Pub Date : 2024-10-01DOI: 10.1016/j.cnsns.2024.108376
Li Peng , Yong Zhou
This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some estimates in the framework of spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.
{"title":"Characterization of solutions in Besov spaces for fractional Rayleigh–Stokes equations","authors":"Li Peng , Yong Zhou","doi":"10.1016/j.cnsns.2024.108376","DOIUrl":"10.1016/j.cnsns.2024.108376","url":null,"abstract":"<div><div>This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></mrow></math></span> estimates in the framework of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421763","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 : 2024-10-01DOI: 10.1016/j.cnsns.2024.108378
Sai Ganga, Ziya Uddin, Rishi Asthana
This article analyses the flow of Maxwell hybrid nanofluid induced by an exponentially stretching and rotating cylinder. The presence of non-linear convection, non-linear radiation, and magnetic field is also assumed. The factors covered in the study has a wide spectrum of application in various disciplines, and therefore we analyse the influence of different flow parameters after numerically solving the set of modelled differential equations. A data-free physics-informed neural network using a wavelet activation function is used to approximate the numerical solution. The reliability of the used methodology is validated by comparing the results of the limiting case with the available results. The paper demonstrates the effectiveness of using PINN in an unsupervised fashion to tackle fluid flow problems, showcasing their ability to provide reliable and accurate solutions without the need for extensive datasets. This approach highlights the potential of PINN to address complex fluid dynamics problems by utilizing physical laws within the neural network framework. From the numerical study, it is observed that hybrid nanofluid has a better rate of heat transfer compared to the nanofluid. Furthermore, radiation parameter and maxwell flow parameter is seen to exhibit significant impact of the flow profiles.
{"title":"Exploring swirling flow dynamics: Unsupervised machine learning in Maxwell hybrid nanofluid convection over an exponentially stretching cylinder with non-linear radiation effects","authors":"Sai Ganga, Ziya Uddin, Rishi Asthana","doi":"10.1016/j.cnsns.2024.108378","DOIUrl":"10.1016/j.cnsns.2024.108378","url":null,"abstract":"<div><div>This article analyses the flow of Maxwell hybrid nanofluid induced by an exponentially stretching and rotating cylinder. The presence of non-linear convection, non-linear radiation, and magnetic field is also assumed. The factors covered in the study has a wide spectrum of application in various disciplines, and therefore we analyse the influence of different flow parameters after numerically solving the set of modelled differential equations. A data-free physics-informed neural network using a wavelet activation function is used to approximate the numerical solution. The reliability of the used methodology is validated by comparing the results of the limiting case with the available results. The paper demonstrates the effectiveness of using PINN in an unsupervised fashion to tackle fluid flow problems, showcasing their ability to provide reliable and accurate solutions without the need for extensive datasets. This approach highlights the potential of PINN to address complex fluid dynamics problems by utilizing physical laws within the neural network framework. From the numerical study, it is observed that hybrid nanofluid has a better rate of heat transfer compared to the nanofluid. Furthermore, radiation parameter and maxwell flow parameter is seen to exhibit significant impact of the flow profiles.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421759","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 : 2024-10-01DOI: 10.1016/j.cnsns.2024.108377
Chan Li , Li-Jun Wu , Yunchuan Chen , Jia-Yi Li
In this paper, we study the long-time behaviors of wave equations subject to boundary memory damping and friction damping. Different from assumptions that memory kernel is a nonnegative, monotone function in the previous literatures, we assume that the primitive of the memory kernel is a generalized positive definite kernel (abbreviated to GPDK), which may vary sign or oscillate. The key to the problem lies in establishing the connection between memory damping and energy terms. By combining the properties of the positive definite kernel with classical multiplier methods, and constructing auxiliary systems, we ultimately establish the asymptotic stability, exponential stability and polynomial stability of systems featuring boundary memory damping and friction damping. To illustrate our theoretical results, we provide some numerical simulations.
