SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2415-2417, October 2024. Abstract. We correct the proofs of Theorems 3.3 and 5.2 in [Y. A. P. Osborne and I. Smears, SIAM J. Numer. Anal., 62 (2024), pp. 138–166]. With the corrected proofs, Theorems 3.3 and 5.2 are shown to be valid without change to their hypotheses or conclusions.
SIAM 数值分析期刊》第 62 卷第 5 期第 2415-2417 页，2024 年 10 月。 摘要。我们对定理 3.3 和 5.2 的证明进行了修正 [Y. A. P. Osborne and I. Smears, SIAM J. No.A. P. Osborne and I. Smears, SIAM J. Numer.Anal., 62 (2024), pp.］在修正证明后，定理 3.3 和 5.2 被证明是有效的，其假设和结论没有改变。
{"title":"Erratum: Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions","authors":"Yohance A. P. Osborne, Iain Smears","doi":"10.1137/24m165123x","DOIUrl":"https://doi.org/10.1137/24m165123x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2415-2417, October 2024. <br/> Abstract. We correct the proofs of Theorems 3.3 and 5.2 in [Y. A. P. Osborne and I. Smears, SIAM J. Numer. Anal., 62 (2024), pp. 138–166]. With the corrected proofs, Theorems 3.3 and 5.2 are shown to be valid without change to their hypotheses or conclusions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142487327","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2393-2414, October 2024. Abstract. We consider the problem of estimating an expectation [math] by quasi-Monte Carlo (QMC) methods, where [math] is an unbounded smooth function and [math] is a standard normal random vector. While the classical Koksma–Hlawka inequality cannot be directly applied to unbounded functions, we establish a novel framework to study the convergence rates of QMC for unbounded smooth integrands. We propose a projection method to modify the unbounded integrands into bounded and smooth ones, which differs from the low variation extension strategy of avoiding the singularities along the boundary of the unit cube [math] in Owen [SIAM Rev., 48 (2006), pp. 487–503]. The total error is then bounded by the quadrature error of the transformed integrand and the projection error. We prove that if the function [math] and its mixed partial derivatives do not grow too fast as the Euclidean norm [math] tends to infinity, then projection-based QMC and randomized QMC (RQMC) methods achieve an error rate of [math] with a sample size [math] and an arbitrarily small [math]. However, the error rate turns out to be only [math] when the functions grow exponentially as [math] with [math]. Remarkably, we find that using importance sampling with [math]-distribution as the proposal can dramatically improve the root mean squared error of RQMC from [math] to [math].
{"title":"Achieving High Convergence Rates by Quasi-Monte Carlo and Importance Sampling for Unbounded Integrands","authors":"Du Ouyang, Xiaoqun Wang, Zhijian He","doi":"10.1137/23m1622489","DOIUrl":"https://doi.org/10.1137/23m1622489","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2393-2414, October 2024. <br/> Abstract. We consider the problem of estimating an expectation [math] by quasi-Monte Carlo (QMC) methods, where [math] is an unbounded smooth function and [math] is a standard normal random vector. While the classical Koksma–Hlawka inequality cannot be directly applied to unbounded functions, we establish a novel framework to study the convergence rates of QMC for unbounded smooth integrands. We propose a projection method to modify the unbounded integrands into bounded and smooth ones, which differs from the low variation extension strategy of avoiding the singularities along the boundary of the unit cube [math] in Owen [SIAM Rev., 48 (2006), pp. 487–503]. The total error is then bounded by the quadrature error of the transformed integrand and the projection error. We prove that if the function [math] and its mixed partial derivatives do not grow too fast as the Euclidean norm [math] tends to infinity, then projection-based QMC and randomized QMC (RQMC) methods achieve an error rate of [math] with a sample size [math] and an arbitrarily small [math]. However, the error rate turns out to be only [math] when the functions grow exponentially as [math] with [math]. Remarkably, we find that using importance sampling with [math]-distribution as the proposal can dramatically improve the root mean squared error of RQMC from [math] to [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"125 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142486666","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2370-2392, October 2024. Abstract. Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to either be novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [Numer. Math., 75 (1997), pp. 379–395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular we show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara’s lower bound for general numerical integration rules, introducing a new open problem. Moreover, the Gauss–Hermite quadrature is proven suboptimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss–Legendre and Clenshaw–Curtis quadratures, we generalize a recent result of Trefethen [SIAM Rev., 64 (2022), pp. 132–150] for the upper error bounds in terms of the decay conditions.
