Energy Stable Schemes Research Articles

A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model

A Second Order Accurate in Time, Energy Stable Finite Element Scheme for a Liquid Thin Film Coarsening Model

Decoupled and energy stable schemes for phase-field surfactant model based on mobility operator splitting technique

Decoupled and energy stable schemes for phase-field surfactant model based on mobility operator splitting technique

A Relaxed Decoupled Second-Order Energy-Stable Scheme for the Binary Phase-Field Crystal Model

A Relaxed Decoupled Second-Order Energy-Stable Scheme for the Binary Phase-Field Crystal Model

Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows

The scalar auxiliary variable (SAV) approach was considered by Shen et al. in Shen et al. (2019) and has been widely used to simulate a series of gradient flows. However, the SAV-based schemes are known for the stability of a ‘modified’ energy. In this paper, we construct a series of modified SAV approaches with unconditional energy dissipation law based on several improvements to the classic SAV approach. Firstly, by introducing the three-step technique, we can reduce the number of constant coefficient linear equations that need to be solved at each time step, while retaining all of its other advantages. Secondly, the addition of energy-optimal technique and Lagrange multiplier technique can make the numerical schemes have the advantage of preserving the original energy dissipation. Thirdly, we use the first-order approximation of the energy balance equation in the GSAV approach, instead of discretizing the dynamic equation of the auxiliary variable, so that we can construct the high-order unconditional original energy stable numerical schemes. Finally, representative numerical examples show that the efficiency and accuracy of the proposed schemes are improved.

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Unconditional Energy Stable Runge-Kutta Schemes for a Phase Field Model for Diblock Copolymers

Two linear energy stable lumped mass finite element schemes for the viscous Cahn–Hilliard equation on curved surfaces in 3D

The evolution of a dynamic system on complex curved 3D surfaces is essential for the understanding of natural phenomena, the development of new materials, and engineering design optimization. In this work, we study the viscous Cahn–Hilliard equation on curved surfaces and develop two linear energy stable finite element schemes based on the lumped mass method. Two stabilizing terms are added to ensure both the unique solvability and unconditional energy stability. We prove rigorously that two schemes are unconditionally energy stable . Numerical experiments are presented to verify theoretical results and to show the robustness and accuracy of the proposed method.

Numerical Analysis for a Non-isothermal Incompressible Navier–Stokes–Allen–Cahn System

In this paper we develop the numerical analysis for a non-isothermal diffuse-interface model, in dimension N=2,3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2, 3,$$\\end{document} 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.

High-order energy stable discrete variational derivative schemes for gradient flows

Abstract The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.

Step-by-step solving virtual element schemes based on scalar auxiliary variable with relaxation for Allen–Cahn-type gradient flows

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.

Efficient energy stable schemes for incompressible flows with variable density

We present novel numerical schemes for the incompressible Navier-Stokes equations with variable density and address two critical concerns, i.e., the preservation of lower density bounds and the preservation of energy inequality under the gravitational force. Spatial discretization is performed using upwind discontinuous Galerkin methods for density, and continuous finite element methods are used for velocity and pressure. Additionally, velocity fields are projected onto divergence-free Raviart-Thomas finite element space in the transport equation. This relaxes using divergence-free finite elements and therefore simplifies the implementation. The proposed schemes are tested on benchmark examples to demonstrate the excellent performance in accurately capturing the complex dynamics of these problems. Overall, the proposed numerical schemes exhibit enhanced stability, efficiency, and accuracy, making them suitable for modeling and simulating incompressible flows with variable density in practical applications.

Energy-stable auxiliary variable viscosity splitting (AVVS) method for the incompressible Navier–Stokes equations and turbidity current system

In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

A second-order linear unconditionally energy-stable scheme for the phase field crystal equation

This paper presents a linear implicit scheme for numerically solving the phase field crystal (PFC) equation. The scheme is constructed by the scalar auxiliary variable (SAV) approach and the leapfrog scheme in temporal direction. The proposed scheme is decoupled, and it is easy to be implemented. In contrast, the previous SAV-type schemes for the PFC equation are usually coupled. The scheme satisfies the energy stability at the discrete level. Moreover, it is proved that the scheme is convergent with second-order accuracy in the temporal direction. Several numerical examples are carried out to confirm the theoretical results.

Structure-preserving and efficient numerical simulation for diffuse interface model of two-phase magnetohydrodynamics

In this paper, we propose a novel diffuse interface model of two-phase magnetohydrodynamics (MHD) based on a magnetic vector potential formulation in the three-dimensional case. This model ensures an exact divergence-free approximation of the magnetic field by introducing a magnetic vector potential A and defining the magnetic field by B=curlA. The resulting framework constitutes a highly coupled, nonlinear saddle point system consisting of the Cahn–Hilliard system and MHD potential system. To solve the model efficiently, we present two fully decoupled, first-order, linear, and unconditionally energy-stable schemes and strictly prove their well-posedness and energy stability. Finally, we present several numerical examples that demonstrate the stability and effectiveness of our schemes.

High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes

This paper develops high-order accurate well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers M⩾2) shallow water equations (SWEs) with non-flat bottom topography on both fixed and adaptive moving meshes, extending our previous work on the single-layer shallow water magnetohydrodynamics [25] and single-layer SWEs on adaptive moving meshes [58]. To obtain an energy inequality, the convexity of an energy function for an arbitrary M is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into M identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit strong stability preserving Runge-Kutta methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [58]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.

A class of efficient high-order time-stepping methods for the anisotropic phase-field dendritic crystal growth model

In this paper, we develop and analyze a novel class of arbitrarily high-order and unconditionally energy stable schemes for the anisotropic phase-field dendritic crystal growth model, which is a highly nonlinear system that combines the anisotropic Allen–Cahn equation with the thermal equation. The proposed schemes are based on an extrapolated and linearized Runge–Kutta method for an auxiliary variable reformulation of the crystal growth model. A delicate implementation demonstrates that the proposed method can be realized in a very efficient way, requiring only the solution of a coupled linear elliptic system at each time step. We prove that the constructed schemes satisfy the energy dissipation property and demonstrate the convergence order through a consistency error analysis. To the best of our knowledge, this is the first unconditionally stable scheme of arbitrarily high-order for the anisotropic phase-field dendritic crystal growth model. Numerical experiments for two and three dimensional dendritic crystal growth are simulated to verify our theoretical accuracy as well as the efficiency of the proposed method. Furthermore, to emphasize the superiority of the high-order schemes, we present a numerical comparison that directly contrasts its performance with the lower-order schemes.

Error analysis of a fully discrete projection method for Cahn–Hilliard Inductionless MHD problems

This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations

In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.

An Energy Stable Alternative WENO Scheme with a New Smoothness Indicator for Hyperbolic Conservation Laws

An Energy Stable Alternative WENO Scheme with a New Smoothness Indicator for Hyperbolic Conservation Laws

Second-order energy-stable scheme and superconvergence for the finite difference method on non-uniform grids for the viscous Cahn–Hilliard equation

Second-order energy-stable scheme and superconvergence for the finite difference method on non-uniform grids for the viscous Cahn–Hilliard equation