{"title":"Long-time behaviors of wave equations stabilized by boundary memory damping and friction damping","authors":"Chan Li , Li-Jun Wu , Yunchuan Chen , Jia-Yi Li","doi":"10.1016/j.cnsns.2024.108377","DOIUrl":"10.1016/j.cnsns.2024.108377","url":null,"abstract":"<div><div>In this paper, we study the long-time behaviors of wave equations subject to boundary memory damping and friction damping. Different from assumptions that memory kernel is a nonnegative, monotone function in the previous literatures, we assume that the primitive of the memory kernel is a generalized positive definite kernel (abbreviated to GPDK), which may vary sign or oscillate. The key to the problem lies in establishing the connection between memory damping and energy terms. By combining the properties of the positive definite kernel with classical multiplier methods, and constructing auxiliary systems, we ultimately establish the asymptotic stability, exponential stability and polynomial stability of systems featuring boundary memory damping and friction damping. To illustrate our theoretical results, we provide some numerical simulations.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421760","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 : 2024-09-28DOI: 10.1016/j.cnsns.2024.108366
B. Rushi Kumar, C.M. Mohana
This research investigates the flow and heat transfer characteristics of a ternary hybrid nanofluid in a rotating system between two parallel stretching surfaces. It examines the impact of nanoparticle shape factor and irreversibility on suspensions containing AlO, CuO, and ZnO nanoparticles in water. Different physical and thermal conditions are taken into account, such as porous medium, suction/injection, radiation, heat source/sink, variable viscosity and thermal conductivity. Ternary hybrid nanofluids exhibit better thermophysical stability and performance than traditional nanofluids. The application of ternary hybrid nanofluids in rotating systems with two parallel stretching surfaces has industrial applications such as material treatment, manufacturing processes, and cooling systems. By introducing similarity variables, the governing partial differential equations are transformed into ordinary differential equations, which are then solved numerically using the Keller Box Method based on implicit finite differences. The study finds that as the viscosity parameter increases, there is a decrease in fluid velocity and an increase in temperature. Additionally, increasing values of radiation and thermal conductivity parameter lead to enhanced temperature and entropy generation rate. The entropy generation rate is higher for platelet-shaped nanoparticles and lower for spherical-shaped nanoparticles. Platelet shapes exhibit lower friction during suction and injection, while spherical shapes exhibit higher friction. Furthermore, the heat transfer rates of ternary hybrid nanofluids containing sphere, brick, cylinder, platelet, and blade-shaped nanoparticles suspended in water are 3.27%, 6.41%, 11.14%, 13.56%, and 14.20%, respectively, when injection is performed at the top surface of the sheet. For suction, the heat transfer rates are 16.91% for the sphere, 19.48% for the brick, 23.83% for the cylinder, 26.84% for the platelet, and 39.69% for the blade.
{"title":"Thermal and entropy analysis of ternary hybrid nanofluid using Keller Box method","authors":"B. Rushi Kumar, C.M. Mohana","doi":"10.1016/j.cnsns.2024.108366","DOIUrl":"10.1016/j.cnsns.2024.108366","url":null,"abstract":"<div><div>This research investigates the flow and heat transfer characteristics of a ternary hybrid nanofluid in a rotating system between two parallel stretching surfaces. It examines the impact of nanoparticle shape factor and irreversibility on suspensions containing Al<span><math><msub><mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span>O<span><math><msub><mrow></mrow><mrow><mn>3</mn></mrow></msub></math></span>, CuO, and ZnO nanoparticles in water. Different physical and thermal conditions are taken into account, such as porous medium, suction/injection, radiation, heat source/sink, variable viscosity and thermal conductivity. Ternary hybrid nanofluids exhibit better thermophysical stability and performance than traditional nanofluids. The application of ternary hybrid nanofluids in rotating systems with two parallel stretching surfaces has industrial applications such as material treatment, manufacturing processes, and cooling systems. By introducing similarity variables, the governing partial differential equations are transformed into ordinary differential equations, which are then solved numerically using the Keller Box Method based on implicit finite differences. The study finds that as the viscosity parameter increases, there is a decrease in fluid velocity and an increase in temperature. Additionally, increasing values of radiation and thermal conductivity parameter lead to enhanced temperature and entropy generation rate. The entropy generation rate is higher for platelet-shaped nanoparticles and lower for spherical-shaped nanoparticles. Platelet shapes exhibit lower friction during suction and injection, while spherical shapes exhibit higher friction. Furthermore, the heat transfer rates of ternary hybrid nanofluids containing sphere, brick, cylinder, platelet, and blade-shaped nanoparticles suspended in water are 3.27%, 6.41%, 11.14%, 13.56%, and 14.20%, respectively, when injection is performed at the top surface of the sheet. For suction, the heat transfer rates are 16.91% for the sphere, 19.48% for the brick, 23.83% for the cylinder, 26.84% for the platelet, and 39.69% for the blade.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142329577","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 : 2024-09-28DOI: 10.1016/j.cnsns.2024.108374
Liya Liu , Songxiao Li , Bing Tan
Bregman distance methods play a key role in solving problems in nonlinear analysis and optimization theory, since the Bregman distance is a useful substitute for the metric. The main purpose of this paper is to investigate two new iterative algorithms based on the Bregman distance and the Bregman projection for solving split feasibility problems in real Hilbert spaces. The algorithms are constructed around these methods: Byrne’s CQ method, Polyak’s gradient method, Halpern method, and hybrid projection method. The proposed methods involve inertial extrapolation terms and self-adaptive step sizes. We prove that the proposed iterations converge strongly to the Bregman projection of the initial point onto the solution set. Some numerical examples are provided to illustrate the computational effectiveness of our algorithms. The main results extend and improve the recent results related to the split feasibility problem.