{"title":"How Sharp Are Error Bounds? –Lower Bounds on Quadrature Worst-Case Errors for Analytic Functions–","authors":"Takashi Goda, Yoshihito Kazashi, Ken’ichiro Tanaka","doi":"10.1137/24m1634163","DOIUrl":"https://doi.org/10.1137/24m1634163","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2370-2392, October 2024. <br/> Abstract. Numerical integration over the real line for analytic functions is studied. Our main focus is on the sharpness of the error bounds. We first derive two general lower estimates for the worst-case integration error, and then apply these to establish lower bounds for various quadrature rules. These bounds turn out to either be novel or improve upon existing results, leading to lower bounds that closely match upper bounds for various formulas. Specifically, for the suitably truncated trapezoidal rule, we improve upon general lower bounds on the worst-case error obtained by Sugihara [Numer. Math., 75 (1997), pp. 379–395] and provide exceptionally sharp lower bounds apart from a polynomial factor, and in particular we show that the worst-case error for the trapezoidal rule by Sugihara is not improvable by more than a polynomial factor. Additionally, our research reveals a discrepancy between the error decay of the trapezoidal rule and Sugihara’s lower bound for general numerical integration rules, introducing a new open problem. Moreover, the Gauss–Hermite quadrature is proven suboptimal under the decay conditions on integrands we consider, a result not deducible from upper-bound arguments alone. Furthermore, to establish the near-optimality of the suitably scaled Gauss–Legendre and Clenshaw–Curtis quadratures, we generalize a recent result of Trefethen [SIAM Rev., 64 (2022), pp. 132–150] for the upper error bounds in terms of the decay conditions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449547","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2349-2369, October 2024. Abstract. In this article, we impose fractal features onto classical multiquadric (MQ) functions. This generates a novel class of fractal functions, called fractal MQ functions, where the symmetry of the original MQ function with respect to the origin is maintained. This construction requires a suitable extension of the domain and similar partitions on the left side with the same choice of scaling parameters. Smooth fractal MQ functions are proposed to solve initial value problems via a collocation method. Our numerical computations suggest that fractal MQ functions offer higher accuracy and more flexibility for the solutions compared to the existing classical MQ functions. Some approximation results associated with fractal MQ functions are also presented.
{"title":"Fractal Multiquadric Interpolation Functions","authors":"D. Kumar, A. K. B. Chand, P. R. Massopust","doi":"10.1137/23m1578917","DOIUrl":"https://doi.org/10.1137/23m1578917","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2349-2369, October 2024. <br/> Abstract. In this article, we impose fractal features onto classical multiquadric (MQ) functions. This generates a novel class of fractal functions, called fractal MQ functions, where the symmetry of the original MQ function with respect to the origin is maintained. This construction requires a suitable extension of the domain and similar partitions on the left side with the same choice of scaling parameters. Smooth fractal MQ functions are proposed to solve initial value problems via a collocation method. Our numerical computations suggest that fractal MQ functions offer higher accuracy and more flexibility for the solutions compared to the existing classical MQ functions. Some approximation results associated with fractal MQ functions are also presented.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"80 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142449548","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2331-2348, October 2024. Abstract. From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of H1 and H(Curl). The functions can be expressed as sums of tensor products of Jacobi polynomials with maximal two summands.
{"title":"High Order Biorthogonal Functions in [math](curl)","authors":"Tim Haubold, Sven Beuchler, Joachim Schöberl","doi":"10.1137/23m1606794","DOIUrl":"https://doi.org/10.1137/23m1606794","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2331-2348, October 2024. <br/> Abstract. From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of H1 and H(Curl). The functions can be expressed as sums of tensor products of Jacobi polynomials with maximal two summands.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142431676","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024. Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.
