R. Araya, C. Bertoglio, Cristian Cárcamo, D. Nolte, S. Uribe
Magnetic Resonance Imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier-Stokes equations, thereby avoiding the use of invasive technologies . However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state- of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier-Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field . However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails . Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements.
{"title":"Convergence Analysis of Pressure Reconstruction Methods from discrete velocities","authors":"R. Araya, C. Bertoglio, Cristian Cárcamo, D. Nolte, S. Uribe","doi":"10.1051/m2an/2023021","DOIUrl":"https://doi.org/10.1051/m2an/2023021","url":null,"abstract":"Magnetic Resonance Imaging allows the measurement of the three-dimensional velocity field in blood flows. Therefore, several methods have been proposed to reconstruct the pressure field from such measurements using the incompressible Navier-Stokes equations, thereby avoiding the use of invasive technologies . However, those measurements are obtained at limited spatial resolution given by the voxel sizes in the image. In this paper, we propose a strategy for the convergence analysis of state- of-the-art pressure reconstruction methods. The methods analyzed are the so-called Pressure Poisson Estimator (PPE) and Stokes Estimator (STE). In both methods, the right-hand side corresponds to the terms that involving the field velocity in the Navier-Stokes equations, with a piecewise linear interpolation of the exact velocity. In the theoretical error analysis, we show that many terms of different order of convergence appear. These are certainly dominated by the lowest-order term, which in most cases stems from the interpolation of the velocity field . However, the numerical results in academic examples indicate that only the PPE may profit of increasing the polynomial order, and that the STE presents a higher accuracy than the PPE, but the interpolation order of the velocity field always prevails . Furthermore, we compare the pressure estimation methods on real MRI data, assessing the impact of different spatial resolutions and polynomial degrees on each method. Here, the results are consistent with the academic test cases in terms of sensitivity to polynomial order as well as the STE showing to be potentially more accurate when compared to reference pressure measurements.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87784473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddlepoint problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuˇska–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Neˇcas–Babuˇska theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ďk. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the balance-preserving properties and applicability of the proposed family of methods.
本文采用基于Banach空间的框架，引入并分析了用于描述带电不可压缩流体动力学的非线性模型Stokes和Poisson-Nernst-Planck耦合方程数值解的混合有限元新方法。由于不可压缩性条件和流体伪应力作为辅助未知数的结合，流体的压力从系统中消除(尽管随后通过后处理公式计算)。反过来，除了静电势和电离粒子的浓度外，我们还使用电场(势的重新标度梯度)和总离子通量作为新的未知数。由此得到的Banach空间中的完全混合变分公式可以写成由两个鞍点问题组成的耦合系统，每个鞍点问题都具有依赖于剩余未知数的非线性源项，以及一个具有线性源项的扰动鞍点问题，该问题反过来又被双线性形式扰动。连续公式的适定性是与Banach定理、Babu - ska - brezzi理论、抽象摄动鞍点问题的可解性以及Banach - ne - cas-Babu - ska定理相结合的不动点策略的结果。为此，我们还对数据采用了较小的假设。利用Brouwer定理和Banach定理，并在任意有限元子空间上调用合适的稳定性条件，用一种类似的方法得出了相关Galerkin格式解的存在唯一性。推导了先验误差估计，以及适合该理论的离散空间的示例，包括，例如，k阶的Raviart-Thomas元素以及阶为ďk的分段多项式。最后，给出了收敛速度，并通过数值实验验证了理论误差范围。这些测试还说明了所提出的方法族的保平衡特性和适用性。
{"title":"New mixed finite element methods for the coupled Stokes and Poisson–Nernst–Planck equations in Banach spaces","authors":"G. Gatica, Claudio Correa, R. Ruiz-Baier","doi":"10.1051/m2an/2023024","DOIUrl":"https://doi.org/10.1051/m2an/2023024","url":null,"abstract":"In this paper we employ a Banach spaces-based framework to introduce and analyze new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson–Nernst–Planck equations, which is a nonlinear model describing the dynamics of electrically charged incompressible fluids. The pressure of the fluid is eliminated from the system (though computed afterwards via a postprocessing formula) thanks to the incompressibility condition and the incorporation of the fluid pseudostress as an auxiliary unknown. In turn, besides the electrostatic potential and the concentration of ionized particles, we use the electric field (rescaled gradient of the potential) and total ionic fluxes as new unknowns. The resulting fully mixed variational formulation in Banach spaces can be written as a coupled system consisting of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddlepoint problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuˇska–Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach–Neˇcas–Babuˇska theorem. For this we also employ smallness assumptions on the data. An analogous approach, but using now both the Brouwer and Banach theorems, and invoking suitable stability conditions on arbitrary finite element subspaces, is employed to conclude the existence and uniqueness of solution for the associated Galerkin scheme. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart–Thomas elements of order k along with piecewise polynomials of degree ďk. Finally, rates of convergence are specified and several numerical experiments confirm the theoretical error bounds. These tests also illustrate the balance-preserving properties and applicability of the proposed family of methods.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73318249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. Three-point entropy stable schemes are extended for partial differential equations of the degenerate convection-diffusion type where a discontinuous space-dependent function is incorporated in the convective flux. Using the compensated compactness theory, convergence of the proposed entropy stable approximations to the entropy weak solution is proved. Assuming the so-called potential condition in the jump discontinuities, an estimate for entropy functions is demonstrated.Finally, using benchmark tests a validation of the efficiency of the entropy stable scheme is provided by comparison with an upwind-type solution.
{"title":"Convergence of entropy stable schemes for degenerate parabolic equations with a discontinuous convection term","authors":"Claudia Acosta, S. Jerez","doi":"10.1051/m2an/2023018","DOIUrl":"https://doi.org/10.1051/m2an/2023018","url":null,"abstract":"Abstract. Three-point entropy stable schemes are extended for partial differential equations of the degenerate convection-diffusion type where a discontinuous space-dependent function is incorporated in the convective flux. Using the compensated compactness theory, convergence of the proposed entropy stable approximations to the entropy weak solution is proved. Assuming the so-called potential condition in the jump discontinuities, an estimate for entropy functions is demonstrated.Finally, using benchmark tests a validation of the efficiency of the entropy stable scheme is provided by comparison with an upwind-type solution.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83172319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aaron Brunk, Egger Herbert, Oliver Habrich, M. Lukácová-Medvidová
The stability of solutions to the Cahn-Hilliard equation with concentration dependent mobility with respect to perturbations is studied by means of relative energy estimates. As a by-product of this analysis, a weak-strong uniqueness principle is derived on the continuous level under realistic regularity assumptions on strong solutions. The stability estimates are further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a-priori error estimates for semi- and fully discrete approximation schemes. Numerical tests are presented for illustration of the theoretical results.
{"title":"Stability and discretization error analysis for the Cahn-Hilliard system via relative energy estimates","authors":"Aaron Brunk, Egger Herbert, Oliver Habrich, M. Lukácová-Medvidová","doi":"10.1051/m2an/2023017","DOIUrl":"https://doi.org/10.1051/m2an/2023017","url":null,"abstract":"The stability of solutions to the Cahn-Hilliard equation with concentration dependent mobility with respect to perturbations is studied by means of relative energy estimates.\u0000\u0000As a by-product of this analysis, a weak-strong uniqueness principle is derived on the continuous level under realistic regularity assumptions on strong solutions.\u0000\u0000The stability estimates are further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a-priori error estimates for semi- and fully discrete approximation schemes. \u0000\u0000Numerical tests are presented for illustration of the theoretical results.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78756250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L 2 -boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.
{"title":"Stable reconstruction of discontinuous solutions to the Cauchy problem in steady-state anisotropic heat conduction with non-smooth coefficients","authors":"M. Bucataru, Iulian Cîmpean, L. Marin","doi":"10.1051/m2an/2023014","DOIUrl":"https://doi.org/10.1051/m2an/2023014","url":null,"abstract":"We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L 2 -boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91220427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor-Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.