{"title":"Strong convergence of Bregman projection algorithms for solving split feasibility problems","authors":"Liya Liu , Songxiao Li , Bing Tan","doi":"10.1016/j.cnsns.2024.108374","DOIUrl":"10.1016/j.cnsns.2024.108374","url":null,"abstract":"<div><div>Bregman distance methods play a key role in solving problems in nonlinear analysis and optimization theory, since the Bregman distance is a useful substitute for the metric. The main purpose of this paper is to investigate two new iterative algorithms based on the Bregman distance and the Bregman projection for solving split feasibility problems in real Hilbert spaces. The algorithms are constructed around these methods: Byrne’s CQ method, Polyak’s gradient method, Halpern method, and hybrid projection method. The proposed methods involve inertial extrapolation terms and self-adaptive step sizes. We prove that the proposed iterations converge strongly to the Bregman projection of the initial point onto the solution set. Some numerical examples are provided to illustrate the computational effectiveness of our algorithms. The main results extend and improve the recent results related to the split feasibility problem.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421762","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 : 2024-09-28DOI: 10.1016/j.cnsns.2024.108372
Xiaojuan Wu , Siqing Gan
This article is concerned with numerical approximations of the Heston 3/2-model from mathematical finance, which takes values in and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed, which can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size . Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is highly non-trivial, by noting that the diffusion coefficient grows super-linearly. The above theoretical results can be then used to justify the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model. Indeed, the unconditional positivity-preserving property is particularly desirable in the MLMC setting, where large discretization time steps are used. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity for the MLMC approach, where is the required target accuracy. Numerical experiments are finally reported to confirm the above results.
{"title":"An explicit positivity-preserving scheme for the Heston 3/2-model with order-one strong convergence","authors":"Xiaojuan Wu , Siqing Gan","doi":"10.1016/j.cnsns.2024.108372","DOIUrl":"10.1016/j.cnsns.2024.108372","url":null,"abstract":"<div><div>This article is concerned with numerical approximations of the Heston 3/2-model from mathematical finance, which takes values in <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></math></span> and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed, which can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size <span><math><mrow><mi>h</mi><mo>></mo><mn>0</mn></mrow></math></span>. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is highly non-trivial, by noting that the diffusion coefficient grows super-linearly. The above theoretical results can be then used to justify the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model. Indeed, the unconditional positivity-preserving property is particularly desirable in the MLMC setting, where large discretization time steps are used. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> for the MLMC approach, where <span><math><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></math></span> is the required target accuracy. Numerical experiments are finally reported to confirm the above results.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142358759","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 : 2024-09-27DOI: 10.1016/j.cnsns.2024.108371
Linying Xiang , Shuwei Yao , Xiao Wang , Zeya Zhu
This paper investigates optimal target control of complex networks through modifications in network topology. A novel edge-addition algorithm is proposed to ensure structural target controllability in directed networks with a single input. An edge-addition cost is introduced to measure the efficiency of achieving target control objectives. The relationships between the target node selection and the average node degree, respectively, and the edge-addition cost are examined by numerical simulations. It concludes that both the choice of target nodes and the average node degree significantly influence the cost-effectiveness of achieving target control. These findings offer valuable insights for the design of optimal networks in practical applications.
