reference for purge operations and improving hydrogen utilization. [ABSTRACT FROM AUTHOR]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [ABSTRACT FROM AUTHOR]

We propose and analyze an H(div) conforming discontinuous galerkin (CDG) method for the three-field Biot's consolidation model with the displacement reconstruction technique. The displacement is discretized by the k th-order Brezzi-Douglas-Marini (BDM) element, and the fluid flux and pore pressure are approximated by the k-1 th-order Raviart-Thomas-Nedelec element pairs. The H(div) CDG method is derived from the H(div) conforming formulation by replacing the classical gradient operator with a weak gradient operator, where the weak one is locally computed by the k th-order Raviart-Thomas (RT) element. We prove optimal a-priori error estimates for both semi-discrete scheme and fully-discrete scheme with the backward Euler discretization in time. The implicit constants in the error estimates are robust for the arbitrarily large Lamé coefficient and the arbitrarily small constrained specific storage coefficient. Our method has no stabilizers, which simplifies the finite element formulations. Meanwhile, it can also overcome the poroelasticity locking mathematically and preserve the pointwise mass conservation on the discrete level. The validity and accuracy of the proposed method are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

We discuss the formulation of a numerical scheme based on the homotopy method to solve different aggregation–fragmentation models including the simultaneous event. Several test cases are considered and analyzed qualitatively and quantitatively to ascertain the improved accuracy and efficiency of the proposed model over the existing semi‐analytical models. The generalized solution of the truncated problem is obtained for some test cases, which in the limiting sense tends to the exact solution. A detailed convergence analysis of the scheme is also studied. [ABSTRACT FROM AUTHOR]

We develop a class of explicit unconditionally maximum-principle-preserving multi-step methods for the Allen-Cahn-type semilinear parabolic equations. Based on the usual second-order finite difference space discretization, we first show that the strong-stability-preserving (SSP) integrating factor multi-step method can preserve the maximum-principle under the condition τ ≤ C τ F E , where C is the SSP coefficient and τ F E is independent of the space step size, thus avoiding the strict parabolic Courant-Friedrichs-Lewy condition τ = O (h 2). By introducing the stabilization technique and replacing the exponential functions with two different approximations, we obtain two types of improved stabilized integrating factor multi-step (isIFMS) schemes. The unconditional maximum-principle-preservation, mass-conservation, linear stability, and error estimates are analyzed for the isIFMS schemes. Since explicit SSP multi-step methods have no order barrier, the proposed framework offers a concise but effective approach for developing temporal high-order, unconditionally maximum-principle-preserving schemes for Allen-Cahn-type equations, and mass-conserving schemes for their conservative reformulations. Finally, numerical experiments validate theoretical results of the proposed isIFMS schemes. [ABSTRACT FROM AUTHOR]

We consider a mass conservative type method for semiconductor device problem by employing mixed finite element method (FEM) for electric potential equation and mass conservative characteristic FEM for both electron and hole density equations. The boundedness of numerical solution without certain time‐step restriction and optimal L2$$ {L}^2 $$ error estimates of full discrete scheme are proved. Numerical experiment is presented to verify the effectiveness and unconditional stability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a strongly mass conservative and stabilizer-free scheme is designed and analyzed for the coupled Brinkman--Darcy flow and transport. The flow equations are discretized by using a strongly mass conservative scheme in mixed formulation with a suitable incorporation of the interface conditions. In particular, the interface conditions can be incorporated into the discrete formulation naturally without introducing additional variables. Moreover, the proposed scheme behaves uniformly robust for various values of viscosity. A novel upwinding staggered discontinuous Galerkin scheme in mixed form is exploited to solve the transport equation, where the boundary correction terms are added to improve the stability. A rigorous convergence analysis is carried out for the approximation of the flow equations. The velocity error is shown to be independent of the pressure and thus confirms the pressure-robustness. Stability and a priori error estimates are also obtained for the approximation of the transport equation; moreover, we are able to achieve sharp stability and convergence error estimates thanks to the strong mass conservation preserved by our scheme. In particular, the stability estimate depends only on the true velocity on the inflow boundary rather than on the approximated velocity. Several numerical experiments are presented to verify the theoretical findings and demonstrate the performances of the method. [ABSTRACT FROM AUTHOR]

It has been proposed that conservation laws might not be beneficial for accurate hydrological modelling due to errors in input (precipitation) and target (streamflow) data (particularly at the event time scale), and this might explain why deep learning models (which are not based on enforcing closure) can out‐perform catchment‐scale conceptual and process‐based models at predicting streamflow. We test this hypothesis with two forcing datasets that disagree in total, long‐term precipitation. We analyse the roll of strictly enforced mass conservation for matching a long‐term mass balance between precipitation input and streamflow output using physics‐informed (mass conserving) machine learning and find that: (1) enforcing closure in the rainfall‐runoff mass balance does appear to harm the overall skill of hydrological models; (2) deep learning models learn to account for spatiotemporally variable biases in data (3) however this 'closure' effect accounts for only a small fraction of the difference in predictive skill between deep learning and conceptual models. [ABSTRACT FROM AUTHOR]

The possible occurrence of instantaneous gelation to the Oort–Hulst–Safronov (OHS) coagulation equation is investigated for a certain class of unbounded coagulation kernels. The existence of instantaneous gelation is confirmed by showing the non-existence of mass-conserving weak solutions. Finally, it is shown that for such kernels, there is no weak solution to the OHS coagulation equation at any time interval. [ABSTRACT FROM AUTHOR]