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Solving reaction-diffusion problems  with explicit Runge-Kutta exponential methods without order reduction 用不降阶的显式 Runge-Kutta 指数法解决反应扩散问题
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2024-02-08 DOI: 10.1051/m2an/2024011
Begoña Cano, María Jesús Moreta
{"title":"Solving reaction-diffusion problems  with explicit Runge-Kutta exponential methods without order reduction","authors":"Begoña Cano, María Jesús Moreta","doi":"10.1051/m2an/2024011","DOIUrl":"https://doi.org/10.1051/m2an/2024011","url":null,"abstract":"Abstract. In this paper a technique is given to recover the classical order of the method when explicit exponential Runge-Kutta methods integrate reaction-diffusion problems. In the literature, methods of high enough stiff order for problems with vanishing boundary conditions have been constructed, but that implies restricting the coefficients and thus, the number of stages and the computational cost may significantly increase with respect to other methods without those restrictions. In contrast, the technique which is suggested here is cheaper because it just needs, for any method, to add some terms with information only on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139794399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Surface boundary layers   through  a scalar equation with  an eddy viscosity vanishing at the ground 通过标量方程计算地面涡流粘度消失的地表边界层
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2024-02-05 DOI: 10.1051/m2an/2024009
R. Lewandowski, François Legeais, L. Berselli
{"title":"Surface boundary layers   through  a scalar equation with  an eddy viscosity vanishing at the ground","authors":"R. Lewandowski, François Legeais, L. Berselli","doi":"10.1051/m2an/2024009","DOIUrl":"https://doi.org/10.1051/m2an/2024009","url":null,"abstract":"We introduce a scalar elliptic equation defined on a boundary layer given by $Pi_2 times [0, z_{top}]$, where $Pi_2$ is a two dimensional torus, with an eddy vertical eddy viscosity of order \u0000$z^alpha$, $alpha in [0, 1]$, an homogeneous boundary condition at $z=0$, and a Robin condition at $z=z_{top}$. We show the existence of weak solutions to this boundary problem, distinguishing the cases \u0000$0 le alpha <1$ and $alpha = 1$.  Then we carry out several numerical simulations, showing the ability of our model to accurately reproduce profiles close to those predicted by the Monin-Obukhov theory, by calculating stabilizing functions.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139805312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Surface boundary layers   through  a scalar equation with  an eddy viscosity vanishing at the ground 通过标量方程计算地面涡流粘度消失的地表边界层
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2024-02-05 DOI: 10.1051/m2an/2024009
R. Lewandowski, François Legeais, L. Berselli
{"title":"Surface boundary layers   through  a scalar equation with  an eddy viscosity vanishing at the ground","authors":"R. Lewandowski, François Legeais, L. Berselli","doi":"10.1051/m2an/2024009","DOIUrl":"https://doi.org/10.1051/m2an/2024009","url":null,"abstract":"We introduce a scalar elliptic equation defined on a boundary layer given by $Pi_2 times [0, z_{top}]$, where $Pi_2$ is a two dimensional torus, with an eddy vertical eddy viscosity of order \u0000$z^alpha$, $alpha in [0, 1]$, an homogeneous boundary condition at $z=0$, and a Robin condition at $z=z_{top}$. We show the existence of weak solutions to this boundary problem, distinguishing the cases \u0000$0 le alpha <1$ and $alpha = 1$.  Then we carry out several numerical simulations, showing the ability of our model to accurately reproduce profiles close to those predicted by the Monin-Obukhov theory, by calculating stabilizing functions.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139865120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation 针对半线性抛物方程的高效双网格高阶紧凑差分方案与变步长 BDF2 方法
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2024-01-30 DOI: 10.1051/m2an/2024008
Bingyin Zhang, Hongfei Fu
{"title":"An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation","authors":"Bingyin Zhang, Hongfei Fu","doi":"10.1051/m2an/2024008","DOIUrl":"https://doi.org/10.1051/m2an/2024008","url":null,"abstract":"Due to the lack of  corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a  variable-step high-order nonlinear difference algorithm using compact difference technique in space and  the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea  and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding  error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies $ r_{k} := tau_{k}/tau_{k-1} < 4.8645 $. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction $ r_{k} < 4.8645 $  on the variable temporal stepsize,  unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140481596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk 圆盘上亥姆霍兹第一类积分方程的准局部和频率稳健预处理器
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2024-01-10 DOI: 10.1051/m2an/2023105
François Alouges, Martin Averseng
{"title":"Quasi-local and frequency robust preconditioners for the Helmholtz first-kind integral equations on the disk","authors":"François Alouges, Martin Averseng","doi":"10.1051/m2an/2023105","DOIUrl":"https://doi.org/10.1051/m2an/2023105","url":null,"abstract":"We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in $R^3$.  Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolutions and can thus be evaluated cheaply in the context of iterative methods.  \u0000    For the Laplace equation (i.e. for the wavenumber $k=0$) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of $k$ and on locally refined meshes.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139440625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problem 针对界面斯托克斯问题的增强质量守恒低阶虚构域法
