{"title":"A New Mixed Method for the Stokes Equations Based on Stress-Velocity-Vorticity Formulation","authors":"E. Miglio, M. Penati","doi":"10.4208/jms.v52n3.19.05","DOIUrl":"https://doi.org/10.4208/jms.v52n3.19.05","url":null,"abstract":"","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43388531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A parameterized generalized successive overrelaxation (PGSOR) method for a class of block two-by-two linear system is established in this paper. The convergence theorem of the method is proved under suitable assumptions on iteration parameters. Besides, we obtain a functional equation between the parameters and the eigenvalues of the iteration matrix for this method. Furthermore, an accelerated variant of the PGSOR (APGSOR) method is also presented in order to raise the convergence rate. Finally, numerical experiments are carried out to confirm the theoretical analysis as well as the feasibility and the efficiency of the PGSOR method and its variant. AMS subject classifications: 65F10, 65F50, 65F08
{"title":"Parameterized GSOR Method for a Class of Complex Symmetric Systems of Linear Equations","authors":"Yujiang Wu","doi":"10.4208/JMS.V52N1.19.02","DOIUrl":"https://doi.org/10.4208/JMS.V52N1.19.02","url":null,"abstract":"A parameterized generalized successive overrelaxation (PGSOR) method for a class of block two-by-two linear system is established in this paper. The convergence theorem of the method is proved under suitable assumptions on iteration parameters. Besides, we obtain a functional equation between the parameters and the eigenvalues of the iteration matrix for this method. Furthermore, an accelerated variant of the PGSOR (APGSOR) method is also presented in order to raise the convergence rate. Finally, numerical experiments are carried out to confirm the theoretical analysis as well as the feasibility and the efficiency of the PGSOR method and its variant. AMS subject classifications: 65F10, 65F50, 65F08","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49075309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A remeshed Vortex method is proposed in this work to simulate threedimensional incompressible flows. The convection equation is solved on particles, using a Vortex method, which are then remeshed on a Cartesian underlying grid. The other differential operators involved in the governing incompressible Navier-Stokes equations are discretized on the grid, through finite differences method or in spectral space. In the present work, the redistribution of the particles on the Cartesian mesh is performed using a directional splitting, allowing to save significant computational efforts especially in the case of 3D flows. A coupling of this semi-Lagrangian method with an immersed boundary method, namely the Brinkman penalization technique, is proposed in this paper in order to efficiently take into account the presence of solid and porous obstacles in the fluid flow and then to perform passive flow control using porous medium. This method, which combines the robustness of particle methods and the flexibility of penalization method, is validated and exploited in the context of different flow physics. AMS subject classifications: 65M22, 35Q30, 76S05
{"title":"A Semi-Langrangian Vortex Penalization Method for 3D Incompressible Flows","authors":"Chlo Mimeau sci","doi":"10.4208/jms.v52n3.19.04","DOIUrl":"https://doi.org/10.4208/jms.v52n3.19.04","url":null,"abstract":"A remeshed Vortex method is proposed in this work to simulate threedimensional incompressible flows. The convection equation is solved on particles, using a Vortex method, which are then remeshed on a Cartesian underlying grid. The other differential operators involved in the governing incompressible Navier-Stokes equations are discretized on the grid, through finite differences method or in spectral space. In the present work, the redistribution of the particles on the Cartesian mesh is performed using a directional splitting, allowing to save significant computational efforts especially in the case of 3D flows. A coupling of this semi-Lagrangian method with an immersed boundary method, namely the Brinkman penalization technique, is proposed in this paper in order to efficiently take into account the presence of solid and porous obstacles in the fluid flow and then to perform passive flow control using porous medium. This method, which combines the robustness of particle methods and the flexibility of penalization method, is validated and exploited in the context of different flow physics. AMS subject classifications: 65M22, 35Q30, 76S05","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42049153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Triviality of a Certain Kind of Shrinking Solitons","authors":"Zhuhong