Energy Stable Numerical Scheme Research Articles

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.

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.

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.

Convergence analysis of a decoupled pressure-correction SAV-FEM for the Cahn–Hilliard–Navier–Stokes model

In this paper, a linear, decoupled and unconditional energy-stable numerical scheme of Cahn–Hilliard–Navier–Stokes (CHNS) model is constructed and analyzed. We reformulate the CHNS model into an equivalent system based on the scalar auxiliary variable approach. The Euler implicit/explicit scheme is used for time discretization, that is, linear terms are implicitly processed, nonlinear terms are explicitly processed, and pressure-correction method of the Navier–Stokes equations is also adopted. In this way, we achieve the decoupling of phase field, velocity and pressure, and only need to solve a series of linear equations with constant coefficients at each time step, thereby improving computational efficiency. Then the finite element method is used for space discretization, and the finite element spaces of phase field, velocity and pressure are taken as Pl−Pl−Pl−1 respectively. We also verify that the proposed scheme satisfies the discrete energy dissipation law. The optimal error estimates of the fully discrete scheme is proved, which the phase field, velocity and pressure satisfy the first-order accuracy in time and the (ℓ+1,ℓ+1,ℓ)th order accuracy in space, respectively. Finally, numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Efficient energy stable numerical schemes for Cahn–Hilliard equations with dynamical boundary conditions

In this paper, we propose a unified framework for studying the Cahn–Hilliard equation with two distinct types of dynamic boundary conditions, namely, the Allen–Cahn and Cahn–Hilliard types. Using this unified framework, we develop a linear, second-order, and energy-stable scheme based on the multiple scalar auxiliary variables (MSAV) approach. We design efficient and decoupling algorithms for solving the corresponding linear system in which the unknown variables are intricately coupled both in the bulk and at the boundary. Several numerical experiments are shown to validate the proposed scheme, and to investigate the effect of different dynamical boundary conditions on the dynamics of phase evolution under different scenarios.

An effective numerical method for the vector-valued nonlocal Allen–Cahn equation

Nonlocal Allen–Cahn model and their numerical schemes have received great attention in the literature as nonlocal model becomes popular in various fields. Our main idea in this work is to consider the vector-valued nonlocal Allen–Cahn model, which is a coupled system of nonlinear partial differential equations. Then, with the help of the operator splitting method and finite difference method, a fully-decoupled and energy stable numerical scheme for the vector-valued nonlocal Allen–Cahn model is proposed. Furthermore, we prove the modified energy dissipation law is unconditionally guaranteed and it is closely related to classical energy up to O(τ). Finally, numerical examples are carried out to verify the efficiency and accuracy of the proposed scheme.

A second-order accurate and unconditionally energy stable numerical scheme for nonlinear sine-Gordon equation

In this work, a second-order finite difference method is proposed to solve a nonlinear sine-Gordon equation. The constructed implicit scheme is proved to be unconditionally energy stable. A linear iteration algorithm is used to solve this nonlinear numerical scheme, and we prove that this iteration algorithm is convergent with a negligible constraint for time step. By constructing a suitable high-precision numerical solution, and using the inverse inequality and refinement constraint Δt≤Ch, the error estimate in L∞(0,T;L∞) norm of the fully discrete scheme is obtained. Several numerical examples are given to confirm the sharpness of our theoretical analysis.

A third-order positivity-preserving and energy stable numerical scheme for the Cahn-Hilliard equation with logarithmic potential

A third-order positivity-preserving and energy stable numerical scheme for the Cahn-Hilliard equation with logarithmic potential

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

电荷输运相场模型二阶、 线性、 解捐、 能量稳 定的数值格式

电荷输运相场模型二阶、 线性、 解捐、 能量稳 定的数值格式

A ternary mixture model with dynamic boundary conditions.

The influence of short-range interactions between a multi-phase, multi-component mixture and a solid wall in confined geometries is crucial in life sciences and engineering. In this work, we extend the Cahn-Hilliard model with dynamic boundary conditions from a binary to a ternary mixture, employing the Onsager principle, which accounts for the cross-coupling between forces and fluxes in both the bulk and surface. Moreover, we have developed a linear, second-order and unconditionally energy-stable numerical scheme for solving the governing equations by utilizing the invariant energy quadratization method. This efficient solver allows us to explore the impacts of wall-mixture interactions and dynamic boundary conditions on phenomena like spontaneous phase separation, coarsening processes and the wettability of droplets on surfaces. We observe that wall-mixture interactions influence not only surface phenomena, such as droplet contact angles, but also patterns deep within the bulk. Additionally, the relaxation rates control the droplet spreading on surfaces. Furthermore, the cross-coupling relaxation rates in the bulk significantly affect coarsening patterns. Our work establishes a comprehensive framework for studying multi-component mixtures in confined geometries.

An energy-stable and conservative numerical method for multicomponent Maxwell–Stefan model with rock compressibility

Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this paper, taking molar densities as the primary unknowns rather than the pressure and molar fractions, we propose an alternative formulation of multicomponent Maxwell–Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation has a distinct feature that it follows an energy dissipation law, and namely, it is consistent with the second law of thermodynamics. Additionally, the formulation obeys the famous Onsager's reciprocal principle. An efficient energy-stable numerical scheme is constructed using the stabilized energy factorization approach for the Helmholtz free energy density and certain carefully designed formulations involving explicit and implicit mixed treatments for the coupling between molar densities, pressure, and porosity. We rigorously prove that the scheme inherits the energy dissipation law at the discrete level. The fully discrete scheme has the ability to ensure the mass conservation law for each component as well as preserve the Onsager's reciprocal principle. Numerical tests are conducted to verify our theories, and in particular, to demonstrate the good performance of the proposed scheme in energy stability and mass conservation as expected from our theories.

