Pub Date : 2024-10-10DOI: 10.1016/j.cnsns.2024.108392
Lidia Aceto , Pietro Antonio Grassi
The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We provide a short review of graph theory and its relation with network -models adapted to our analysis. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.
{"title":"A charge-preserving method for solving graph neural diffusion networks","authors":"Lidia Aceto , Pietro Antonio Grassi","doi":"10.1016/j.cnsns.2024.108392","DOIUrl":"10.1016/j.cnsns.2024.108392","url":null,"abstract":"<div><div>The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We provide a short review of graph theory and its relation with network <span><math><mi>σ</mi></math></span>-models adapted to our analysis. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142536032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.cnsns.2024.108390
Xiaoliang Li , Bo Li , Zohreh Eskandari
In this paper, we focus on the impact of one firm’s information advantage over the other on the dynamic behavior of duopolistic competition. We establish conditions for the local stability of the model and find that more information does not necessarily lead to more stability. A bifurcation analysis shows that the Nash equilibrium may lose its stability through period-doubling, Neimark–Sacker, 1:2 resonance, 1:3 resonance, and 1:4 resonance bifurcations. In addition, we explore through numerical simulations complex dynamics such as chaos and multistability that may occur in the model.
{"title":"Impact of information advantage on dynamics of duopolistic competition under nonlinear demand","authors":"Xiaoliang Li , Bo Li , Zohreh Eskandari","doi":"10.1016/j.cnsns.2024.108390","DOIUrl":"10.1016/j.cnsns.2024.108390","url":null,"abstract":"<div><div>In this paper, we focus on the impact of one firm’s information advantage over the other on the dynamic behavior of duopolistic competition. We establish conditions for the local stability of the model and find that more information does not necessarily lead to more stability. A bifurcation analysis shows that the Nash equilibrium may lose its stability through period-doubling, Neimark–Sacker, 1:2 resonance, 1:3 resonance, and 1:4 resonance bifurcations. In addition, we explore through numerical simulations complex dynamics such as chaos and multistability that may occur in the model.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142536033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.cnsns.2024.108385
Mengjie Liu , Mingyan He , Pengtao Sun
The Bazant–Storey–Kornyshev (BSK) theory [1], [2], [3] recently developed an important continuum framework to expound the nonlocal dielectric permittivity of ionic liquids due to electrostatic correlations, leading to a fourth-order modified Poisson–Fermi equation to model the electrostatic potential field in the solvent (e.g., the electrolyte), while the standard second-order Poisson–Fermi equation is still valid in modeling the electrostatic potential field in the solute. Thus an interface problem is formed between the fourth-order and second-order Poisson–Fermi equations through the interface of solvent and solute with jump coefficients, which has exerted tremendous impacts on applications of electrokinetics, electrochemistry, biophysics, and etc. In this paper, a type of mixed finite element method is developed to solve the proposed interface problem once for all variables: the electrostatic potential, electric field, electric displacement field, electrostatic stress as well as interactional force in the electrolyte in an accurate fashion, and its optimal convergence properties are analyzed for all variables in their respective norms. Numerical experiments are carried out through self-defined mathematical examples to validate all attained theoretical results. Furthermore, as a part of modeling verification for the presented interface problem that models the BSK theory, a practically physical example is investigated to validate the necessity of introducing the fourth-order modified Poisson–Fermi equation to describe the electrostatic correlation effects due to charge reversal phenomenon.
