Daniele Boffi, Sining Gong, Johnny Guzmán, Michael Neilan
We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey–Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.
{"title":"Convergence of Lagrange finite element methods for Maxwell eigenvalue problem in 3D","authors":"Daniele Boffi, Sining Gong, Johnny Guzmán, Michael Neilan","doi":"10.1093/imanum/drad053","DOIUrl":"https://doi.org/10.1093/imanum/drad053","url":null,"abstract":"We prove convergence of the Maxwell eigenvalue problem using quadratic or higher Lagrange finite elements on Worsey–Farin splits in three dimensions. To do this, we construct two Fortin-like operators to prove uniform convergence of the corresponding source problem. We present numerical experiments to illustrate the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509813","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}
Abstract This paper presents a novel approach for solving convex constrained minimization problems by introducing a special subclass of quasi-nonexpansive operators and combining them with the superiorization methodology that utilizes subgradient vectors. Superiorization methodology tries to reduce a target function while seeking a feasible point for the given constraints. We begin by introducing a new class of operators, which includes many well-known operators used for solving convex feasibility problems. Next, we demonstrate how the superiorization methodology can be combined with the introduced class of operators to obtain superiorized operators. To achieve this, we present a new formula for the step size of the perturbations in the superiorized operators. Finally, we propose an iterative method that utilizes the superiorized operators to solve convex constrained minimization problems. We provide examples of image reconstruction from projections (tomography) to demonstrate the capabilities of our proposed iterative method.
{"title":"A new step size selection strategy for the superiorization methodology using subgradient vectors and its application for solving convex constrained optimization problems","authors":"Mokhtar Abbasi, Mahdi Ahmadinia, Ali Ahmadinia","doi":"10.1093/imanum/drad070","DOIUrl":"https://doi.org/10.1093/imanum/drad070","url":null,"abstract":"Abstract This paper presents a novel approach for solving convex constrained minimization problems by introducing a special subclass of quasi-nonexpansive operators and combining them with the superiorization methodology that utilizes subgradient vectors. Superiorization methodology tries to reduce a target function while seeking a feasible point for the given constraints. We begin by introducing a new class of operators, which includes many well-known operators used for solving convex feasibility problems. Next, we demonstrate how the superiorization methodology can be combined with the introduced class of operators to obtain superiorized operators. To achieve this, we present a new formula for the step size of the perturbations in the superiorized operators. Finally, we propose an iterative method that utilizes the superiorized operators to solve convex constrained minimization problems. We provide examples of image reconstruction from projections (tomography) to demonstrate the capabilities of our proposed iterative method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135146275","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}
Abstract In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori$L^{infty }$ estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.
{"title":"Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator","authors":"Xuelong Gu, Wenjun Cai, Yushun Wang, Chaolong Jiang","doi":"10.1093/imanum/drad067","DOIUrl":"https://doi.org/10.1093/imanum/drad067","url":null,"abstract":"Abstract In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori$L^{infty }$ estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135859525","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}
Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-alpha }$ for some constant $alpha in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $alpha = alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.
第二类弱奇异Volterra积分方程(VIEs)的分段多项式配置问题在文献中得到了广泛的研究，其中考虑了$(t-s)^{-alpha}$形式的积分核对于某常数$alpha in(0,1)$。变阶分数阶微分方程是目前研究的热点。Wang(2020)，均匀或梯度网格上变阶空间分数扩散方程的最优阶数值逼近。SIAM J. number。分析的， 58, 330-352)，将该问题转化为弱奇异VIE，其核具有上述形式，变量$alpha = alpha (t)$，然后通过分段线性配置进行数值求解，但尚不清楚这种分析是否可以推广到更一般的问题或更高次的多项式。本文利用新技术，建立了变指数弱奇异vie的一般理论(解的存在性、唯一性和正则性)。然后，这些结果支持可以使用任意程度的分段多项式的搭配方法的误差分析。这种误差分析也是新颖的，它没有使用通常的解析表示，这是当前研究文献中vie搭配方法误差分析的关键技术。此外，上述对标量VIE的分析可以推广到某些非线性VIE和VIE系统。数值算例表明，所得到的配置方法的理论误差界限是清晰的。
{"title":"A general collocation analysis for weakly singular Volterra integral equations with variable exponent","authors":"Hui Liang, Martin Stynes","doi":"10.1093/imanum/drad072","DOIUrl":"https://doi.org/10.1093/imanum/drad072","url":null,"abstract":"Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-alpha }$ for some constant $alpha in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $alpha = alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135108645","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}
Abstract In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard $L^2(varOmega )$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.
{"title":"Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems","authors":"Siyu Cen, Bangti Jin, Qimeng Quan, Zhi Zhou","doi":"10.1093/imanum/drad073","DOIUrl":"https://doi.org/10.1093/imanum/drad073","url":null,"abstract":"Abstract In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard $L^2(varOmega )$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135208123","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}
Christoph Reisinger, Wolfgang Stockinger, Yufei Zhang
Abstract Fully coupled McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the $L^2$-approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator.
