Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry gave suboptimal conditions for existence of solutions and proposed an $H^{2}$-conformal finite element method to approximate them. In this paper the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method, is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness.
{"title":"Computation of Miura surfaces with gradient Dirichlet boundary conditions","authors":"Frédéric Marazzato","doi":"10.1093/imanum/draf033","DOIUrl":"https://doi.org/10.1093/imanum/draf033","url":null,"abstract":"Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry gave suboptimal conditions for existence of solutions and proposed an $H^{2}$-conformal finite element method to approximate them. In this paper the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method, is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144153379","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}
In this paper we propose and analyse a second-order accurate (in both time and space) numerical scheme for the Poisson–Nernst–Planck–Navier–Stokes system, which describes the ion electro-diffusion in fluids. In particular, the Poisson–Nernst–Planck (PNP) equation is reformulated as a nonconstant mobility gradient flow in the energetic variational approach. The marker and cell finite difference method is chosen as the spatial discretization, which facilitates the analysis for the fluid part. In the temporal discretization the mobility function is computed by a second-order extrapolation formula for the sake of unique solvability analysis, while a modified Crank–Nicolson approximation is applied to the singular logarithmic nonlinear term. Nonlinear artificial regularization terms are added in the chemical potential part, so that the positivity-preserving property could be theoretically proved. Meanwhile, a second-order accurate, semi-implicit approximation is applied to the convective term in the PNP evolutionary equation, and the fluid momentum equation is similarly computed. In addition, an optimal rate convergence analysis is provided, based on the higher order asymptotic expansion for the numerical solution, and the rough and refined error estimate techniques. The following combined theoretical properties have been established for the second-order accurate numerical method: (i) second-order accuracy, (ii) unique solvability and positivity, (iii) total energy stability and (iv) optimal rate convergence. A few numerical results are displayed to validate the theoretical analysis.
{"title":"A second-order accurate, positivity-preserving numerical scheme for the Poisson–Nernst–Planck–Navier–Stokes system","authors":"Yuzhe Qin, Cheng Wang","doi":"10.1093/imanum/draf027","DOIUrl":"https://doi.org/10.1093/imanum/draf027","url":null,"abstract":"In this paper we propose and analyse a second-order accurate (in both time and space) numerical scheme for the Poisson–Nernst–Planck–Navier–Stokes system, which describes the ion electro-diffusion in fluids. In particular, the Poisson–Nernst–Planck (PNP) equation is reformulated as a nonconstant mobility gradient flow in the energetic variational approach. The marker and cell finite difference method is chosen as the spatial discretization, which facilitates the analysis for the fluid part. In the temporal discretization the mobility function is computed by a second-order extrapolation formula for the sake of unique solvability analysis, while a modified Crank–Nicolson approximation is applied to the singular logarithmic nonlinear term. Nonlinear artificial regularization terms are added in the chemical potential part, so that the positivity-preserving property could be theoretically proved. Meanwhile, a second-order accurate, semi-implicit approximation is applied to the convective term in the PNP evolutionary equation, and the fluid momentum equation is similarly computed. In addition, an optimal rate convergence analysis is provided, based on the higher order asymptotic expansion for the numerical solution, and the rough and refined error estimate techniques. The following combined theoretical properties have been established for the second-order accurate numerical method: (i) second-order accuracy, (ii) unique solvability and positivity, (iii) total energy stability and (iv) optimal rate convergence. A few numerical results are displayed to validate the theoretical analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"244 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144165275","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}
In this paper we devise and analyze a Banach-spaced mixed virtual element scheme for the steady motion of $rho $-type Brinkman–Forchheimer equation with strongly symmetric stress. Our approach introduces stress and vorticity as additional variables, enabling the elimination of pressure from the original unknowns, which can later be recovered using a postprocessing formula based solely on the stress. Consequently, a mixed variational formulation of the velocity and these new unknowns has been obtained within a Banach space framework. We then propose the $mathbb{H}({mathbf{div}}_varrho ;varOmega )$-conforming virtual element method, where $varrho $ is the conjugate of $rho $, to discretize this formulation and establish the existence and uniqueness of the discrete solution, along with stability bounds, using the Browder–Minty theorem without imposing any assumptions on the data. Furthermore, convergence analysis for all variables in their natural norms is conducted, demonstrating an optimal rate of convergence. Finally, several numerical experiments are presented to illustrate the efficiency and validity of the proposed method.