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2023-12-19 DOI: 10.1051/m2an/2023103
Daniele Corti, Guillaume Delay, Miguel A. Fernández, Fabien Vergnet, Marina Vidrascu
{"title":"Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problem","authors":"Daniele Corti, Guillaume Delay, Miguel A. Fernández, Fabien Vergnet, Marina Vidrascu","doi":"10.1051/m2an/2023103","DOIUrl":"https://doi.org/10.1051/m2an/2023103","url":null,"abstract":"One of the main difficulties that has to be faced with fictitious domain approximation of incompressible flows with immersed interfaces is related to the potential lack of mass conservation across the interface. In this paper, we propose and analyze a low order fictitious domain stabilized finite element method which mitigates this issue with the addition of a single velocity constraint. We provide a complete a priori numerical analysis of the method under minimal regularity assumptions. A comprehensive numerical study illustrates the capabilities of the proposed method, including comparisons with alternative fitted and unfitted mesh methods.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139171761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete entropy inequalities via an optimization process 通过优化过程实现离散熵不等式
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2023-12-15 DOI: 10.1051/m2an/2023098
N. Aguillon, Emmanuel Audusse, Vivien Desveaux, Julien Salomon
{"title":"Discrete entropy inequalities via an optimization process","authors":"N. Aguillon, Emmanuel Audusse, Vivien Desveaux, Julien Salomon","doi":"10.1051/m2an/2023098","DOIUrl":"https://doi.org/10.1051/m2an/2023098","url":null,"abstract":"The solutions of hyperbolic systems may contain discontinuities. These weak solutions verify not only the original PDEs, but also an entropy inequality that acts as a selection criterion determining whether a discontinuity is physical or not. Obtaining a discrete version of these entropy inequalities when approximating the solutions numerically is crucial to avoid convergence to unphysical solutions or even unstability. However such a task is difficult in general, if not impossible for schemes of order 2 or more. In this paper, we introduce an optimization framework that enables us to quantify a posteriori the decrease or increase of entropy of a given scheme, locally in space and time. We use it to obtain maps of numerical diffusion and to prove that some schemes do not have a discrete entropy inequality. A special attention is devoted to the widely used second order MUSCL scheme for which almost no theoretical results are known.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139177687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix 各向异性介质中麦克斯韦系统的 PWDG 方法，系数矩阵为片断常数
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2023-11-30 DOI: 10.1051/m2an/2023097
Long Yuan, Qiya Hu
{"title":"A PWDG method for the Maxwell system in anisotropic media with piecewise constant coefficient matrix","authors":"Long Yuan, Qiya Hu","doi":"10.1051/m2an/2023097","DOIUrl":"https://doi.org/10.1051/m2an/2023097","url":null,"abstract":"In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions.  We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local $hp-$refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139200337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface 带自由表面的纳维-斯托克斯方程有限元近似的稳定性分析
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2023-11-29 DOI: 10.1051/m2an/2023096
G. Barrenechea, E. Audusse, A. Decoene, Pierrick Quemar
{"title":"Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface","authors":"G. Barrenechea, E. Audusse, A. Decoene, Pierrick Quemar","doi":"10.1051/m2an/2023096","DOIUrl":"https://doi.org/10.1051/m2an/2023096","url":null,"abstract":"In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139213670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
PDE-convergence in Euclidean norm of AMF-W methods for linear multidimensional parabolic problems 线性多维抛物线问题的 AMF-W 方法在欧氏规范下的 PDE 收敛性
ESAIM: Mathematical Modelling and Numerical Analysis Pub Date : 2023-11-16 DOI: 10.1051/m2an/2023094
S. González-Pinto, Ernst Hairer, D. Hernández-Abreu
{"title":"PDE-convergence in Euclidean norm of AMF-W methods for linear multidimensional parabolic problems","authors":"S. González-Pinto, Ernst Hairer, D. Hernández-Abreu","doi":"10.1051/m2an/2023094","DOIUrl":"https://doi.org/10.1051/m2an/2023094","url":null,"abstract":"This work considers space-discretised parabolic problems on a rectangular domain subject to Dirichlet boundary conditions. For the time integration s-stage AMF-W-methods, which are ADI (alternating direc- tion implicit) type integrators, are considered. They are particularly efficient when the space dimension m of the problem is large. Optimal results on PDE-convergence have recently been obtained in [J. Comput. Appl. Math., 417:114642, 2023] for the case m = 2. The aim of the present work is to extend these results to arbitrary space dimension m ≥ 3. It is explained which order statements carry over from the case m = 2 to m ≥ 3, and which do not.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139270404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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