{"title":"Convergence Analysis of the Parareal Algorithm with Nonuniform Fine Time Grid","authors":"Shu-Lin Wu, Tao Zhou","doi":"10.1137/23m1592481","DOIUrl":"https://doi.org/10.1137/23m1592481","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024. <br/> Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142398211","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2276-2307, October 2024. Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-squares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in [math] for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.
{"title":"Numerical Reconstruction of Diffusion and Potential Coefficients from Two Observations: Decoupled Recovery and Error Estimates","authors":"Siyu Cen, Zhi Zhou","doi":"10.1137/23m1590743","DOIUrl":"https://doi.org/10.1137/23m1590743","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2276-2307, October 2024. <br/> Abstract. The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-squares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in [math] for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142369295","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2249-2275, October 2024. Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the [math]-greedy target-data-independent selection rule and can additionally be proven to be optimal when they fully exploit adaptivity ([math]-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in reproducing kernel Hilbert spaces, as they allow us to compare adaptive interpolation with nonadaptive best nonlinear approximation.
{"title":"On the Optimality of Target-Data-Dependent Kernel Greedy Interpolation in Sobolev Reproducing Kernel Hilbert Spaces","authors":"Gabriele Santin, Tizian Wenzel, Bernard Haasdonk","doi":"10.1137/23m1587956","DOIUrl":"https://doi.org/10.1137/23m1587956","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2249-2275, October 2024. <br/> Abstract. Kernel interpolation is a versatile tool for the approximation of functions from data, and it can be proven to have some optimality properties when used with kernels related to certain Sobolev spaces. In the context of interpolation, the selection of optimal function sampling locations is a central problem, both from a practical perspective and as an interesting theoretical question. Greedy interpolation algorithms provide a viable solution for this task, being efficient to run and provably accurate in their approximation. In this paper we close a gap that is present in the convergence theory for these algorithms by employing a recent result on general greedy algorithms. This modification leads to new convergence rates which match the optimal ones when restricted to the [math]-greedy target-data-independent selection rule and can additionally be proven to be optimal when they fully exploit adaptivity ([math]-greedy). Other than closing this gap, the new results have some significance in the broader setting of the optimality of general approximation algorithms in reproducing kernel Hilbert spaces, as they allow us to compare adaptive interpolation with nonadaptive best nonlinear approximation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142313711","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2222-2248, October 2024. Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.
{"title":"Analysis of Local Discontinuous Galerkin Methods with Implicit-Explicit Time Marching for Linearized KdV Equations","authors":"Haijin Wang, Qi Tao, Chi-Wang Shu, Qiang Zhang","doi":"10.1137/24m1635818","DOIUrl":"https://doi.org/10.1137/24m1635818","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2222-2248, October 2024. <br/> Abstract. In this paper, we present the stability and error analysis of two fully discrete IMEX-LDG schemes, combining local discontinuous Galerkin spatial discretization with implicit-explicit Runge–Kutta temporal discretization, for the linearized one-dimensional KdV equations. The energy stability analysis begins with a series of temporal differences about stage solutions. Then by exploring the stability mechanism from the temporal differences, and by constructing the seminegative definite symmetric form related to the discretization of the dispersion term, and by adopting the important relationships between the auxiliary variables with the prime variable to control the antidissipation terms, we derive the unconditional stability for a discrete energy involving the prime variable and all the auxiliary variables, in the sense that the time step is bounded by a constant that is independent of the spatial mesh size. We also propose a new projection technique and adopt the technique of summation by parts in the time direction to achieve the optimal order of accuracy. The new projection technique can serve as an analytical tool to be applied to general odd order wave equations. Finally, numerical experiments are shown to test the stability and accuracy of the considered schemes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"119 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142276030","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024. Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.
{"title":"Some Grönwall Inequalities for a Class of Discretizations of Time Fractional Equations on Nonuniform Meshes","authors":"Yuanyuan Feng, Lei Li, Jian-Guo Liu, Tao Tang","doi":"10.1137/24m1631614","DOIUrl":"https://doi.org/10.1137/24m1631614","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2196-2221, October 2024. <br/> Abstract. We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have restriction on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen–Cahn equations for illustration.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245218","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号