Energy Stable Schemes Research Articles

Highly Efficient and Energy Stable Schemes for the 2D/3D Diffuse Interface Model of Two-Phase Magnetohydrodynamics

Highly Efficient and Energy Stable Schemes for the 2D/3D Diffuse Interface Model of Two-Phase Magnetohydrodynamics

A High-Order and Unconditionally Energy Stable Scheme for the Conservative Allen–Cahn Equation with a Nonlocal Lagrange Multiplier

A High-Order and Unconditionally Energy Stable Scheme for the Conservative Allen–Cahn Equation with a Nonlocal Lagrange Multiplier

A chemorepulsion model with superlinear production: analysis of the continuous problem and two approximately positive and energy-stable schemes

We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying1∂tu−Δu−∇⋅(u∇v)=0inΩ,t>0,∂tv−Δv+v=upinΩ,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\left\\{ \\begin{array}{l} \\partial_t u - {\\Delta} u - \\nabla\\cdot (u\\nabla v)=0 \\ \\ \\text{ in}\\ {\\Omega},\\ t>0,\\\\ \\partial_t v - {\\Delta} v + v = u^p \\ \\ { in}\\ {\\Omega},\\ t>0, \\end{array} \\right. $$\\end{document}with p ∈ (1, 2), {Omega }subseteq mathbb {R}^{d} a bounded domain (d = 1, 2, 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for d = 1, 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables (u,v) under structured meshes, and the second one by using the auxiliary variable σ = ∇v and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

A novel fully-decoupled, linear, and unconditionally energy-stable scheme of the conserved Allen–Cahn phase-field model of a two-phase incompressible flow system with variable density and viscosity

In this article, we first use the nonlocal conserved Allen–Cahn dynamics to reformulate the phase-field model with variable density and viscosity for the two-phase incompressible flow system, and then construct a novel fully-decoupled, linear, and unconditionally energy stable scheme to solve the model. The development of the scheme is based on a combination of a novel decoupling technique, the penalty method of the Navier–Stokes equation, the quadratization method of linearizing the double-well potential, and the operator Strang-splitting method. The scheme is highly efficient, and it only needs to solve a series of fully-decoupled linear elliptic equations at each time step, in which the Allen–Cahn equation and the pressure Poisson equation only have constant coefficients. We rigorously prove the unconditional energy stability of the scheme and conduct numerical simulations in 2D and 3D to demonstrate the accuracy, stability, and effectiveness of the scheme numerically.

An energy stable linear numerical method for thermodynamically consistent modeling of two-phase incompressible flow in porous media

In this paper, we consider numerical approximation of a thermodynamically consistent model of two-phase flow in porous media, which obeys an intrinsic energy dissipation law. The model under consideration is newly-developed, so there is no energy stable numerical scheme proposed for it at present. This model consists of two nonlinear degenerate parabolic equations and a saturation constraint, but lacking an independent equation for the pressure, so for the purpose of designing efficient numerical scheme, we reformulate the model forms as well as the free energy function, and further prove the corresponding energy dissipation inequality. Based on the alternative reformulations, using the invariant energy quadratization approach and subtle semi-implicit treatments for the pressure and saturation, we for the first time propose a linear and energy stable numerical method for this model. The fully discrete scheme is devised combining the upwind approach for the phase mobilities and the cell-centered finite difference method. The unique solvability of numerical solutions and unconditional energy stability are rigorously proved for both the semi-discrete time scheme and the fully discrete scheme. Moreover, the scheme can guarantee the local mass conservation for both phases. We also show that the upwind mobility approach plays an essential role in preserving energy stability of the fully discrete scheme. Numerical results are presented to demonstrate the performance of the proposed scheme.

A linearly second-order, unconditionally energy stable scheme and its error estimates for the modified phase field crystal equation

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme, which is based on the Lagrange Multiplier approach and could also be viewed as a special case of the invariant energy quadratization approach, for the modified phase field crystal equation. At each time step, the proposed scheme only requires solving several linear equations. Rigorous proofs are presented to demonstrate the unique solvability, mass conservation, and unconditional energy stability of the scheme. Various numerical experiments in 2D and 3D are performed to validate the accuracy, unconditional energy stability and mass conservation of the proposed numerical strategy.