{"title":"Mixed finite element analysis for a modified Poisson–Fermi interface problem accounting for electrostatic correlations","authors":"Mengjie Liu , Mingyan He , Pengtao Sun","doi":"10.1016/j.cnsns.2024.108385","DOIUrl":"10.1016/j.cnsns.2024.108385","url":null,"abstract":"<div><div>The Bazant–Storey–Kornyshev (BSK) theory <span><span>[1]</span></span>, <span><span>[2]</span></span>, <span><span>[3]</span></span> recently developed an important continuum framework to expound the nonlocal dielectric permittivity of ionic liquids due to electrostatic correlations, leading to a fourth-order modified Poisson–Fermi equation to model the electrostatic potential field in the solvent (e.g., the electrolyte), while the standard second-order Poisson–Fermi equation is still valid in modeling the electrostatic potential field in the solute. Thus an interface problem is formed between the fourth-order and second-order Poisson–Fermi equations through the interface of solvent and solute with jump coefficients, which has exerted tremendous impacts on applications of electrokinetics, electrochemistry, biophysics, and etc. In this paper, a type of mixed finite element method is developed to solve the proposed interface problem once for all variables: the electrostatic potential, electric field, electric displacement field, electrostatic stress as well as interactional force in the electrolyte in an accurate fashion, and its optimal convergence properties are analyzed for all variables in their respective norms. Numerical experiments are carried out through self-defined mathematical examples to validate all attained theoretical results. Furthermore, as a part of modeling verification for the presented interface problem that models the BSK theory, a practically physical example is investigated to validate the necessity of introducing the fourth-order modified Poisson–Fermi equation to describe the electrostatic correlation effects due to charge reversal phenomenon.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.cnsns.2024.108384
Shanshan Peng , Yanping Chen
In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken (or ), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.
本文针对非线性中性延迟-反应-扩散方程开发了一种空间虚拟元素方法，同时在时间上采用了加权隐式-显式方案。非线性项采用牛顿线性化方法。利用隐式方案分析线性项，利用显式方案处理非线性项，从而提高了计算效率。然后，创造性地将时空误差分割技术和 G 稳定性结合起来，严格分析了数值方案的无条件最优收敛结果，对网格比例没有任何限制。最后，提供了一组多边形网格的数值实例来证实理论结果。特别是当加权参数取θ=12（或θ=1）时，该方法退化为 Crank-Nicolson（或二阶后向微分公式（BDF2））方案。
{"title":"Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation","authors":"Shanshan Peng , Yanping Chen","doi":"10.1016/j.cnsns.2024.108384","DOIUrl":"10.1016/j.cnsns.2024.108384","url":null,"abstract":"<div><div>In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken <span><math><mrow><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> (or <span><math><mrow><mi>θ</mi><mo>=</mo><mn>1</mn></mrow></math></span>), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-08DOI: 10.1016/j.cnsns.2024.108383
Huaijun Yang, Xu Jia, Jinjin Yang
In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in -norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., and , where denotes the temporal size and is the spatial size. Subsequently, the global superconvergence error estimate in -norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.
{"title":"Unconditionally superconvergence error analysis of an energy-stable finite element method for Schrödinger equation with cubic nonlinearity","authors":"Huaijun Yang, Xu Jia, Jinjin Yang","doi":"10.1016/j.cnsns.2024.108383","DOIUrl":"10.1016/j.cnsns.2024.108383","url":null,"abstract":"<div><div>In this paper, the unconditionally superconvergence analysis is studied for the cubic Schrödinger equation with an energy-stable finite element method. A different approach is proposed to obtain the unconditionally superclose error estimate in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm firstly without using the time splitting technique required in the previous literature. The key to the analysis is to use a priori boundedness of the numerical solution in energy norm and control the nonlinear terms rigorously by two cases, i.e., <span><math><mrow><mi>τ</mi><mo>≤</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>τ</mi><mo>≥</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mi>τ</mi></math></span> denotes the temporal size and <span><math><mi>h</mi></math></span> is the spatial size. Subsequently, the global superconvergence error estimate in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm is derived by an effective interpolation post-processing approach. Finally, some numerical experiments are carried out to confirm the theoretical findings.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-05DOI: 10.1016/j.cnsns.2024.108382
Jaume Llibre , Paulo Santana
In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.
