Mass Conservation Properties Research Articles

A split-step finite element method for the space-fractional Schrödinger equation in two dimensions

In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability.

Energetic spectral-element time marching methods for phase-field nonlinear gradient systems

We propose two efficient energetic spectral-element methods in time for marching nonlinear gradient systems with the phase-field Allen–Cahn equation as an example: one fully implicit nonlinear method and one semi-implicit linear method. Different from other spectral methods in time using spectral Petrov-Galerkin or weighted Galerkin approximations, the presented implicit method employs an energetic variational Galerkin form that can maintain the mass conservation and energy dissipation property of the continuous dynamical system. Another advantage of this method is its superconvergence. A high-order extrapolation is adopted for the nonlinear term to get the semi-implicit method. The semi-implicit method does not have superconvergence, but can be improved by a few Picard-like iterations to recover the superconvergence of the implicit method. Numerical experiments verify that the method using Legendre elements of degree three outperforms the 4th-order implicit-explicit backward differentiation formula and the 4th-order exponential time difference Runge-Kutta method, which were known to have best performances in solving phase-field equations. In addition to the standard Allen–Cahn equation, we also apply the method to a conservative Allen–Cahn equation, in which the conservation of discrete total mass is verified. The applications of the proposed methods are not limited to phase-field Allen–Cahn equations. They are suitable for solving general, large-scale nonlinear dynamical systems.

Mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces.

A space-time Petrov-Galerkin method for the two-dimensional regularized long-wave equation

In this work, a space-time Petrov-Galerkin (STPG) method is used to numerically analyze the two-dimensional regularized long-wave (RLW) equation. The STPG method is a nonstandard finite element method, that is, both of the spatial and temporal variables of this method are discretized by finite element method. Therefore, it can display the superiority of the finite element method in both spatial and temporal directions. In particular, it is easy to obtain space-time high accuracy and is unconditionally stable, thus this method is very appropriate for solving the nonlinear convection-diffusion equations. We prove the existence and uniqueness of the approximate solution and derive an optimal order error estimate in the maximum norm without requiring any compatibility between the space and time mesh size. In addition, some numerical experiments are presented to validate the accuracy and efficiency of the proposed method. Also, we provide several numerical experiments to confirm the conservation properties of discrete mass, energy and momentum for both the single and double-soliton waves.

On the existence and uniqueness of solution to a stochastic Chemotaxis–Navier–Stokes model

In this article, we study a mathematical system modelling the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis–Navier–Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is positive and satisfies the mass conservation property and an energy inequality.

Correction: Islam et al. Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property. Mathematics 2023, 11, 2770

There were some errors in the original publication [...]

A Level Set-Based Solver for Two-Phase Incompressible Flows: Extension to Magnetic Fluids

Development of a two-phase incompressible solver for magnetic flows in the magnetostatic case is presented. The proposed numerical toolkit couples the Navier–Stokes equations of hydrodynamics with Maxwell's equations of electromagnetism to model the behaviour of magnetic flows in the presence of a magnetic field. To this end, a rigorous implementation of a second-order two-phase solver for incompressible nonmagnetic flows is introduced first. This solver is implemented in the finite-difference framework, where a fifth-order conservative level set method is employed to capture the evolution of the interface, along with an incompressible solver based on the projection scheme to model the fluids. The solver demonstrates excellent performance even with high density ratios across the interface (Atwood number ≈ 1 ), while effectively preserving the mass conservation property. Subsequently, the numerical discretisation of Maxwell's equations under the magnetostatic assumption is described in detail, utilising the vector potential formulation. The primary second-order solver for two-phase flows is extended to the case of magnetic flows, by incorporating the Lorentz force into the momentum equation, accounting for high magnetic permeability ratios across the interface. The implemented solver is then utilised for examining the deformation of ferrofluid droplets in both quiescent and shear flow regimes across various susceptibility values of the droplets. The results suggest that increasing the susceptibility value of the ferrofluid droplet can affect its deformation and rotation in low capillary regimes. In higher capillary flows, increasing the magnetic permeability jump across the interface can further lead to droplet breakup as well. The effect of this property is also investigated for the Rayleigh–Taylor instability growth in magnetic fluids.

