Auxiliary Variable Approach Research Articles

Local energy-preserving scalar auxiliary variable approaches for general multi-symplectic Hamiltonian PDEs

We develop two classes of general-purpose second-order integrators for the general multi-symplectic Hamiltonian system by incorporating a scalar auxiliary variable. Unlike the previous methods introduced in [22,31], these new approaches do not impose constraints on the state function of multi-symplectic system, and can preserve the original local/global energy conservation laws exactly. Moreover, the approaches are computationally efficient, as they only require solving linear equations with the same constant coefficients at each time step along with some additional scalar nonlinear equations. We employ the proposed methods to solve various equations, and the numerical results validate their solution accuracy, effectiveness, robustness, and energy-preserving ability.

Linearly Implicit Conservative Schemes for the Nonlocal Schrödinger Equation

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach and the symplectic Runge–Kutta method. By integrating these advanced methods, the proposed schemes aim to significantly enhance computational accuracy and efficiency, while maintaining the essential conservative properties necessary for accurate physical modeling. This offers a structured approach to handle auxiliary variables, ensuring stability and conservation, while the symplectic Runge–Kutta method provides a robust framework with high accuracy. Together, these techniques offer a powerful and reliable approach for researchers dealing with complex quantum mechanical systems described by the nonlocal Schrödinger equation, ensuring both accuracy and stability in their numerical simulations.

Structure-preserving weighted BDF2 methods for anisotropic Cahn–Hilliard model: Uniform/variable-time-steps

In this paper, we innovatively develop uniform/variable-time-step weighted and shifted BDF2 (WSBDF2) methods for the anisotropic Cahn–Hilliard (CH) model, combining the scalar auxiliary variable approach with two types of stabilized techniques. Using the concept of G stability, the uniform-time-step WSBDF2 method is theoretically proved to be energy-stable. Due to the inapplicability of the relevant G-stability properties, another technique is adopted in this work to demonstrate the energy stability of the variable-time-step WSBDF2 method. In addition, the two numerical schemes are all mass-conservative. Finally, numerous numerical simulations are presented to demonstrate the stability and accuracy of these schemes.

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.

Cylindrical cavity expansion-contraction solutions in undrained MCC soils with the auxiliary variable approach

Cylindrical cavity exhibits non-self-similarity during contraction process following expansion. Previous studies solve this problem with total strain approach and simple constitutive models, but the approach is not applicable when using an advanced constitutive model. This paper presents a semi-analytical solution for a cylindrical cavity undergoing expansion-contraction in undrained soils with auxiliary variable approach, incorporating the Modified Cam-Clay (MCC) model. The stress states around the cavity are formed by the superposition of initial and superimposed stress states. By treating superimposed effective stresses as self-similar, a semi-analytical solution is derived for solving the cavity expansion-contraction problem. The elastoplastic stress-strain relationship is formulated as a set of first-order differential equations, which can be solved as an initial value problem though Runge-Kutta (RK) method. Then the stress distribution around the cavity during expansion-contraction process can be determined. To validate the proposed approach, a series of well-conduced self-boring pressuremeter (SBP) tests are used to verify the proposed approach, which shows good agreements. Additionally, a FEM simulation incorporating the MCC model is performed, and the simulation results are presented to carry out parametric studies on soil parameters. A significant influence on the range of the plastic and reverse plastic zones is shown for overconsolidation ratio, while the in-situ coefficient of the earth pressure only quantitatively affects the stress distribution.

A generalized scalar auxiliary variable approach for the Navier–Stokes-[formula omitted]/Navier–Stokes-[formula omitted] equations based on the grad-div stabilization

In this article, based on the grad-div stabilization, we propose a generalized scalar auxiliary variable approach for solving a fluid–fluid interaction problem governed by the Navier–Stokes-ω/Navier–Stokes-ω equations. We adopt the backward Euler scheme and mixed finite element approximation for temporal-spatial discretization, and explicit treatment for the interface terms and nonlinear terms. The proposed scheme is almost unconditionally stable and requires solving only the linear equation with constant coefficient at each time step. It can also penalize for lack of mass conservation and improve the accuracy. Finally, a series of numerical experiments are carried out to illustrate the stability and effectiveness of the proposed scheme.