Decoupled second-order energy stable scheme for an electrohydrodynamic model with variable electrical conductivity

The motion of ions in the complex fluids widely appears in hydrodynamics, geodynamics and geophysics. Setting up suitable mathematical model, performing accurate and efficient numerical simulations are essential to understand the underlying physical principle in these phenomena. In this paper, we present a linear, second-order accurate in time, decoupled and unconditionally energy stable numerical scheme for solving an electrohydrodynamic power-law model with variable electrical conductivity. Based on a logarithm transformation for conductivity and the zero energy contribution property of the nonlinear coupling terms in the model, we derive an equivalent system by introducing a nonlocal auxiliary variable. The two-step backward differentiation scheme and the finite element method are used for the temporal and spatial discretizations, respectively. To decouple the system, the nonlocal splitting technique is employed to yield several linear subsystems, which can be solved very efficiently. Numerical simulations are carried out to demonstrate the second-order convergence in time and the unconditional energy stability. Moreover, the effects of Coulombic force and power-law fluid exponents are numerically investigated.

A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential

A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential

High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model

AbstractWe propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.

Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.

Modeling and inversion in acoustic-elastic coupled media using energy-stable summation-by-parts operators

Seismic data acquired at the seafloor are valuable in characterizing the subsurface and monitoring producing hydrocarbon fields. To fully use such data, a stable, accurate, and efficient numerical scheme is needed that accounts for acoustic and elastic wave propagation, their interaction at the seafloor interface, and for sources and receivers placed on either side of that interface. Existing methods either make incorrect assumptions or have high implementation and computational costs. We have developed a high-order finite-difference summation-by-parts framework for the acoustic-elastic wave equations in second-order form (in terms of displacements in the solid and an extension of velocity potential in the fluid). Our modified discretization of the elastic operator overcomes the dispersion errors known to plague displacement-based schemes in the high VP/ VS limit. We weakly impose boundary and interface conditions using simultaneous approximation terms, leading to an energy-stable numerical scheme that rigorously handles point injection and extraction. The fully discrete system is self-adjoint after a time reversal and a sign flip and is furthermore a high-order accurate discretization of the continuous problem. The self-adjointness ensures that forward and adjoint wavefields are computed with similar accuracy and simplifies gradient computation for inversion purposes. We find that our numerical scheme achieves accuracy comparable to a more computationally expensive spectral-element method and demonstrate its application to full-waveform inversion using the Marmousi2 model.

A discontinuous Galerkin method for the Camassa‐Holm‐Kadomtsev‐Petviashvili type equations

AbstractThis paper develops a high‐order discontinuous Galerkin (DG) method for the Camassa‐Holm‐Kadomtsev‐Petviashvili (CH‐KP) type equations on Cartesian meshes. The significant part of the simulation for the CH‐KP type equations lies in the treatment for the integration operator . Our proposed DG method deals with it element by element, which is efficient and applicable to most solutions. Using the instinctive energy of the original PDE as a guiding principle, the DG scheme can be proved as an energy stable numerical scheme. In addition, the semi‐discrete error estimates results for the nonlinear case are derived without any priori assumption. Several numerical experiments demonstrate the capability of our schemes for various types of solutions.

A Symmetrized Parametric Finite Element Method for Anisotropic Surface Diffusion of Closed Curves

We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under anisotropic surface diffusion with a general anisotropic surface energy in two dimensions, where is the outward unit normal vector. By introducing a novel surface energy matrix which depends on the Cahn–Hoffman -vector and a stabilizing function , we first reformulate the equation into a conservative form and derive a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energies. Then, a semidiscretization in space for the variational formulation is proposed, and its area conservation and energy dissipation properties are proved. The semidiscretization is further discretized in time by an implicit structural-preserving scheme (SP-PFEM) which can rigorously preserve the enclosed area in the fully discrete level. Furthermore, we prove that the SP-PFEM is unconditionally energy-stable for almost any anisotropic surface energy under a simple and mild condition on . For several commonly used anisotropic surface energies, we construct explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed scheme.

Efficient and energy stable numerical schemes for the two-mode phase field crystal equation

In this paper, we propose four time marching schemes for the two-mode phase field crystal equation with the periodic boundary condition. The first- and second-order schemes are based on the stabilized semi-implicit approach with a combination of the backward Euler method, Crank–Nicolson method, and second-order backward differentiation formula, respectively. Based on the convex splitting method and third-order backward differentiation formula, a third-order scheme is developed. Different stabilized terms are added so that the energy stability of the proposed schemes can be guaranteed. Theoretically, the unique solvability and energy stability of the proposed schemes are rigorously proved. The error estimate of the first-order scheme is also given. Various numerical examples and simulations in the 2D and 3D cases are presented to demonstrate the accuracy, stability, and efficiency of the proposed schemes.