We propose a numerical method based on the Lie-Poisson reduction for a class ofstochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $mathfrak{g}^∗$ of the Liealgebra related to the Lie group manifold where the system is located, which is also the reducedform of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentummapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplecticmethods on the cotangent bundle to integrators on $mathfrak{g}^∗.$ Stochastic generating functions creatingstochastic symplectic methods are used to construct the schemes. An application to the stochasticrigid body system illustrates the theory and provides numerical validation of the method.
{"title":"Lie-Poisson Numerical Method for a Class of Stochastic Lie-Poisson Systems","authors":"Qianqian Liu, Lijin Wang","doi":"10.4208/ijnam2024-1004","DOIUrl":"https://doi.org/10.4208/ijnam2024-1004","url":null,"abstract":"We propose a numerical method based on the Lie-Poisson reduction for a class of\u0000stochastic Lie-Poisson systems. Such system is transformed to SDE on the dual $mathfrak{g}^∗$ of the Lie\u0000algebra related to the Lie group manifold where the system is located, which is also the reduced\u0000form of a stochastic Hamiltonian system on the cotangent bundle of the Lie group by momentum\u0000mapping. Stochastic Poisson integrators are obtained by discretely reducing stochastic symplectic\u0000methods on the cotangent bundle to integrators on $mathfrak{g}^∗.$ Stochastic generating functions creating\u0000stochastic symplectic methods are used to construct the schemes. An application to the stochastic\u0000rigid body system illustrates the theory and provides numerical validation of the method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083350","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 prove the almost sure convergence of the explicit-in-time Finite Volume Methodwith monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locallyLipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use thestandard CFL condition and a martingale exponential inequality on sets whose probabilities areconverging towards one. Then, with the help of stopping times on those sets, we apply theoremsof convergence for approximate kinetic solutions of balance laws with stochastic forcing.
{"title":"Convergence of the Finite Volume Method for Stochastic Hyperbolic Scalar Conservation Laws: A Proof By Truncation on the Sample-Time Space","authors":"Sylvain Dotti","doi":"10.4208/ijnam2024-1005","DOIUrl":"https://doi.org/10.4208/ijnam2024-1005","url":null,"abstract":"We prove the almost sure convergence of the explicit-in-time Finite Volume Method\u0000with monotone fluxes towards the unique solution of the scalar hyperbolic balance law with locally\u0000Lipschitz continuous flux and additive noise driven by a cylindrical Wiener process. We use the\u0000standard CFL condition and a martingale exponential inequality on sets whose probabilities are\u0000converging towards one. Then, with the help of stopping times on those sets, we apply theorems\u0000of convergence for approximate kinetic solutions of balance laws with stochastic forcing.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"70 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083514","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 work we derive a posteriori error estimates for the convection-diffusion-reactionequation coupled with the Darcy-Forchheimer problem by a nonlinear external source dependingon the concentration of the fluid. We introduce the variational formulation associated to theproblem, and discretize it by using the finite element method. We prove optimal a posteriorierrors with two types of calculable error indicators. The first one is linked to the linearization andthe second one to the discretization. Then we find upper and lower error bounds under additionalregularity assumptions on the exact solutions. Finally, numerical computations are performed toshow the effectiveness of the obtained error indicators.
{"title":"A Posteriori Error Estimates for Darcy-Forchheimer’s Problem Coupled with the Convection-Diffusion-Reaction Equation","authors":"Faouzi Triki,Toni Sayah, Georges Semaan","doi":"10.4208/ijnam2024-1003","DOIUrl":"https://doi.org/10.4208/ijnam2024-1003","url":null,"abstract":"In this work we derive a posteriori error estimates for the convection-diffusion-reaction\u0000equation coupled with the Darcy-Forchheimer problem by a nonlinear external source depending\u0000on the concentration of the fluid. We introduce the variational formulation associated to the\u0000problem, and discretize it by using the finite element method. We prove optimal a posteriori\u0000errors with two types of calculable error indicators. The first one is linked to the linearization and\u0000the second one to the discretization. Then we find upper and lower error bounds under additional\u0000regularity assumptions on the exact solutions. Finally, numerical computations are performed to\u0000show the effectiveness of the obtained error indicators.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139083411","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 develop a novel meshless, ray-based deep neural network algorithm for solving the high-frequency Helmholtz scattering problem in the unbounded domain. While our recent work [44] designed a deep neural network method for solving the Helmholtz equation over (cid:12)nite bounded domains, this paper deals with the more general and di(cid:14)cult case of unbounded regions. By using the perfectly matched layer method, the original mathematical model in the unbounded domain is transformed into a new format of second-order system in a (cid:12)nite bounded domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz equation in the bounded domain, the new system is equipped with variable coe(cid:14)cients. Then, a deep neural network algorithm is designed for the new system, where the rays in various random directions are used as the basis of the numerical solution. Various numerical examples have been carried out to demonstrate the accuracy and e(cid:14)ciency of the proposed numerical method. The proposed method has the advantage of easy implementation and meshless while maintaining high accuracy. To the best of the author’s knowledge, this is the (cid:12)rst deep neural network method to solve the Helmholtz equation in the unbounded domain.