{"title":"An Lρ spaces-based mixed virtual element method for the steady ρ-type Brinkman–Forchheimer problem based on the velocity–stress–vorticity formulation","authors":"Zeinab Gharibi, Mehdi Dehghan","doi":"10.1093/imanum/draf029","DOIUrl":"https://doi.org/10.1093/imanum/draf029","url":null,"abstract":"In this paper we devise and analyze a Banach-spaced mixed virtual element scheme for the steady motion of $rho $-type Brinkman–Forchheimer equation with strongly symmetric stress. Our approach introduces stress and vorticity as additional variables, enabling the elimination of pressure from the original unknowns, which can later be recovered using a postprocessing formula based solely on the stress. Consequently, a mixed variational formulation of the velocity and these new unknowns has been obtained within a Banach space framework. We then propose the $mathbb{H}({mathbf{div}}_varrho ;varOmega )$-conforming virtual element method, where $varrho $ is the conjugate of $rho $, to discretize this formulation and establish the existence and uniqueness of the discrete solution, along with stability bounds, using the Browder–Minty theorem without imposing any assumptions on the data. Furthermore, convergence analysis for all variables in their natural norms is conducted, demonstrating an optimal rate of convergence. Finally, several numerical experiments are presented to illustrate the efficiency and validity of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066911","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}
Consider the interaction of biharmonic waves with a periodic array of cavities, characterized by the Kirchhoff–Love model. This paper investigates the perfectly matched layer (PML) formulation and its numerical solution to the governing biharmonic wave equation. The study establishes the well-posedness of the associated variational problem employing the Fredholm alternative theorem. Based on the examination of an auxiliary problem in the PML layer, exponential convergence of the PML solution is attained. Moreover, it develops and compares three decomposition methods alongside their corresponding mixed finite element formulations, incorporating interior penalty techniques for solving the PML problem. Numerical experiments validate the effectiveness of the proposed methods in absorbing outgoing waves within the PML layers and suppressing oscillations in the bending moment of biharmonic waves near the cavity’s surface.
{"title":"Numerical solution to the PML problem of the biharmonic wave scattering in periodic structures","authors":"Peijun Li, Xiaokai Yuan","doi":"10.1093/imanum/draf025","DOIUrl":"https://doi.org/10.1093/imanum/draf025","url":null,"abstract":"Consider the interaction of biharmonic waves with a periodic array of cavities, characterized by the Kirchhoff–Love model. This paper investigates the perfectly matched layer (PML) formulation and its numerical solution to the governing biharmonic wave equation. The study establishes the well-posedness of the associated variational problem employing the Fredholm alternative theorem. Based on the examination of an auxiliary problem in the PML layer, exponential convergence of the PML solution is attained. Moreover, it develops and compares three decomposition methods alongside their corresponding mixed finite element formulations, incorporating interior penalty techniques for solving the PML problem. Numerical experiments validate the effectiveness of the proposed methods in absorbing outgoing waves within the PML layers and suppressing oscillations in the bending moment of biharmonic waves near the cavity’s surface.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"45 16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers robust solutions to a class of nonlinear least squares problems using a min-max optimization approach. We give an explicit formula for the value function of the inner maximization problem and show the existence of global minimax points. We establish error bounds from any solution of the nonlinear least squares problem to the solution set of the robust nonlinear least squares problem. Moreover, we propose a smoothing method for finding a global minimax point of the min-max problem by using the formula and show that finding an $varepsilon $ minimax critical point of the min-max problem needs at most $O(varepsilon ^{-2} +delta ^{2} varepsilon ^{-3})$ evaluations of the function value and gradients of the objective function, where $delta $ is the tolerance of the noise. Numerical results of integral equations with uncertain data demonstrate the robustness of solutions of our approach and unstable behavior of least squares solutions disregarding uncertainties in the data.
