Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.
{"title":"Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions","authors":"Víctor Domínguez, Catalin Turc","doi":"10.1093/imanum/drae078","DOIUrl":"https://doi.org/10.1093/imanum/drae078","url":null,"abstract":"Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142597484","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 define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.
{"title":"Positive definite functions on a regular domain","authors":"Martin Buhmann, Yuan Xu","doi":"10.1093/imanum/drae074","DOIUrl":"https://doi.org/10.1093/imanum/drae074","url":null,"abstract":"We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142588647","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}
Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom
In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.
{"title":"An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation","authors":"Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom","doi":"10.1093/imanum/drae076","DOIUrl":"https://doi.org/10.1093/imanum/drae076","url":null,"abstract":"In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490355","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 studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.
{"title":"Time-dependent electromagnetic scattering from dispersive materials","authors":"Jörg Nick, Selina Burkhard, Christian Lubich","doi":"10.1093/imanum/drae071","DOIUrl":"https://doi.org/10.1093/imanum/drae071","url":null,"abstract":"This paper studies time-dependent electromagnetic scattering from obstacles that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave–material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490356","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}
Claudine von Hallern, Ricarda Missfeldt, Andreas Rössler
For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.
{"title":"An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type","authors":"Claudine von Hallern, Ricarda Missfeldt, Andreas Rössler","doi":"10.1093/imanum/drae064","DOIUrl":"https://doi.org/10.1093/imanum/drae064","url":null,"abstract":"For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge–Kutta type that allows for convergence with a temporal order of up to $frac{3}/{2}$ and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444062","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}
Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions, but rather local approximate solutions to the governing PDE. In order to develop quasi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation this work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. The polynomial basis shows significant promise as it does not suffer from the ill-conditioning issue inherent to wave-like bases.
{"title":"Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties","authors":"Lise-Marie Imbert-Gérard, Guillaume Sylvand","doi":"10.1093/imanum/drae060","DOIUrl":"https://doi.org/10.1093/imanum/drae060","url":null,"abstract":"Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions, but rather local approximate solutions to the governing PDE. In order to develop quasi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation this work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. The polynomial basis shows significant promise as it does not suffer from the ill-conditioning issue inherent to wave-like bases.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142405029","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 discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. (2018, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys., 353, 407–416) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable this approach allows to derive a linear scheme while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen–Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski’s generalization of Skorokhod’s theorem. A generalization of the Gyöngy–Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen–Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.
{"title":"A convergent stochastic scalar auxiliary variable method","authors":"Stefan Metzger","doi":"10.1093/imanum/drae065","DOIUrl":"https://doi.org/10.1093/imanum/drae065","url":null,"abstract":"We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. (2018, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys., 353, 407–416) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable this approach allows to derive a linear scheme while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen–Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski’s generalization of Skorokhod’s theorem. A generalization of the Gyöngy–Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen–Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142405030","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 new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.
{"title":"A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow","authors":"Lina Zhao","doi":"10.1093/imanum/drae067","DOIUrl":"https://doi.org/10.1093/imanum/drae067","url":null,"abstract":"In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142369108","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 extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $varDelta tle O(h^{2})$. We show that we only need the lower bound $varDelta tge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $varDelta tle C$ (for globally Lipschitz) or $varDelta tle O(h^{gamma })$ (for locally Lipschitz) for some constant $C>0$ arising from the PDE, or some $gamma < 2$, respectively. In most situations in practical models, the latter condition becomes $varDelta t le O( h^{2/3} )$ in 2D and $varDelta t le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.
在本文中，我们扩展了之前在 Faragó, I., Karátson, J. 和 Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems.IMA J. Numer.Anal.，32，1541-1573）关于非线性抛物 PDE 系统的离散最大最小原理（DMP）。我们提出了一种线性隐式方案，只需解决时间层上的线性问题。我们得到的 DMP 没有限制性条件 $varDelta tle O(h^{2})$。我们证明，我们只需要下界 $varDelta tge O(h^{2})$，此外，根据给定非线性的 Lipschitz 条件，对于由 PDE 产生的某个常数 $C>0$，或者某个 $gamma < 2$，上界仅仅是 $varDelta tle C$（对于全局 Lipschitz）或 $varDelta tle O(h^{gamma})$（对于局部 Lipschitz）。在实际模型的大多数情况下，后一个条件在二维模型中变成 $varDelta t le O( h^{2/3} )$ ，在三维模型中变成 $varDelta t le O( h )$ 。此外，还介绍了各种现实生活中的例子，在这些例子中，可以应用这些结果来获得与物理相关的数值解。
{"title":"Discrete maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions","authors":"István Faragó, Róbert Horváth, János Karátson","doi":"10.1093/imanum/drae072","DOIUrl":"https://doi.org/10.1093/imanum/drae072","url":null,"abstract":"In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $varDelta tle O(h^{2})$. We show that we only need the lower bound $varDelta tge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $varDelta tle C$ (for globally Lipschitz) or $varDelta tle O(h^{gamma })$ (for locally Lipschitz) for some constant $C&gt;0$ arising from the PDE, or some $gamma &lt; 2$, respectively. In most situations in practical models, the latter condition becomes $varDelta t le O( h^{2/3} )$ in 2D and $varDelta t le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328968","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}
An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0<r_{k}:=tau _{k}/tau _{k-1}< r_{max }approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.
{"title":"An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach","authors":"Yifan Wei, Jiwei Zhang, Chengchao Zhao, Yanmin Zhao","doi":"10.1093/imanum/drae057","DOIUrl":"https://doi.org/10.1093/imanum/drae057","url":null,"abstract":"An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0&lt;r_{k}:=tau _{k}/tau _{k-1}&lt; r_{max }approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142321471","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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