Zhang","doi":"10.4208/JMS.V52N2.19.04","DOIUrl":"https://doi.org/10.4208/JMS.V52N2.19.04","url":null,"abstract":"","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47740116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a monolithic algorithm for solving fluid-structure interaction. The Updated Lagrangian framework is used for the incompressible neo-hookean structure and Arbitrary Lagrangian Eulerian coordinate is employed for the Navier-Stokes equations. The algorithm uses a global mesh for the fluid-structure domain which is compatible with the fluid-structure interface. At each time step, a non-linear system is solved in a domain corresponding to the precedent time step. It is a semi-implicit algorithm in the sense that the velocity, the pressure are computed implicitly, but the domain is updated explicitly. Using one velocity field defined over the fluid-structure mesh, and globally continuous finite elements, the continuity of the velocity at the interface is automatically verified. The equation of the continuity of the stress at the interface does not appear in this formulation due to action and reaction principle. The stability in time is proved. A second algorithm is introduced where at each time step, only a linear system is solved in order to find the velocity and the pressure. Numerical experiments are presented. AMS subject classifications: 74F10, 65M12
{"title":"Stable Semi-Implicit Monolithic Scheme for Interaction Between Incompressible Neo-hookean Structure and Navier-Stokes Fluid","authors":"Cornel Marius Murea sci","doi":"10.4208/jms.v52n4.19.05","DOIUrl":"https://doi.org/10.4208/jms.v52n4.19.05","url":null,"abstract":"We present a monolithic algorithm for solving fluid-structure interaction. The Updated Lagrangian framework is used for the incompressible neo-hookean structure and Arbitrary Lagrangian Eulerian coordinate is employed for the Navier-Stokes equations. The algorithm uses a global mesh for the fluid-structure domain which is compatible with the fluid-structure interface. At each time step, a non-linear system is solved in a domain corresponding to the precedent time step. It is a semi-implicit algorithm in the sense that the velocity, the pressure are computed implicitly, but the domain is updated explicitly. Using one velocity field defined over the fluid-structure mesh, and globally continuous finite elements, the continuity of the velocity at the interface is automatically verified. The equation of the continuity of the stress at the interface does not appear in this formulation due to action and reaction principle. The stability in time is proved. A second algorithm is introduced where at each time step, only a linear system is solved in order to find the velocity and the pressure. Numerical experiments are presented. AMS subject classifications: 74F10, 65M12","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45790073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we use the topological and shape gradient framework, to optimize a current carrying multicables. The geometry of the multicables is modeled as a coated inclusions with different conductivities and the problem we are interested is the location of the inclusions to get a suitable thermal environnent. We solve numerically the optimization problem using topological and shape gradient strategy. Finally, we present some numerical experiments. AMS subject classifications: 49Q10, 49Q12, 65K10, 68W25.
{"title":"Optimization of Current Carrying Muticables using Topological and Shape Sensitivity","authors":"Zakaria Belhachmi sci","doi":"10.4208/jms.v52n4.19.04","DOIUrl":"https://doi.org/10.4208/jms.v52n4.19.04","url":null,"abstract":"In this paper, we use the topological and shape gradient framework, to optimize a current carrying multicables. The geometry of the multicables is modeled as a coated inclusions with different conductivities and the problem we are interested is the location of the inclusions to get a suitable thermal environnent. We solve numerically the optimization problem using topological and shape gradient strategy. Finally, we present some numerical experiments. AMS subject classifications: 49Q10, 49Q12, 65K10, 68W25.","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46361573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a generalized multivariate fractional Taylor’s and Cauchy’s mean value theorem of the kind f (x,y)= n ∑ j=0 Djα f (x0,y0) Γ(jα+1) +Rn(ξ,η), f (x,y)− n ∑ j=0 Djα f (x0,y0) Γ(jα+1) g(x,y)− n ∑ j=0 Dg(x0,y0) Γ(jα+1) = Rn(ξ,η) Tα n (ξ,η) , where 0< α≤ 1, is established. Such expression is precisely the classical Taylor’s and Cauchy’s mean value theorem in the particular case α=1. In addition, detailed expressions for Rn(ξ,η) and Tα n (ξ,η) involving the sequential Caputo fractional derivative are also given. AMS subject classifications: 65M70, 65L60, 41A10, 60H35