{"title":"Optimizing target control in complex networks using edge-addition cost","authors":"Linying Xiang , Shuwei Yao , Xiao Wang , Zeya Zhu","doi":"10.1016/j.cnsns.2024.108371","DOIUrl":"10.1016/j.cnsns.2024.108371","url":null,"abstract":"<div><div>This paper investigates optimal target control of complex networks through modifications in network topology. A novel edge-addition algorithm is proposed to ensure structural target controllability in directed networks with a single input. An edge-addition cost is introduced to measure the efficiency of achieving target control objectives. The relationships between the target node selection and the average node degree, respectively, and the edge-addition cost are examined by numerical simulations. It concludes that both the choice of target nodes and the average node degree significantly influence the cost-effectiveness of achieving target control. These findings offer valuable insights for the design of optimal networks in practical applications.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421830","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 : 2024-09-27DOI: 10.1016/j.cnsns.2024.108375
Chao Tan , Yong Liang , Min Zou , Tong Lei , Mingwei Liu
In this paper, we investigate the control for Hermite-Gaussian (HG) solitons in the nonlinear fractional Schrödinger equation (FSE) by sequentially applying power function modulations, cosine modulations, parabolic potentials, and quadratic phase modulations (QPM). In the photorefractive media, the HG beam forms scattered breathing solitons when the fractional diffraction effect equilibrates with nonlinear effect. Under the power function modulation, the soliton maintains an equidistant linear transmission along the z-axis, and the number of solitons is equal to the mode. In the cosine modulation, the soliton distorts and its energy rapidly decreases after a certain distance of transmission. The time of the distortion varies with the Lévy index, photorefractive coefficient, modulation frequency and order. The freak spots exhibit a “flower” shape pattern. If a parabolic potential is introduced, the HG beam forms crawling soliton pairs or merges into a single bounded breathing soliton by adjusting the correlation among the Lévy index, nonlinear and parabolic coefficients. By increasing the nonlinear coefficient in the negative QPM regime, the defocusing HG beam emits several “filiform” breathing solitons during its propagation, which move in a parallel straight line to each other. The HG beam is transformed into a single fine breathing soliton after being focused under a positive QPM. The time of the formation and breathing rate varies with the Lévy index, QPM and nonlinear coefficients. Moreover, the number of solitons changes irregularly with modes. These results are significant for applications in optical communication, optical device design, and optical signal processing.
{"title":"The control for multiple kinds of solitons generated in the nonlinear fractional Schrödinger optical system based on Hermite-Gaussian beams","authors":"Chao Tan , Yong Liang , Min Zou , Tong Lei , Mingwei Liu","doi":"10.1016/j.cnsns.2024.108375","DOIUrl":"10.1016/j.cnsns.2024.108375","url":null,"abstract":"<div><div>In this paper, we investigate the control for Hermite-Gaussian (HG) solitons in the nonlinear fractional Schrödinger equation (FSE) by sequentially applying power function modulations, cosine modulations, parabolic potentials, and quadratic phase modulations (QPM). In the photorefractive media, the HG beam forms scattered breathing solitons when the fractional diffraction effect equilibrates with nonlinear effect. Under the power function modulation, the soliton maintains an equidistant linear transmission along the <em>z</em>-axis, and the number of solitons is equal to the mode. In the cosine modulation, the soliton distorts and its energy rapidly decreases after a certain distance of transmission. The time of the distortion varies with the Lévy index, photorefractive coefficient, modulation frequency and order. The freak spots exhibit a “flower” shape pattern. If a parabolic potential is introduced, the HG beam forms crawling soliton pairs or merges into a single bounded breathing soliton by adjusting the correlation among the Lévy index, nonlinear and parabolic coefficients. By increasing the nonlinear coefficient in the negative QPM regime, the defocusing HG beam emits several “filiform” breathing solitons during its propagation, which move in a parallel straight line to each other. The HG beam is transformed into a single fine breathing soliton after being focused under a positive QPM. The time of the formation and breathing rate varies with the Lévy index, QPM and nonlinear coefficients. Moreover, the number of solitons changes irregularly with modes. These results are significant for applications in optical communication, optical device design, and optical signal processing.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142421764","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 : 2024-09-26DOI: 10.1016/j.cnsns.2024.108370
Chaobao Huang , Na An , Xijun Yu , Hu Chen
In this work, the time-fractional sine-Gordon equation with Neumann boundary conditions is considered, where the solutions exhibit typical weak singularities at initial time. By introducing an intermediate variable, the original problem can be equivalently written as a low-order coupled system. Utilizing the nonuniform corrected L1 scheme in time and the finite difference scheme in space, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability of the proposed scheme is rigorously established. Meanwhile, a sharp pointwise-in-time error analysis is developed. In particular, the deriving convergent result implies that the error away from the initial time reaches the optimal convergence rate of by merely taking the grading parameter for any . Finally, numerical results are provided to verify that the proposed scheme achieves optimal convergence rates in both temporal and spatial directions.