{"title":"Limit cycles and chaos in planar hybrid systems","authors":"Jaume Llibre , Paulo Santana","doi":"10.1016/j.cnsns.2024.108382","DOIUrl":"10.1016/j.cnsns.2024.108382","url":null,"abstract":"<div><div>In this paper we study the family of planar hybrid differential systems formed by two linear centers and a polynomial reset map of any degree. We study their limit cycles and also provide examples of these hybrid systems exhibiting chaotic dynamics.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-05DOI: 10.1016/j.cnsns.2024.108386
Rong An, Weiwen Wan
Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step and mesh size . In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform error estimate of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.
{"title":"Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization","authors":"Rong An, Weiwen Wan","doi":"10.1016/j.cnsns.2024.108386","DOIUrl":"10.1016/j.cnsns.2024.108386","url":null,"abstract":"<div><div>Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step <span><math><mi>τ</mi></math></span> and mesh size <span><math><mi>h</mi></math></span>. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> error estimate <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>τ</mi><mo>+</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1016/j.cnsns.2024.108381
Yunfeng Jia , Jingjing Wang
We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.
{"title":"Effects of extra resource and harvesting on the pattern formation for a predation system","authors":"Yunfeng Jia , Jingjing Wang","doi":"10.1016/j.cnsns.2024.108381","DOIUrl":"10.1016/j.cnsns.2024.108381","url":null,"abstract":"<div><div>We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is focused on investigating a stochastic fractional differential variational inequalities general system possessing Lévy jumps (SFDVIGS possessing Lévy jumps), which consists of two systems, i.e., a stochastic variational inequalities general system (SVIGS) and a stochastic fractional differential equations general system (SFDEGS) possessing Lévy jumps. Applying Picard successively iterative technique and the projection method, we derive the unique existence of solutions to SFDVIGS possessing Lévy jumps under certain mild assumptions.
{"title":"On stochastic fractional differential variational inequalities general system with Lévy jumps","authors":"Lu-Chuan Ceng , X.Z. Huan , Yunshui Liang , Jen-Chih Yao","doi":"10.1016/j.cnsns.2024.108373","DOIUrl":"10.1016/j.cnsns.2024.108373","url":null,"abstract":"<div><div>This paper is focused on investigating a stochastic fractional differential variational inequalities general system possessing Lévy jumps (SFDVIGS possessing Lévy jumps), which consists of two systems, i.e., a stochastic variational inequalities general system (SVIGS) and a stochastic fractional differential equations general system (SFDEGS) possessing Lévy jumps. Applying Picard successively iterative technique and the projection method, we derive the unique existence of solutions to SFDVIGS possessing Lévy jumps under certain mild assumptions.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-02DOI: 10.1016/j.cnsns.2024.108379
Yanping Chen , Zhengguang Liu , Xiaoqing Meng
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.
Shen 等人（2019）考虑了标量辅助变量（SAV）方法，并广泛用于模拟一系列梯度流。然而，基于 SAV 的方案以 "修正 "能量的稳定性著称。本文基于对经典 SAV 方法的若干改进，构建了一系列具有无条件能量耗散规律的修正 SAV 方法。首先，通过引入三步技术，我们可以减少每个时间步需要求解的常数系数线性方程的数量，同时保留其所有其他优点。其次，能量优化技术和拉格朗日乘法器技术的加入可以使数值方案具有保留原有能量消耗的优势。第三，我们在 GSAV 方法中使用了能量平衡方程的一阶近似，而不是将辅助变量的动态方程离散化，从而可以构建高阶无条件原始能量稳定数值方案。最后，有代表性的数值示例表明，所提方案的效率和精度都有所提高。
{"title":"Partially and fully implicit multi-step SAV approaches with original dissipation law for gradient flows","authors":"Yanping Chen , Zhengguang Liu , Xiaoqing Meng","doi":"10.1016/j.cnsns.2024.108379","DOIUrl":"10.1016/j.cnsns.2024.108379","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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