{"title":"On Multivariate Fractional Taylor’s and Cauchy’ Mean Value Theorem","authors":"Jinfa Cheng","doi":"10.4208/JMS.V52N1.19.04","DOIUrl":"https://doi.org/10.4208/JMS.V52N1.19.04","url":null,"abstract":"In this paper, a generalized multivariate fractional Taylor’s and Cauchy’s mean value theorem of the kind f (x,y)= n ∑ j=0 Djα f (x0,y0) Γ(jα+1) +Rn(ξ,η), f (x,y)− n ∑ j=0 Djα f (x0,y0) Γ(jα+1) g(x,y)− n ∑ j=0 Dg(x0,y0) Γ(jα+1) = Rn(ξ,η) Tα n (ξ,η) , where 0< α≤ 1, is established. Such expression is precisely the classical Taylor’s and Cauchy’s mean value theorem in the particular case α=1. In addition, detailed expressions for Rn(ξ,η) and Tα n (ξ,η) involving the sequential Caputo fractional derivative are also given. AMS subject classifications: 65M70, 65L60, 41A10, 60H35","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44670778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the reduced-order modelling for Allen-Cahn equation. First, a collection of phase field data, i.e., an ensemble of snapshots of at some time instances is obtained from numerical simulation using a time-space discretization. The full discretization makes use of a temporal scheme based on the scalar auxiliary variable approach and a spatial spectral Galerkin method. It is shown that the time stepping scheme is unconditionally stable. Then a reduced order method is developed using by proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). It is well-known that the Allen-Cahn equations have a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. Our numerical experiments show that the discretized Allen-Cahn system resulting from the POD-DEIM method inherits this favorable property by using the scalar auxiliary variable approach. A few numerical results are presented to illustrate the performance of the proposed reduced order method. In particular, the numerical results show that the computational efficiency is significantly enhanced as compared to directly solving the full order system. AMS subject classifications: 76T10, 78M34, 74S25
{"title":"Reduced-Order Modelling for the Allen-Cahn Equation Based on Scalar Auxiliary Variable Approaches","authors":"Xiaolan Zhou sci","doi":"10.4208/jms.v52n3.19.03","DOIUrl":"https://doi.org/10.4208/jms.v52n3.19.03","url":null,"abstract":"In this article, we study the reduced-order modelling for Allen-Cahn equation. First, a collection of phase field data, i.e., an ensemble of snapshots of at some time instances is obtained from numerical simulation using a time-space discretization. The full discretization makes use of a temporal scheme based on the scalar auxiliary variable approach and a spatial spectral Galerkin method. It is shown that the time stepping scheme is unconditionally stable. Then a reduced order method is developed using by proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). It is well-known that the Allen-Cahn equations have a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. Our numerical experiments show that the discretized Allen-Cahn system resulting from the POD-DEIM method inherits this favorable property by using the scalar auxiliary variable approach. A few numerical results are presented to illustrate the performance of the proposed reduced order method. In particular, the numerical results show that the computational efficiency is significantly enhanced as compared to directly solving the full order system. AMS subject classifications: 76T10, 78M34, 74S25","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43120521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global Existence and Blow-Up in a p(x)-Laplace Equation with Dirichlet Boundary Conditions","authors":"Yuhua Yang","doi":"10.4208/JMS.V52N2.19.01","DOIUrl":"https://doi.org/10.4208/JMS.V52N2.19.01","url":null,"abstract":"","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48093220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We first get an existence and uniqueness result for a nonlinear eigenvalue problem. Then, we establish the constant rank theorem for the problem and use it to get a convexity property of the solution. AMS subject classifications: 35J15, 35P30, 52A99
{"title":"The Convexity of a Fully Nonlinear Operator and Its Related Eigenvalue Problem","authors":"Jiuzhou Huang","doi":"10.4208/JMS.V52N1.19.07","DOIUrl":"https://doi.org/10.4208/JMS.V52N1.19.07","url":null,"abstract":"We first get an existence and uniqueness result for a nonlinear eigenvalue problem. Then, we establish the constant rank theorem for the problem and use it to get a convexity property of the solution. AMS subject classifications: 35J15, 35P30, 52A99","PeriodicalId":43526,"journal":{"name":"Journal of Mathematical Study","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47870891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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