A highly efficient semi-implicit corrective SPH scheme for 2D/3D tumor growth model

In this work, a semi-implicit corrective SPH method is proposed to solve the multi-component Cahn–Hilliard equation with fourth-order derivatives, and it is further used to predict the high-dimensional tumor growth model. The scheme is motivated by: (a) the fourth-order spatial derivative is discretized continuously by a corrective SPH formula for approximating second-order derivative twice, and the Neumann boundary is imposed by a ghost technique; (b) the temporal direction is approximated by the implicit scheme, and an iterative concept is employed to handle the above implicit form; (c) the multi-CPUs MPI parallelization is adopted to reduce the computing cost. Firstly, the second-order convergence rate of the proposed method for 2D/3D equation is shown and discussed by two analytical examples, and the mass conservation and energy properties are also demonstrated. Secondly, the efficiency of the proposed approach for multi-phase separation phenomenon is illustrated in an irregular domain. Finally, the 2D/3D tumor growth evolution at a short time is predicted by the proposed method and qualitatively compared with other numerical results. The numerical experiments show that the proposed scheme for phase-separation phenomenon or tumor growth model is highly efficient and reliable.

Spectral deferred correction method for Landau–Brazovskii model with convex splitting technique

The Landau–Brazovskii model is a well-known Landau model for finding the complex phase structures in microphase-separating systems ranging from block copolymers to liquid crystals. It is critical to design efficient numerical schemes for the Landau–Brazovskii model with energy dissipation and mass conservation properties. Here, we propose a mass conservative and energy stable scheme by combining the spectral deferred correction (SDC) method with the convex splitting technique to solve the Landau–Brazovskii model efficiently. An adaptive correction strategy for the SDC method is implemented to reduce the cost time and preserve energy stability. Numerical experiments, including two- and three-dimensional periodic crystals in Landau–Brazovskii model, are presented to show the efficiency of the proposed numerical method.

Generalized Moment Method for Smoluchowski Coagulation Equation and Mass Conservation Property

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.

A mass‐ and energy‐preserving numerical scheme for solving coupled Gross–Pitaevskii equations in high dimensions

AbstractThis article is concerned with numerical study of a coupled system of Gross–Pitaevskii equations which describes the spin‐orbit‐coupled Bose–Einstein condensates. Due to the fact that this system possesses the total mass and energy conservation property and often appears in high dimensions, it brings a significant burden in designing and analyzing a suitable numerical scheme for solving the coupled Gross–Pitaevskii equations (CGPEs). In this article, an implicit finite difference scheme is proposed to solve the CGPEs, which is proved to be uniquely solvable, mass‐ and energy‐conservative in the discrete sense. In particular, it is proved in a rigorous way that, without any grid‐ratio restriction, the scheme is stable and convergent at the rate of with time step and mesh size in the maximum norm, while previous works often require certain restriction on the grid ratio and only give the error estimates in the discrete norm or norm which could not imply the maximum error estimate. Numerical results are carried out to underline the error estimate and conservation laws, and investigate several dynamics of the CGPEs.

A novel approach of unconditional optimal error estimate of linearized and conservative Galerkin FEM for Klein–Gordon–Schrödinger equations

This paper is devoted to the unconditional optimal error analysis of a linearized, decoupled and conservative Galerkin finite element method (FEM) for the Klein–Gordon–Schödinger equations. The present work has two main contributions: One is that the mass and energy conservation properties of the scheme are proved, and a priori boundedness of the numerical solutions is derived. The other one is that a novel approach is developed and analyzed, which can avoid the use of popular error splitting technique and the restrictions between the time step size and mesh size required in the previous works, and may lead to the unconditional optimal error estimates for the related variables. Some numerical results are also provided to confirm the theoretical analysis and show the efficiency of the proposed scheme.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