{"title":"Robust solutions of nonlinear least squares problems via min-max optimization","authors":"Xiaojun Chen, C T Kelley","doi":"10.1093/imanum/draf026","DOIUrl":"https://doi.org/10.1093/imanum/draf026","url":null,"abstract":"This paper considers robust solutions to a class of nonlinear least squares problems using a min-max optimization approach. We give an explicit formula for the value function of the inner maximization problem and show the existence of global minimax points. We establish error bounds from any solution of the nonlinear least squares problem to the solution set of the robust nonlinear least squares problem. Moreover, we propose a smoothing method for finding a global minimax point of the min-max problem by using the formula and show that finding an $varepsilon $ minimax critical point of the min-max problem needs at most $O(varepsilon ^{-2} +delta ^{2} varepsilon ^{-3})$ evaluations of the function value and gradients of the objective function, where $delta $ is the tolerance of the noise. Numerical results of integral equations with uncertain data demonstrate the robustness of solutions of our approach and unstable behavior of least squares solutions disregarding uncertainties in the data.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"96 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066912","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}
Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.
{"title":"A walk-on-sphere-motivated finite-difference method for the fractional Poisson equation on a bounded d-dimensional domain","authors":"Daxin Nie, Jing Sun, Weihua Deng","doi":"10.1093/imanum/draf031","DOIUrl":"https://doi.org/10.1093/imanum/draf031","url":null,"abstract":"Inspired by the idea of ‘walk-on-sphere’ algorithm, we propose a novel finite-difference framework for solving the fractional Poisson equation under the help of the Feynman-Kac representation of its solution, i.e., walk-on-sphere-motivated finite-difference scheme. By choosing suitable basis functions in interpolatory quadrature and using graded meshes, the convergence rates can achieve up to $O(h^{2})$ in arbitrary $d$-dimensional bounded Lipschitz domain satisfying the exterior ball condition, where $d>1$; while the convergence rate can reach $O(h^{10})$ in 1-dimensional bounded domain under some regularity assumptions on the source term $f$. Furthermore, we propose a strict convergence analysis and several numerical examples in different domains, including circle, L-shape, pentagram and ball, are provided to illustrate the effectiveness of the above built scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066936","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}
We present and analyze three distinct semidiscrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under $H^{1}$-norm for the first scheme and linear convergence under $H^{1}$-norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under $L^infty $-norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.
{"title":"Convergence analysis of three semidiscrete numerical schemes for nonlocal geometric flows including perimeter terms","authors":"Wei Jiang, Chunmei Su, Ganghui Zhang","doi":"10.1093/imanum/draf015","DOIUrl":"https://doi.org/10.1093/imanum/draf015","url":null,"abstract":"We present and analyze three distinct semidiscrete schemes for solving nonlocal geometric flows incorporating perimeter terms. These schemes are based on the finite difference method, the finite element method and the finite element method with a specific tangential motion. We offer rigorous proofs of quadratic convergence under $H^{1}$-norm for the first scheme and linear convergence under $H^{1}$-norm for the latter two schemes. All error estimates rely on the observation that the error of the nonlocal term can be controlled by the error of the local term. Furthermore, we explore the relationship between the convergence under $L^infty $-norm and manifold distance. Extensive numerical experiments are conducted to verify the convergence analysis, and demonstrate the accuracy of our schemes under various norms for different types of nonlocal flows.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"52 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143893397","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}
In this work we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained, not only on the inputs and labels, but also on the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of the BSDE solution themselves satisfy another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient and the Hessian matrix, represented by the triple of processes $left (Y, Z, varGamma right ).$ All the integrals within this system are discretized by using the Euler–Maruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process $Y$ and the other the process $Z$. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments of up to $50$ dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient in terms of computation time or accuracy compared with other contemporary deep learning-based methodologies.