{"title":"Pointwise-in-time error analysis of the corrected L1 scheme for a time-fractional sine-Gordon equation","authors":"Chaobao Huang , Na An , Xijun Yu , Hu Chen","doi":"10.1016/j.cnsns.2024.108370","DOIUrl":"10.1016/j.cnsns.2024.108370","url":null,"abstract":"<div><div>In this work, the time-fractional sine-Gordon equation with Neumann boundary conditions is considered, where the solutions exhibit typical weak singularities at initial time. By introducing an intermediate variable, the original problem can be equivalently written as a low-order coupled system. Utilizing the nonuniform corrected L1 scheme in time and the finite difference scheme in space, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability of the proposed scheme is rigorously established. Meanwhile, a sharp pointwise-in-time error analysis is developed. In particular, the deriving convergent result implies that the error away from the initial time reaches the optimal convergence rate of <span><math><mrow><mn>2</mn><mo>−</mo><mi>α</mi><mo>/</mo><mn>2</mn></mrow></math></span> by merely taking the grading parameter <span><math><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></math></span> for any <span><math><mrow><mn>1</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></math></span>. Finally, numerical results are provided to verify that the proposed scheme achieves optimal convergence rates in both temporal and spatial directions.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142329692","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 : 2024-09-26DOI: 10.1016/j.cnsns.2024.108368
F. Guerrero , E. Castillo , F. Galarce , D.R.Q. Pacheco
Non-Newtonian fluids are of interest in industrial sectors, biological problems and other natural phenomena. This work proposes rheologically-dependent, spatially and temporally high-order non-residual stabilized finite element formulations. The accuracy of the methods is assessed by tackling highly-convective time-dependent power-law flows. The spatial approximation uses Lagrangian finite elements up to fourth order. The temporal integration is done via backward differentiation formulas of order one, two and three. A key aspect of our work is using a non-residual orthogonal variational multiscale (VMS) formulation to stabilize dominant convection and to allow equal-order interpolation of velocity and pressure. Our VMS method uses dynamic nonlinear subscales, which have not been tested so far for generalized Newtonian fluids. In this work, the use of high-order temporal discretizations for the subscale components is systematically evaluated. Numerical experiments consider the flow over a confined cylinder for Reynolds numbers between 40 and 400 and power-law indices between 0.4 and 1.8. Numerical testing demonstrates the method to be stable in all combinations of spatial and temporal orders. Our results show that using high-order spatial discretizations more accurately approximates boundary layers and viscosity fields. Moreover, higher temporal orders allow using larger time steps while still capturing highly dynamic behaviors with better resolution in frequency spectra.
{"title":"Spatially and temporally high-order dynamic nonlinear variational multiscale methods for generalized Newtonian flows","authors":"F. Guerrero , E. Castillo , F. Galarce , D.R.Q. Pacheco","doi":"10.1016/j.cnsns.2024.108368","DOIUrl":"10.1016/j.cnsns.2024.108368","url":null,"abstract":"<div><div>Non-Newtonian fluids are of interest in industrial sectors, biological problems and other natural phenomena. This work proposes rheologically-dependent, spatially and temporally high-order non-residual stabilized finite element formulations. The accuracy of the methods is assessed by tackling highly-convective time-dependent power-law flows. The spatial approximation uses Lagrangian finite elements up to fourth order. The temporal integration is done via backward differentiation formulas of order one, two and three. A key aspect of our work is using a non-residual orthogonal variational multiscale (VMS) formulation to stabilize dominant convection and to allow equal-order interpolation of velocity and pressure. Our VMS method uses dynamic nonlinear subscales, which have not been tested so far for generalized Newtonian fluids. In this work, the use of high-order temporal discretizations for the subscale components is systematically evaluated. Numerical experiments consider the flow over a confined cylinder for Reynolds numbers between 40 and 400 and power-law indices between 0.4 and 1.8. Numerical testing demonstrates the method to be stable in all combinations of spatial and temporal orders. Our results show that using high-order spatial discretizations more accurately approximates boundary layers and viscosity fields. Moreover, higher temporal orders allow using larger time steps while still capturing highly dynamic behaviors with better resolution in frequency spectra.</div></div>","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":null,"pages":null},"PeriodicalIF":3.4,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142329579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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