{"title":"Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch IgA","authors":"J. Sogn, Stefan Takacs","doi":"10.1051/m2an/2023011","DOIUrl":"https://doi.org/10.1051/m2an/2023011","url":null,"abstract":"We are interested in a fast solver for the Stokes equations, discretized with multi-patch Isogeometric Analysis. In the last years, several inf-sup stable discretizations for the Stokes problem have been proposed, often the analysis was restricted to single-patch domains. We focus on one of the simplest approaches, the isogeometric Taylor-Hood element. We show how stability results for single-patch domains can be carried over to multi-patch domains. While this is possible, the stability strongly depends on the shape of the geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solver that does not suffer from that effect. We give a convergence analysis and provide numerical tests.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86960243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn-Hilliard-Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn-Hilliard-Navier-Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn-Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn-Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.
{"title":"Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities","authors":"Yali Gao, Xiaoming He, Tao Lin, Yanping Lin","doi":"10.1051/m2an/2023012","DOIUrl":"https://doi.org/10.1051/m2an/2023012","url":null,"abstract":"In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn-Hilliard-Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn-Hilliard-Navier-Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn-Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn-Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82228619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.
{"title":"A flux globalization based well-balanced path-conservative central-upwind scheme for the shallow water flows in channels","authors":"Yiming Chen, A. Kurganov, Ming-Ye Na","doi":"10.1051/m2an/2023009","DOIUrl":"https://doi.org/10.1051/m2an/2023009","url":null,"abstract":"We develop a flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the one-dimensional shallow water flows in channels. Challenges in developing numerical methods for the studied system are mainly related to the presence of nonconservative terms modeling the flow when the channel width and bottom topography are discontinuous. We use the path-conservative technique to treat these nonconservative product terms and implement this technique within the flux globalization framework, for which the friction and aforementioned nonconservative terms are incorporated into the global flux: This results in a quasi-conservative system, which is numerically solved using the Riemann-problem-solver-free central-upwind scheme. The WB property of the resulting scheme (that is, its ability to exactly preserve both still- and moving-water equilibria at the discrete level) is ensured by performing piecewise linear reconstruction for the equilibrium variables rather than the conservative variables, and then evaluating the global flux using the obtained point values of the equilibrium quantities. The robustness and excellent performance of the proposed flux globalization based WB PCCU scheme are demonstrated in several numerical examples with both continuous and discontinuous channel width and bottom topography. In these examples, we clearly demonstrate the advantage of the proposed scheme over its simpler counterparts.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83310592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxil- iary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions [11, Huang and Shen, SIAM J. Numer. Anal., 2021] to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H 1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solv- ing Navier-Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.
{"title":"A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier-Stokes equations","authors":"Yana Di, Yuheng Ma, Jie Shen, Jiwei Zhang","doi":"10.1051/m2an/2023007","DOIUrl":"https://doi.org/10.1051/m2an/2023007","url":null,"abstract":"We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxil- iary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions [11, Huang and Shen, SIAM J. Numer. Anal., 2021] to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H 1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solv- ing Navier-Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83099500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Imad El Bouchairi, J. Fadili, Abderrahim El Moataz
In this paper we study continuum limits of the discretized [[EQUATION]] -Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-called [[EQUATION]] -graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the [[EQUATION]] -Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.
{"title":"Continuum limit of [[EQUATION]]-Laplacian evolution problems on graphs: [[EQUATION]]graphons and sparse graphs","authors":"Imad El Bouchairi, J. Fadili, Abderrahim El Moataz","doi":"10.1051/m2an/2023006","DOIUrl":"https://doi.org/10.1051/m2an/2023006","url":null,"abstract":"In this paper we study continuum limits of the discretized [[EQUATION]] -Laplacian evolution problem on sparse graphs with homogeneous Neumann boundary conditions. This goes far beyond known results by handling much more general class of kernels, possibly singular, and graph sequences whose limit are the so-called [[EQUATION]] -graphons. More precisely, we derive a bound on the distance between two continuous-in-time trajectories defined by two different evolution systems (i.e. with different kernels, second member and initial data). Similarly, we provide a bound in the case that one of the trajectories is discrete-in-time and the other is continuous. In turn, these results lead us to establish error estimates of the full discretization of the [[EQUATION]] -Laplacian problem on sparse random graphs. In particular, we provide rate of convergence of solutions for the discrete models to the solution of the continuous problem as the number of vertices grows.","PeriodicalId":50499,"journal":{"name":"Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique","volume":null,"pages":null},"PeriodicalIF":1.9,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87782629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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