{"title":"<i>A posteriori</i> error estimates for fully coupled McKean–Vlasov forward-backward SDEs","authors":"Christoph Reisinger, Wolfgang Stockinger, Yufei Zhang","doi":"10.1093/imanum/drad060","DOIUrl":"https://doi.org/10.1093/imanum/drad060","url":null,"abstract":"Abstract Fully coupled McKean–Vlasov forward-backward stochastic differential equations (MV-FBSDEs) arise naturally from large population optimization problems. Judging the quality of given numerical solutions for MV-FBSDEs, which usually require Picard iterations and approximations of nested conditional expectations, is typically difficult. This paper proposes an a posteriori error estimator to quantify the $L^2$-approximation error of an arbitrarily generated approximation on a time grid. We establish that the error estimator is equivalent to the global approximation error between the given numerical solution and the solution of a forward Euler discretized MV-FBSDE. A crucial and challenging step in the analysis is the proof of stability of this Euler approximation to the MV-FBSDE, which is of independent interest. We further demonstrate that, for sufficiently fine time grids, the accuracy of numerical solutions for solving the continuous MV-FBSDE can also be measured by the error estimator. The error estimates justify the use of residual-based algorithms for solving MV-FBSDEs. Numerical experiments for MV-FBSDEs arising from mean field control and games confirm the effectiveness and practical applicability of the error estimator.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135438201","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}
Abstract We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $ell _2$ norm and non-Lipschitz $ell _p (0<p<1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled Karush–Kuhn–Tucker (KKT) point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Cosmic Microwave Background (CMB) data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.
{"title":"Group sparse optimization for inpainting of random fields on the sphere","authors":"Chao Li, Xiaojun Chen","doi":"10.1093/imanum/drad071","DOIUrl":"https://doi.org/10.1093/imanum/drad071","url":null,"abstract":"Abstract We propose a group sparse optimization model for inpainting of a square-integrable isotropic random field on the unit sphere, where the field is represented by spherical harmonics with random complex coefficients. In the proposed optimization model, the variable is an infinite-dimensional complex vector and the objective function is a real-valued function defined by a hybrid of the $ell _2$ norm and non-Lipschitz $ell _p (0&lt;p&lt;1)$ norm that preserves rotational invariance property and group structure of the random complex coefficients. We show that the infinite-dimensional optimization problem is equivalent to a convexly-constrained finite-dimensional optimization problem. Moreover, we propose a smoothing penalty algorithm to solve the finite-dimensional problem via unconstrained optimization problems. We provide an approximation error bound of the inpainted random field defined by a scaled Karush–Kuhn–Tucker (KKT) point of the constrained optimization problem in the square-integrable space on the sphere with probability measure. Finally, we conduct numerical experiments on band-limited random fields on the sphere and images from Cosmic Microwave Background (CMB) data to show the promising performance of the smoothing penalty algorithm for inpainting of random fields on the sphere.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135396059","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}
Abstract In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the ${L}^{infty }$-convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp., 34, 401–424), we prove optimal error bounds in the $W^{1,infty } times L^infty $-norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the $H^{1} times L^2$-norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments.
摘要本文研究光滑有界区域上的一类特殊的非自治波动方程及其在空间上用等参有限元和时间上用隐式欧拉方法的离散化。在Baker和Dougalis(1980)关于二阶双曲方程伽辽金近似的${L}^{infty }$ -收敛性的基础上。数学。(p.， 34, 401-424)，我们证明了空间半离散化和完全离散化的$W^{1,infty } times L^infty $ -范数的最优误差界。关键的工具是由离散微分算子的逆得到的可积性增益。为此，我们必须对$H^{1} times L^2$ -范数中的误差进行(离散)时间导数，将其简化为微分初始误差的估计。为了证实我们的理论发现，我们也提出了数值实验。
{"title":"Maximum norm error bounds for the full discretization of nonautonomous wave equations","authors":"Benjamin Dörich, Jan Leibold, Bernhard Maier","doi":"10.1093/imanum/drad065","DOIUrl":"https://doi.org/10.1093/imanum/drad065","url":null,"abstract":"Abstract In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the ${L}^{infty }$-convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp., 34, 401–424), we prove optimal error bounds in the $W^{1,infty } times L^infty $-norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the $H^{1} times L^2$-norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136192193","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}
Jeffrey Galkowski, David Lafontaine, Euan A. Spence
Abstract We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e., away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.
{"title":"Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves","authors":"Jeffrey Galkowski, David Lafontaine, Euan A. Spence","doi":"10.1093/imanum/drad058","DOIUrl":"https://doi.org/10.1093/imanum/drad058","url":null,"abstract":"Abstract We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e., away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136107442","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}
Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a, b, c, d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.
{"title":"Error estimates of high-order compact finite difference schemes for the nonlinear <i>abcd</i> Boussinesq systems","authors":"Su-Cheol Yi, Kai Fu, Shusen Xie","doi":"10.1093/imanum/drad069","DOIUrl":"https://doi.org/10.1093/imanum/drad069","url":null,"abstract":"Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a, b, c, d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136299906","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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