{"title":"A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations","authors":"Lorenc Kapllani, Long Teng","doi":"10.1093/imanum/draf022","DOIUrl":"https://doi.org/10.1093/imanum/draf022","url":null,"abstract":"In this work we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained, not only on the inputs and labels, but also on the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of the BSDE solution themselves satisfy another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient and the Hessian matrix, represented by the triple of processes $left (Y, Z, varGamma right ).$ All the integrals within this system are discretized by using the Euler–Maruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process $Y$ and the other the process $Z$. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments of up to $50$ dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient in terms of computation time or accuracy compared with other contemporary deep learning-based methodologies.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"39 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143884791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is devoted to constructing an asymptotic preserving method for the defocusing Davey–Stewartson II equation in the semiclassical limit. First, we introduce the Wentzel–Kramers–Brillouin ansatz $varPsi =A^varepsilon e^{iphi ^varepsilon /varepsilon }$ for the equation and obtain the new system for both $A^varepsilon $ and $phi ^varepsilon $, where the complex-valued amplitude function $A^varepsilon $ can avoid automatically the singularity of the quantum potential in vacuum. Secondly, we prove the local existence of the solutions of the new system for $tin [0,T]$, and show that the solutions of the new system are convergent to the limit when $varepsilon rightarrow 0$. Finally, we construct a second-order time-splitting Fourier spectral method for the new system and numerous numerical experiments show that the method is uniformly accurate with respect to $varepsilon $, i.e., its accuracy does not deteriorate for vanishing $varepsilon $, and it is an asymptotic preserving one. However, it might not be uniformly convergent.
{"title":"An asymptotic preserving scheme for the defocusing Davey–Stewartson II equation in the semiclassical limit","authors":"Dandan Wang, Hanquan Wang","doi":"10.1093/imanum/draf019","DOIUrl":"https://doi.org/10.1093/imanum/draf019","url":null,"abstract":"This article is devoted to constructing an asymptotic preserving method for the defocusing Davey–Stewartson II equation in the semiclassical limit. First, we introduce the Wentzel–Kramers–Brillouin ansatz $varPsi =A^varepsilon e^{iphi ^varepsilon /varepsilon }$ for the equation and obtain the new system for both $A^varepsilon $ and $phi ^varepsilon $, where the complex-valued amplitude function $A^varepsilon $ can avoid automatically the singularity of the quantum potential in vacuum. Secondly, we prove the local existence of the solutions of the new system for $tin [0,T]$, and show that the solutions of the new system are convergent to the limit when $varepsilon rightarrow 0$. Finally, we construct a second-order time-splitting Fourier spectral method for the new system and numerous numerical experiments show that the method is uniformly accurate with respect to $varepsilon $, i.e., its accuracy does not deteriorate for vanishing $varepsilon $, and it is an asymptotic preserving one. However, it might not be uniformly convergent.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"79 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143878130","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}
We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie–Poisson system on the dual of the magnetic extension Lie algebra $mathfrak{f}=mathfrak{su}(N)ltimes mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie–Poisson systems on the dual of semidirect product Lie algebras of the form $mathfrak{f}=mathfrak{g}ltimes mathfrak{g^{*}}$, where $mathfrak{g}$ is a $J$-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie–Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible MHD and Hazeltine’s model.
{"title":"Spatio-temporal Lie–Poisson discretization for incompressible magnetohydrodynamics on the sphere","authors":"Klas Modin, Michael Roop","doi":"10.1093/imanum/draf024","DOIUrl":"https://doi.org/10.1093/imanum/draf024","url":null,"abstract":"We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie–Poisson system on the dual of the magnetic extension Lie algebra $mathfrak{f}=mathfrak{su}(N)ltimes mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie–Poisson systems on the dual of semidirect product Lie algebras of the form $mathfrak{f}=mathfrak{g}ltimes mathfrak{g^{*}}$, where $mathfrak{g}$ is a $J$-quadratic Lie algebra. The time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves a modified Lie–Poisson structure and corresponding Casimir functions, and that the modified structure and Casimirs converge to the continuous ones. The method is demonstrated for two models of magnetic fluids: incompressible MHD and Hazeltine’s model.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143878131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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