Fully decoupled and energy stable BDF schemes for a class of Keller-Segel equations

The Keller-Segel equations are widely used for describing chemotaxis in biology. Recently, first-order and second-order approximations for a class of Keller-Segel equations based on the gradient flow structure were proposed in [48]. Mass conservation, positivity and energy stability were proved for the first-order scheme, whereas for the second-order scheme the energy stability was not provided. Besides, an explicit-implicit treatment is performed to a non-convex and non-concave term −χρϕ, making their decoupled system could only be solved in sequence. In this paper, we propose new BDF schemes of first-order (BDF1) and second-order accuracy (BDF2 and EsBDF2): the coupled term −χρϕ involved in two equations of ρ and ϕ is fully explicitly treated, thus the discrete schemes could be computed in parallel. For the first-order scheme (BDF1) and the second-order scheme (BDF2), the standard backward differentiation formula is applied to approximate the origin continuous equations with a regularization term added to guarantee the unconditional energy stability. For the BDF1 scheme, the discrete system is proved to be energy stable with a constant restriction for the stabilized parameter. For the EsBDF2 scheme, which is different with the standard BDF2 scheme, the differential operators are carefully handled and some regularization terms are added to provide the energy stability. Several numerical examples are presented to verify the theoretical results.

High-order time-accurate, efficient, and structure-preserving numerical methods for the conservative Swift–Hohenberg model

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L2-gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.

A linear second-order in time unconditionally energy stable finite element scheme for a Cahn–Hilliard phase-field model for two-phase incompressible flow of variable densities

We propose a novel second-order BDF time stepping method of variable time step sizes combined with a classical residual-based stabilized finite element spatial discretization using the Streameline-Upwind Petrov–Galerkin (SUPG)/pressure stabilization Petrov–Galerkin (PSPG)/grad-div stabilization for solving the phase–field model for two-phase incompressible flow of different densities and viscosities in the advection dominated regime. In the case of uniform time step size and without extra stabilization, the scheme is shown to satisfy a discrete energy law. Benchmark test of the Rayleigh–Taylor instability under high Reynolds number and Péclect number demonstrates that the scheme captures details of the instability comparable to results in the literature by schemes based on sharp-interface models.

Chemical Potential-Based Modeling of Shale Gas Transport

Shale gas plays an increasingly important role in the current energy industry. Modeling of gas flow in shale media has become a crucial and useful tool to estimate shale gas production accurately. The second law of thermodynamics provides a theoretical criterion to justify any promising model, but it has been never fully considered in the existing models of shale gas. In this paper, a new mathematical model of gas flow in shale formations is proposed, which uses gas density instead of pressure as the primary variable. A distinctive feature of the model is to employ chemical potential gradient rather than pressure gradient as the primary driving force. This allows to prove that the proposed model obeys an energy dissipation law, and thus, the second law of thermodynamics is satisfied. Moreover, on the basis of energy factorization approach for the Helmholtz free energy density, an efficient, linear, energy stable semi-implicit numerical scheme is proposed for the proposed model. Numerical experiments are also performed to validate the model and numerical method.

Efficient Fully decoupled and second-order time-accurate scheme for the Navier–Stokes coupled Cahn–Hilliard Ohta–Kawaski Phase-Field model of Diblock copolymer melt

In this work, we aim to develop a highly efficient numerical scheme for the flow-coupled phase-field model of diblock copolymer melt. Formally, the model is a very complicated nonlinear system that consists of the Navier–Stokes equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through a combination of a novel decoupling technique and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme for this model. The decoupling technique is based on the design of an auxiliary ODE, which plays a vital role in obtaining the full decoupling structure while maintaining energy stability. The high efficiency of the scheme is not only reflected by its linear and decoupled structure but also because it only needs to solve a few elliptic equations at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

In this paper, we construct efficient schemes based on the scalar auxiliary variable (SAV) block-centered finite difference method for the modified phase field crystal (MPFC) equation, which is a sixth-order nonlinear damped wave equation. The schemes are linear, conserve mass and unconditionally dissipate a pseudo energy. We prove rigorously second-order error estimates in both time and space for the phase field variable in discrete norms. We also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy.