Efficiently and consistently energy-stable [formula omitted]-phase-field method for the incompressible ternary fluid problems

Ternary incompressible fluid flows extensively exist in atmospheric science, chemical engineering, and energy and power engineering, etc. The phase-field method is popular in multi-phase fluid modeling thanks to its efficient ability of interface capturing. This work aims to develop an energy dissipation law-preserving and temporally second-order accurate algorithm for a ternary phase-field fluid system. To establish a simple energy estimation, an adapted auxiliary variable approach is used to transform the original model into its equivalent form. Later, a second-order backward difference strategy is used to design the fully decoupled and linear time-marching scheme. To improve the consistency between original and modified discrete energy functionals, a practical energy correction technique is presented. We analytically prove the discrete energy dissipation property and show that the energy estimation can be easily established without considering the complex treatments of nonlinear and coupling terms. To facilitate the interested readers, we briefly describe the numerical implementation in each time step. The numerical tests indicate that the proposed method not only has desired accuracy, but also satisfies the energy stability even if a larger time step is used. Moreover, the proposed method can well simulate various ternary fluid phenomena, such as the liquid lens, phase separation, droplet dynamics, Kelvin–Helmholtz instability, and billowing cloud.

Energy-Conserving Explicit Relaxed Runge–Kutta Methods for the Fractional Nonlinear Schrödinger Equation Based on Scalar Auxiliary Variable Approach

In this paper, we present a novel explicit structure-preserving numerical method for solving nonlinear space-fractional Schrödinger equations based on the concept of the scalar auxiliary variable approach. Firstly, we convert the equations into an equivalent system through the introduction of a scalar variable. Subsequently, a semi-discrete energy-preserving scheme is developed by employing a fourth-order fractional difference operator to discretize the equivalent system in spatial direction, and obtain the fully discrete version by using an explicit relaxed Runge–Kutta method for temporal integration. The proposed method preserves the energy conservation property of the space-fractional nonlinear Schrödinger equation and achieves high accuracy. Numerical experiments are carried out to verify the structure-preserving qualities of the proposed method.

An artificial compressibility SAV finite element method for the time-dependent natural convection problem

In this article, we present, analyze, and test a first-order implicit-explicit type scheme based on the artificial compressibility (AC) method and the scalar auxiliary variable (SAV) approach for solving the time-dependent natural convection problem. The proposed method is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization, and explicit treatment for nonlinear terms. It effectively decouples the velocity and pressure via artificial compressibility, thereby reducing computational complexity and execution time. Moreover, we use the SAV approach for the convective terms, it leads to the scheme is linear, and only require solving a sequence of linear differential equation with constant coefficients at each time step. It is proved that the solution of this method is bounded and satisfying the law of energy dissipation. The rigorous error estimate of velocity, pressure, and temperature are derived. Finally, some numerical tests are implemented to verify the theoretical analysis and illustrate the efficiency of the presented method.

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.

Linear relaxation method with regularized energy reformulation for phase field models

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models.

The high-order exponential semi-implicit scalar auxiliary variable approach for the general nonlocal Cahn-Hilliard equation

The nonlocal Cahn-Hilliard equation with nonlocal diffusion operator is more suitable for the simulation of microstructure phase transition than the local Cahn-Hilliard equation. In this paper, based on the exponential semi-implicit scalar auxiliary variable method, the highly efficient and accurate schemes (in time) with unconditional energy stability for solving the nonlocal Cahn-Hilliard equation are proposed. On the one hand, we have demonstrated the unconditional energy stability and mass conservation for the nonlocal Cahn-Hilliard equation with its high-order numerical schemes in the continuous and discrete level carefully and rigorously. On the other hand, in order to reduce the calculation and storage cost in numerical simulation, we use the fast solver based on fast Fourier transform and conjugate gradient approach for spatial discretization. Some numerical simulations involving the Gaussian kernel are presented and show the stability, accuracy, efficiency and unconditional energy stability of the proposed schemes.

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.

Numerical energy dissipation for time fractional volume-conserved Allen–Cahn model based on the ESAV and R-ESAV approaches

In this study, we consider volume-conserved numerical schemes for the volume-conserved time fractional Allen–Cahn equation. We start with the L1 scheme based on a modified exponential scalar auxiliary variable (ESAV) approach for handling the nonlinear potential term. Then we introduce the fast L1 scheme to significantly reduce the CPU time and to make the long-term computation possible. Further to reduce the error between the discrete energy and the original energy of the ESAV approach, we introduce the fast L1 scheme based on a relaxed modified ESAV (R-ESAV) idea. We rigorously show the discrete energy dissipation properties for the above three schemes, including the discrete energy boundedness law, the discrete fractional energy dissipation law, and the discrete weighted energy dissipation law. Theoretical findings are validated by a few numerical experiments.

Efficiently linear and unconditionally energy-stable time-marching schemes with energy relaxation for the phase-field surfactant model

In this article, we consider consistent, efficient, and accurate numerical approximations for a recently developed Cahn–Hilliard type phase-field surfactant model enjoying the bounded-from-below condition of total energy. The phase-field surfactant model naturally satisfies the energy dissipation property in time. Based on this feature, we construct linearly decoupled and unconditionally energy-stable time-marching schemes based on an efficient variant of scalar auxiliary variable approach. To enhance the consistency of discrete modified energy and discrete original energy, a practical energy relaxation technique is adopted. We analytically proved the unique solvability and unconditional energy stability of the proposed schemes. In each time step, only two linear elliptic type equations with constants coefficients need to be solved and several algebraic operations need to be performed. Therefore, the proposed schemes are highly efficient and easy to implement. The space is discretized by the finite difference method and the linear multigrid algorithm is used to accelerate the convergence. The numerical results indicate that the proposed method not only has desired accuracy in time but also satisfies the energy dissipation law even if larger time steps are used. Furthermore, the surfactant laden spinodal decompositions and surfactant absorptions can be well simulated.

Efficiently high-order time-stepping R-GSAV schemes for the Navier–Stokes–Poisson–Nernst–Planck equations

We propose in this paper two kinds of generalized scalar auxiliary variable (GSAV) approach with relaxation (R-GSAV) for the Navier–Stokes–Poisson–Nernst–Planck equations. By applying positive function transform approach for ion concentration equation, introducing auxiliary variable for energy equation and adopting consistent splitting approach for momentum equations, we construct fully decoupled, linearized and kth-order semi-implicit schemes. The proposed schemes have several advantages: the concentration components of the discrete solution preserve the properties of positivity and mass conservation; these are unconditionally energy stable; the corrected energy is consistent with the original energy; only require solving decoupled linear systems with constant coefficients at each time step. We present some numerical results to validate these schemes and investigate the dynamics with initial discontinuous concentrations.

A stabilized SAV difference scheme and its accelerated solver for spatial fractional Cahn–Hilliard equations

A novel energy-stable scheme is proposed to solve the spatial fractional Cahn–Hilliard equations, using the idea of scalar auxiliary variable (SAV) approach and stabilization technique. Thanks to the stabilization technique, it is shown that larger temporal stepsizes can be applied in numerical simulations. Moreover, the proposed SAV finite difference scheme is non-coupled and linearly implicit, which can be efficiently solved by the preconditioned conjugate gradient (PCG) method with a sine transform based preconditioner. Optimal error estimates of the fully-discrete scheme are obtained rigorously. Numerical examples are given to confirm the theoretical results and show the higher efficiency of the proposed scheme than the previous SAV schemes.

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.

Scalar auxiliary variable pressure-correction method for natural convection problems⋆

In this article, we present first-order standard pressure-correction and second-order rotational pressure-correction algorithms using an exponential scalar auxiliary variable (SAV) approach for the natural convection problems. The SAV pressure-correction method is linear and decoupled with explicit treatment of nonlinear terms, so it only needs to solve a sequence of Poisson equations for velocity and pressure at each time step. Furthermore, we proved the first- and second-order algorithms are unconditionally stable and established error estimates of the velocity, temperature and pressure approximation for the first-order algorithms without any restriction on the time step. Finally, some numerical experiments are provided to support the theoretical analysis and to show the performances of our proposed algorithms.

High-order energy stable variable-step schemes for the time-fractional Cahn–Hilliard model

Based on the Alikhanov formula of the Caputo derivative and the exponential scalar auxiliary variable approach, two different second-order stable numerical schemes are constructed for the time-fractional Cahn–Hilliard model, including the nonlinear L2-1σ scheme and the linear L2-1σ-ESAV scheme. Applying the discrete gradient structure, we construct the asymptotically compatible energies and the associated energy dissipation laws of these two schemes for the time-fractional Cahn–Hilliard model. Several experiments are carried out to demonstrate the accuracy of our schemes and compare their computational efficiency, as well as to investigate the coarsening dynamics of the time-fractional Cahn–Hilliard model with different adaptive time-stepping strategies. In the long-time simulations, the linear L2-1σ-ESAV scheme may be more costly than the nonlinear L2-1σ scheme although the linear scheme may take less time at each time level. It is observed that the energy of the L2-1σ-ESAV scheme prones to produce small fluctuations when large time steps are used.