A high‐order stabilized solver for the volume averaged Navier‐Stokes equations

AbstractThe volume‐averaged Navier‐Stokes equations are used to study fluid flow in the presence of fixed or moving solids such as packed or fluidized beds. We develop a high‐order finite element solver using both forms A and B of these equations. We introduce tailored stabilization techniques to prevent oscillations in regions of sharp gradients, to relax the Ladyzhenskaya–Babuska–Brezzi inf‐sup condition, and to enhance the local mass conservation and the robustness of the formulation. We calculate the void fraction using the particle centroid method. Using different drag models, we calculate the drag force exerted by the solids on the fluid. We implement the method of manufactured solution to verify our solver. We demonstrate that the model preserves the order of convergence of the underlying finite element discretization. Finally, we simulate gas flow through a randomly packed bed and study the pressure drop and mass conservation properties to validate our model.

An efficient discontinuous Galerkin - mixed finite element model for variable density flow in fractured porous media

Modeling variable density flow (VDF) in fractured porous media is computationally challenging. The challenges are mainly related to: (i) the high permeability contrast between the matrix and the fractures and, (ii) the nonlinearity induced by density variations. Due to their local mass conservation property, cell-centered methods, such as finite volumes (FV), mixed finite elements (MFE) or Discontinuous Galerkin (DG) methods are well suited for modeling mass transfer in highly heterogeneous domains. When applied for fractured media, these methods require small grid cells next to fractures because of the cross-flow equilibrium assumption (i.e. the pressure and concentration in the fracture and in the adjacent matrix grid-cells are assumed the same). To avoid this constraint, an efficient model is developed in this work using advanced cell-centered numerical methods for VDF in fractured porous media with cross-flow equilibrium assumed only across the fractures. The new model uses the hybrid MFE method for the flow discretization in the matrix and in the fracture continua. Mass transport in the matrix is modeled using a monotonic upwind MFE scheme. Advection-dominated transport in fractures is discretized with the DG method, which is well adapted for hyperbolic equations. The new model ensures continuity of pressure, concentration, fluid mass flux, advective and dispersive contaminant fluxes at matrix-fracture interfaces as well as at the intersection of several fractures. The time discretization is performed using high-order adaptive time integration techniques via the method of lines (MOL). The developed model is first validated by comparison against a 2D-2D model for a test problem dealing with linear flow and transport in a 2D domain involving a “+”-shaped barrier/fracture network. Then, the MFE-DG model is used for the simulation of the Henry saltwater intrusion problem and the results are validated by comparison against the semi-analytical solution in the case of unfractured and fractured aquifers. Finally, the simulation of a transport problem with variable viscosity in a fractured heterogeneous domain points out the superiority of the new model in terms of accuracy and efficiency when compared to a standard finite element model.

Generalized Allen–Cahn-type phase-field crystal model with FCC ordering structure and its conservative high-order accurate algorithm

In this paper, a generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure is presented. By introducing a space-time dependent Lagrange multiplier, the mass conservation is enforced and the local effect can be reflected. Moreover, the proposed model is lower-order in space and will be more efficient for numerical computation. For the simulation of crystallization, the main challenge is to design practical and stable algorithm which meanwhile guarantees the high-order accuracy. To resolve this problem, we propose a temporally high-order (up to third-order) accurate and linear multi-stage scheme based on implicit-explicit Runge–Kutta approach. The nonlinear term is explicitly treated and the appropriate stabilizer is added to balance the stiffness. By utilizing the Fourier spectral method in space, the numerical calculation is easy to implement. We analytically proved the discrete mass conservation property based on the fully discrete scheme. Extensive numerical experiments are performed to validate the accuracy, efficiency, and capability for phase transition. The effects of parameters and local interaction are investigated in detail. To facilitate the interested readers, we provide MATLAB codes of our proposed method in Appendix A.

A Flux-Form Semi-Lagrangian advection scheme for tracer transport on arbitrary meshes

A variant of the flux-form semi-Lagrangian (FFSL) advection scheme is presented suitable for use on arbitrary unstructured meshes. The time-step used with the scheme is not constrained by the Courant number, resulting in model run-times that may be many times faster than Courant-constrained schemes. The scheme also has the property of mass conservation. This makes the scheme a potential candidate for offline simulation of tracers in unstructured sediment transport, biogeochemical or atmospheric chemistry models. We describe the numerics of the scheme and apply it to a number of idealised and real applications to demonstrate its utility. In these test cases we compare results from the offline FFSL scheme with those generated by the original hydrodynamic model which uses high order conservative tracer transport. We assess the FFSL scheme for accuracy, conservation and computational efficiency. Additionally, traditional semi-Lagrangian advection schemes are included in assessments for comparison. Results indicate that the FFSL scheme produces closer agreement with tracer distributions generated by the hydrodynamic model than the semi-Lagrange scheme, and mass conservation was vastly superior. The FFSL scheme was also an order of magnitude faster than the hydrodynamic model in certain cases. The FFSL scheme presented here therefore offers a viable mass-conserving, computationally-efficient option for use in offline transport models on arbitrary unstructured meshes.

A free surface flow solver based on an efficient improvement to a coupling method for interface computations

In this paper, a free surface flow solver based on an improved coupling method for practical, highly nonlinear, complex free-surface flows is presented. The coupling method which is based on taking the advantages of the continuity and boundedness properties of the conservative level-set (LS) function and the mass conservation property of the VOF approach is integrated into a Navier-Stokes solution to build a new free surface solver. The developed solver is then validated with several practical free surface flows of engineering applications. The test cases are selected so that the developed solver is challenged in simulations of flows with a large scale of density ratio, high-nonlinear, and complicated free surface in a wide range of engineering applications to explore its capability. The simulated flow fields are presented and compared with the available experimental data and of other numerical simulations. Good agreement is achieved implying the capability of the developed solver, and also exhibiting the ability to extend to other applications.

Three dimensional finite element computation of the non-isothermal polymer filling process by the phase field model

In this paper, we numerically study the three dimensional non-isothermal non-Newtonian filling process using the phase field method. The Cahn-Hilliard equation is firstly attempted to capture the moving melt front between the Newtonian and non-Newtonian fluid. The governing equations of the flow field are able to be written in a unified form on the basis of phase field parameter. The unified Navier-Stokes equations are decoupled via the splitting scheme and then the traditional FEM and SUPG method are employed to solve the suitable equations. The Cross-WLF model is utilized to describe the variation of the viscosity of the polymer melt in the non-isothermal condition. We take the thin wall rectangular cavity as example and compare with the experimental data to illustrate the convergence, robustness and accuracy of numerical algorithm. In addition, we consider the thin wall cavity with a cylindrical obstacle and analyze the influences of the inlet velocity and the size of the injection gate on the filling process. The numerical results agree well with the experimental data and also exhibit good mass conservation property.

A Posteriori Error Estimates of Edge Residual-Type of Weak Galerkin Mixed FEM Solving Second-Order Elliptic Problems on Polytopal Mesh

In this paper, we present a posteriori error estimator of edge residual-type Weak Galerkin mixed finite element method (WG-MFEM) solving second-order elliptic problems, where two different ways of a posteriori error estimator are presented, both of which hold on polygonal mesh. The two analyses are based on the standard FEM method and continuous local mass conservation property, respectively. In order to prove the reliability holding on polygonal meshes, the a posteriori error estimator is disposed by the post-processed numerical scalar solution. Compared with some other methods, the proposed a posteriori error estimator does not contain the post-processed solution and the efficiency can be obtained automatically. Numerical experiments are conducted to confirm the efficiency of this estimator.