Unconditionally energy stable second-order numerical scheme for the Allen\u2013Cahn equation with a high-order polynomial free energy

In this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

A highly efficient and accurate exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach for dissipative system

The scalar auxiliary variable (SAV) approach [42] is a very popular and efficient method to simulate various phase field models. To save the computational cost, a new SAV approach is given in [25] by introducing a new variable θ. The new SAV approach can be proved to save nearly half CPU time of the original SAV approach while keeping all its other advantages. In this paper, we propose a novel technique to construct an exponential semi-implicit scalar auxiliary variable (ESI-SAV) approach without introducing any extra variables. The new proposed method also only needs to solve one linear equation with constant coefficients at each time step. Furthermore, the constructed ESI-SAV method does not need the bounded below restriction of nonlinear free energy potential which is more reasonable and effective for various phase field models. Meanwhile it is easy to construct first-order, second-order and higher-order unconditionally energy stable time-stepping schemes. Other than that, the ESI-SAV approach can be proved to be effective to solve the non-gradient but dissipative system such as Navier-Stokes equations. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

High order, semi-implicit, energy stable schemes for gradient flows

We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance.

Numerical error analysis for an energy-stable HDG method for the Allen–Cahn equation

This paper presents an energy-stable hybridizable interior penalty discontinuous Galerkin method for the Allen–Cahn equation. To obtain an unconditionally energy stable scheme, the energy potential is split into a sum of a convex and concave function. Energy stability for the proposed scheme is proven to hold for arbitrary time. Existence and uniqueness for the scheme is also established. Under standard assumptions on the energy potential (Lipschitz continuity), we demonstrate rigorously that the method converges optimally for symmetric schemes, and suboptimally for nonsymmetric schemes. Several examples are provided which numerically verify and validate the method.

A continuum and computational framework for viscoelastodynamics: I. Finite deformation linear models

This work concerns the continuum basis and numerical formulation for deformable materials with viscous dissipative mechanisms. We derive a viscohyperelastic modeling framework based on fundamental thermomechanical principles. Since most large deformation problems exhibit isochoric properties, our modeling work is constructed based on the Gibbs free energy in order to develop a continuum theory using pressure-primitive variables, which is known to be well-behaved in the incompressible limit. A set of general evolution equations for the internal state variables is derived. With that, we focus on a family of free energies that leads to the so-called finite deformation linear model. Our derivation elucidates the origin of the evolution equations of that model, which was originally proposed heuristically and thus lacked formal compatibility with the underlying thermodynamics. In our derivation, the thermodynamic inconsistency is clarified and rectified. A classical model based on the identical polymer chain assumption is revisited and is found to have non-vanishing viscous stresses in the equilibrium limit, which is counter-intuitive in the physical sense. Because of that, we then discuss the relaxation property of the non-equilibrium stress in the thermodynamic equilibrium limit and its implication on the form of free energy. A modified version of the identical polymer chain model is then proposed, with a special case being the model proposed by G. Holzapfel and J. Simo. Based on the consistent modeling framework, a provably energy stable numerical scheme is constructed for incompressible viscohyperelasticity using inf–sup stable elements. In particular, we adopt a suite of smooth generalization of the Taylor–Hood element based on Non-Uniform Rational B-Splines (NURBS) for spatial discretization. The temporal discretization is performed via the generalized-α scheme. We present a suite of numerical results to corroborate the proposed numerical properties, including the nonlinear stability, robustness under large deformation, and the stress accuracy resolved by the higher-order elements. Additionally, the pathological behavior of the original identical polymer chain model is numerically identified with an unbounded energy decaying. This again underlines the importance of demanding vanishing non-equilibrium stress in the equilibrium limit.

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

AbstractWe consider numerical approximations for a modified phase field crystal model with a strong nonlinear vacancy potential. Based on the invariant energy quadratization approach and stabilized strategies, we develop linear, unconditionally energy stable numerical schemes using the first‐order Euler method, the second‐order backward differentiation formulas and the second‐order Crank–Nicolson method, respectively. We rigorously prove the unconditional energy stability, the mass conservation of these three numerical schemes and carry out error estimates in time for the first‐order numerical scheme. Various numerical experiments in 2D and 3D are carried out to validate the accuracy, energy stability, mass conservation, and efficiency of the proposed schemes.

A Second Order Energy Stable BDF Numerical Scheme for the Swift–Hohenberg Equation

A Second Order Energy Stable BDF Numerical Scheme for the Swift–Hohenberg Equation

Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented.