Scalar Auxiliary Variable Approach Research Articles

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.

A linearly implicit energy-stable scheme for critical dissipative surface quasi-geostrophic flows

In this paper, we propose an effective linearly implicit unconditional energy-stable scheme for surface quasi-geostrophic flows based on the scalar auxiliary variable approach and the Fourier spectral Galerkin method. Compared with traditional numerical methods, our scheme has constant coefficient matrices at each time step, and the numerical solutions are consistent with the dissipation laws for modified energy. By treating linear terms implicitly and nonlinear terms explicitly, we derive the dissipation laws for discrete modified surface kinetic energy and Hamiltonian. To reduce the aliasing error induced by the Fourier spectral Galerkin method, we implement a 2/3 de-aliasing technique for the nonlinear terms. Furthermore, the integration concerning energy in our numerical scheme is exact due to the Fourier spectral Galerkin method. Numerical experiments are presented to verify the stability and efficiency of the proposed scheme.

Unconditional H2$$ {H}^2 $$‐stability of the Euler implicit/explicit SAV‐based scheme for the 2D Navier–Stokes equations with smooth or nonsmooth initial data

AbstractIn this article, we propose ‐unconditional stable schemes for solving time‐dependent incompressible Navier–Stokes equations with smooth or nonsmooth initial data, , . The ‐stability analysis is established by leveraging the scalar auxiliary variable (SAV) approach. When dealing with nonsmooth initial data, we utilize a limited number of iteration of the semi‐implicit scheme followed by the SAV scheme. The overall efficiency is greatly enhanced due to the minimal computational cost of the semi‐implicit scheme and the explicit treatment of the nonlinear term within the SAV approach. The proposed schemes investigate two types of scalar auxiliary variables: the energy‐based variable and the exponential‐based variable. Rigorous proofs of the ‐unconditional stability of both schemes have been provided. Notice that both proposed numerical schemes enjoy unconditional long time stability for smooth and nonsmooth initial data when . Numerical experiments have been conducted to demonstrate the theoretical results.

Correction to: High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

Correction to: High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

AbstractThis paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

Linear energy-stable method with correction technique for the Ohta–Kawasaki–Navier–Stokes model of incompressible diblock copolymer melt

An efficiently linear time-marching method is proposed for the incompressible fluid flows coupled Ohta–Kawasaki model of diblock copolymer melt. Although this model satisfies the energy dissipation law under appropriate boundary conditions, the existences of nonlinear terms and coupling terms lead to difficulties of designing energy-stable numerical algorithms. To overcome the difficulties resulting from nonlinear and coupling terms, we herein construct a numerical method based on a highly efficient variant of scalar auxiliary variable approach. The stabilization technique is introduced to maintain the stability with large time steps. To improve the consistency, a simple and practical relaxation strategy is used to correct the difference between time-discretized modified energy and original energy. In each time step, all variables are totally decoupled and we only need to solve linear elliptic equations in a step-by-step manner. The unique solvability and energy stability are analytically proved. Extensive numerical experiments indicate that the proposed method not only achieves desired accuracy, energy stability, and consistency, but also has good capabilities on simulating various hydrodynamically coupled pattern formations.

A computationally optimal relaxed scalar auxiliary variable approach for solving gradient flow systems

In this paper, we give a computationally optimal relaxed scalar auxiliary variable (SAV) approach for solving gradient flow systems, in which we only need to solve an elliptic partial differential equation with constant coefficients at each time step. In addition to being applicable to various types of auxiliary variables, there are several advantages of our methods, including: (i) by modifying the optimization procedure in correction of auxiliary variables into a linear programming problems, the original hard-to-solve nonlinear optimization problem arising from previous relaxed SAV approach introduced in Jiang et al. (2022) [28] can be avoided; (ii) the resulting method yields some novel linear unconditionally energy stable schemes, in which backward Euler and Crank–Nicolson formulas are used to discretize the time so that the accuracy can reach the first- and second-order, respectively; (iii) comparing with the baseline SAV approach, the discrete energy in the energy dissipation law is closer to the original energy. Finally, ample numerical results demonstrate the accuracy, efficiency and energy stability of the proposed scheme.

Energy-dissipative evolutionary deep operator neural networks

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seek numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G, the Branch net encodes different input functions f at a fixed number of sensors xi,i=1,2,⋯,m, and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G(f)(y) at the test point y, DeepONet generates a good approximation of the operator G. A key distinction in our methodology is the utilization of DeepONet for the training of the initial state, which operates as a multi-parametric operator. Further, the evolution of parameters in our model is facilitated by the Scalar Auxiliary Variable (SAV) method, leading to a formulated iterative process for the parameter's progression over time. The SAV approach is adopted to preserve essential physical properties of PDEs, particularly the Energy Dissipation Law. It introduces a kind of modified energy and establishes unconditional energy dissipation law in the discrete level. By treating the parameters of the well-trained DeepONet as a representation of the initial operator and evolving them by a dynamic system, our network can predict the accurate solution at any further time, while the training data is only the initial state. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations, Allen-Cahn equations and a reaction–diffusion equation in three dimensions.

Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator

Abstract In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori$L^{\infty }$ estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

Arbitrarily high-order explicit energy-conserving methods for the generalized nonlinear fractional Schrödinger wave equations

A novel category of explicit conservative numerical methods with arbitrarily high-order is introduced for solving the nonlinear fractional Schrödinger wave equations in one and two dimensions. The proposed method is based on the scalar auxiliary variable approach. The equations studied is first transformed into an equivalent system by introducing a scalar auxiliary variable, and the energy is then reformulated as a sum of three quadratic terms. Applying the explicit relaxation Runge–Kutta methods in temporal and the Fourier pseudo-spectral discretization in spatial, the resulting time–space full discrete scheme is proved to preserve the reformulated energy in the discrete level to machine accuracy. The proposed methods improve the numerical stability during long-term computations, as demonstrated through numerical experiments. Also this idea can be easily extended to other similar equations, such as the nonlinear fractional wave equation and the fractional Klein–Gordon–Schrödinger equation.

Second‐order scalar auxiliary variable Fourier‐spectral method for a liquid thin film coarsening model

In this paper, we propose and analyze a second‐order accurate (in time) numerical scheme for the droplet liquid film coarsening model by using the scalar auxiliary variable (SAV) method, and the Fourier‐spectral method is used in space. The scheme is linear; thus, it could be solved very efficiently. Meanwhile, due to the application of SAV approach, the unconditional energy stability of the numerical scheme is also derived, without any restriction on the time step size. In addition, we also provide a rigorous error estimate that shows that our second‐order numerical scheme with Fourier‐spectral method in space converges with order , where and are time and space step sizes, respectively. Finally, a few numerical experiments are performed, which confirms the efficiency and accuracy of the proposed scheme, including the tests of the convergence, the mass conservation and the decrease of the energy. The simulation of coarsening process with time is also observed.

A highly efficient variant of scalar auxiliary variable (SAV) approach for the phase-field fluid-surfactant model

In this work, we study a phase-field fluid-surfactant system by applying two conservative Allen-Cahn equations and Navier-Stokes equations. A highly efficient variant of scalar auxiliary variable approach is applied in designing a second-order time-accurate numerical scheme. By introducing several scalar auxiliary variables, all variables are decoupled with each other and they can be updated in a step-by-step manner. The proposed method satisfies the mass conservation law, unique solvability and modified energy dissipation property. Several tests are performed to investigate the surfactant-laden phase-separation and the effect of surfactant in two dimensional and three dimensional space.

Spatio‐temporal scalar auxiliary variable approach for the nonlinear convection–diffusion equation with discontinuous Galerkin method

AbstractIn this paper, a scalar auxiliary variable approach combining with a discontinuous Galerkin method is proposed to handle the gradient‐type nonlinear term. The nonlinear convection–diffusion equation is used as the model. The proposed equivalent system can effectively handle the nonlinear convection term by incorporating the spatial and temporal information, globally. With the introduced auxiliary variable, the stability of the system can be simply characterized. In the space, according to the regularity of the system, an optimal accuracy is obtained with the discontinuous Galerkin method. Two different time discretization techniques, that is, backward Euler and linearly extrapolated Crank–Nicolson schemes, are separately considered with first order and second order accuracy. The proposed schemes are unconditionally stable with proper selected parameters. For the error estimates, the optimal convergence rates are rigorously proved. In the numerical experiments, the convergence information is confirmed and a benchmark problem with shock tendency is then followed with robustness demonstration.

A novel class of explicit energy-preserving splitting methods for charged-particle dynamics

In this letter, based on the exponential scalar auxiliary variable technology, we propose and study a new class of explicit energy-preserving splitting methods for solving the charged-particle dynamics. The energy-preserving property of these methods is rigorously analysed. We also provide the error estimates for the new methods. Numerical computations are presented, which confirm the effectiveness and superiority of these novel methods in comparison with the standard scalar auxiliary variable approach.

A linear adaptive second‐order backward differentiation formulation scheme for the phase field crystal equation

AbstractIn this article, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function by setting the positive constant large enough such that The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio . Furthermore, a rigorous error estimate of the second order accuracy of is derived for the proposed scheme on the nonuniform mesh by using the uniform bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and the efficiency of the proposed scheme combined with the time adaptive strategy.

A high-order structure-preserving difference scheme for generalized fractional Schrödinger equation with wave operator

This paper focuses on the construction and analysis of the structure-preserving algorithm for generalized fractional Schrödinger equation with wave operator. A fourth-order energy-conserving difference scheme is developed for the resulting equivalent system based on scalar auxiliary variable approach. The discrete energy conservation law, boundedness and convergence of difference solutions are proved in detail. Numerical experiments are performed to verify our theoretical analysis results.

A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach

In this paper, we consider the numerical approximation of a hydrodynamic Ericksen–Leslie system, which describes the macroscopic continuum of liquid crystals. A numerical challenge for solving the governing system is how to construct a suitable numerical scheme so that it is unconditionally energy stable at the discrete level. We propose a novel linear, decoupled fully discrete scheme, which is based on the design of the scalar auxiliary variable (SAV) for the nonlinear potential and combined with the discontinuous Galerkin (DG) for spatial discretization, the pressure-projection method for the Navier–Stokes equations, and implicit–explicit (IMEX) approaches for the highly nonlinear and coupling terms. We rigorously prove the unconditional energy stability and optimal error estimates of the proposed scheme, especially the optimal error bounds for the pressure. Finally, several numerical examples are performed to numerically demonstrate the accuracy, energy stability, and applicability of the proposed scheme.

A novel relaxed scalar auxiliary variable approach for gradient flows

In this paper, we propose a novel relaxed scalar auxiliary variable (nRSAV) approach to solve a series of gradient flow problems. The proposed nRSAV approach inherits all the advantages of the traditional SAV and RSAV method. Meanwhile, it preserves a quite close original energy dissipative law and provides an improved accuracy than the baseline SAV method. Besides, compared with the RSAV approach, we do not need to solve a quadratic equation with one unknown to obtain the relaxation. All the semi-discrete schemes are proved to be unconditionally energy stable. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.

High-order linearly implicit schemes conserving quadratic invariants

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be maintained relatively easily compared with general invariants, and can be preserved by canonical Runge–Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge–Kutta methods and prove some properties involving accuracy.

A convergent SAV scheme for Cahn–Hilliard equations with dynamic boundary conditions

Abstract The Cahn–Hilliard equation is one of the most common models to describe phase separation processes in mixtures of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for this equation have been proposed. Recently, a family of models using Cahn–Hilliard-type equations on the boundary of the domain to describe adsorption processes was analysed (cf. Knopf, P., Lam, K. F., Liu, C. & Metzger, S. (2021) Phase-field dynamics with transfer of materials: the Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. ESAIM: Math. Model. Numer. Anal., 55, 229–282). This family of models includes the case of instantaneous adsorption processes studied by Goldstein, Miranville and Schimperna (2011, A Cahn–Hilliard model in a domain with non-permeable walls. Phys. D, 240, 754–766) as well as the case of vanishing adsorption rates, which was investigated by Liu and Wu (2019, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal., 233, 167–247). In this paper, we are interested in the numerical treatment of these models and propose an unconditionally stable, linear, fully discrete finite element scheme based on the scalar auxiliary variable approach. Furthermore, we establish the convergence of discrete solutions towards suitable weak solutions of the original model. Thereby, when passing to the limit, we are able to remove the auxiliary variables introduced in the discrete setting completely. Finally, we present simulations based on the proposed linear scheme and compare them to results obtained using a stable, nonlinear scheme to underline the practicality of our scheme.