{"title":"A Novel Deep Neural Network Algorithm for the Helmholtz Scattering Problem In the Unbounded Domain","authors":"Andy L Yang","doi":"10.4208/ijnam2023-1032","DOIUrl":"https://doi.org/10.4208/ijnam2023-1032","url":null,"abstract":". In this paper, we develop a novel meshless, ray-based deep neural network algorithm for solving the high-frequency Helmholtz scattering problem in the unbounded domain. While our recent work [44] designed a deep neural network method for solving the Helmholtz equation over (cid:12)nite bounded domains, this paper deals with the more general and di(cid:14)cult case of unbounded regions. By using the perfectly matched layer method, the original mathematical model in the unbounded domain is transformed into a new format of second-order system in a (cid:12)nite bounded domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz equation in the bounded domain, the new system is equipped with variable coe(cid:14)cients. Then, a deep neural network algorithm is designed for the new system, where the rays in various random directions are used as the basis of the numerical solution. Various numerical examples have been carried out to demonstrate the accuracy and e(cid:14)ciency of the proposed numerical method. The proposed method has the advantage of easy implementation and meshless while maintaining high accuracy. To the best of the author’s knowledge, this is the (cid:12)rst deep neural network method to solve the Helmholtz equation in the unbounded domain.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143752","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 present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.
{"title":"Unconditionally Energy Stable and First-Order Accurate Numerical Schemes for the Heat Equation with Uncertain Temperature-Dependent Conductivity","authors":"Fiordilino, J. A., Winger, M.","doi":"10.4208/ijnam2023-1035","DOIUrl":"https://doi.org/10.4208/ijnam2023-1035","url":null,"abstract":"In this paper, we present first-order accurate numerical methods for solution of the heat equation with uncertain temperature-dependent thermal conductivity. Each algorithm yields a shared coefficient matrix for the ensemble set improving computational efficiency. Both mixed and Robin-type boundary conditions are treated. In contrast with alternative, related methodologies, stability and convergence are unconditional. In particular, we prove unconditional, energy stability and optimal-order error estimates. A battery of numerical tests are presented to illustrate both the theory and application of these algorithms.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219537","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 conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.
{"title":"A Conforming Dg Method for the Biharmonic Equation on Polytopal Meshes","authors":"Xiu Ye null, Shangyou Zhang","doi":"10.4208/ijnam2023-1037","DOIUrl":"https://doi.org/10.4208/ijnam2023-1037","url":null,"abstract":"A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195304","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}
Champike Attanayake, So-Hsiang Chou null, Quanling Deng
Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.
{"title":"High-Order Enriched Finite Element Methods for Elliptic Interface Problems with Discontinuous Solutions","authors":"Champike Attanayake, So-Hsiang Chou null, Quanling Deng","doi":"10.4208/ijnam2023-1038","DOIUrl":"https://doi.org/10.4208/ijnam2023-1038","url":null,"abstract":"Elliptic interface problems whose solutions are $C^0$ continuous have been well studied over the past two decades. The well-known numerical methods include the strongly stable generalized finite element method (SGFEM) and immersed FEM (IFEM). In this paper, we study numerically a larger class of elliptic interface problems where their solutions are discontinuous. A direct application of these existing methods fails immediately as the approximate solution is in a larger space that covers discontinuous functions. We propose a class of high-order enriched unfitted FEMs to solve these problems with implicit or Robin-type interface jump conditions. We design new enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. A linear enriched method in 1D was recently developed using one enrichment function and we generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. The natural tensor product extension to the 2D case is demonstrated. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. We also establish superconvergence at all discretization nodes (including exact nodal values in special cases). Numerical examples are provided to confirm the theory. Finally, to prove the efficiency of the method for practical problems, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195306","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}
P.-A. Gourdain, M. Evans, H. R. Hasson, J. R. Young, I. West-Abdallah null, M. B. Adams
{"title":"A Fully Implicit Method Using Nodal Radial Basis Functions to Solve the Linear Advection Equation","authors":"P.-A. Gourdain, M. Evans, H. R. Hasson, J. R. Young, I. West-Abdallah null, M. B. Adams","doi":"10.4208/ijnam2023-1018","DOIUrl":"https://doi.org/10.4208/ijnam2023-1018","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135421665","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}
Sean Ingimarson, Monika Neda, Leo G. Rebholz, Jorge Reyes null, An Vu
{"title":"Improved Long Time Accuracy for Projection Methods for Navier-Stokes Equations Using Emac Formulation","authors":"Sean Ingimarson, Monika Neda, Leo G. Rebholz, Jorge Reyes null, An Vu","doi":"10.4208/ijnam2023-1008","DOIUrl":"https://doi.org/10.4208/ijnam2023-1008","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136370904","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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