The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problem using C0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which assumed to be a convex polygon. The proofs require local energy estimate and new pointwise Green’s function estimates for the continuous problem which have an independent interest.
{"title":"Pointwise error estimates for C0 interior penalty approximation of biharmonic problems","authors":"D. Leykekhman","doi":"10.1090/mcom/3596","DOIUrl":"https://doi.org/10.1090/mcom/3596","url":null,"abstract":"The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problem using C0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which assumed to be a convex polygon. The proofs require local energy estimate and new pointwise Green’s function estimates for the continuous problem which have an independent interest.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83293560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrodinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^gamma$ for any initial data belonging to $H^gamma$, for any $gamma >frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(|u|_{L^infty([0,T]; H^{gamma})})>0$, such that $$ |u^n-u(t_n)|_{H^gamma(mathbb T)}le C tau, $$ where $u^n$ denotes the numerical solution at $t_n=ntau$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$ left|M(u^n)-M(u_0)right|le Ctau^5. $$ In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0in H^1(mathbb T)$, we rigorously prove that $$ |u^n-u(t_n)|_{H^1(mathbb T)}le Ctau^{frac12-}, $$ where $C= C(|u_0|_{H^1(mathbb T)})>0$.
{"title":"A first-order Fourier integrator for the nonlinear Schrödinger equation on T without loss of regularity","authors":"Yifei Wu, Fangyan Yao","doi":"10.1090/mcom/3705","DOIUrl":"https://doi.org/10.1090/mcom/3705","url":null,"abstract":"In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrodinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^gamma$ for any initial data belonging to $H^gamma$, for any $gamma >frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(|u|_{L^infty([0,T]; H^{gamma})})>0$, such that $$ |u^n-u(t_n)|_{H^gamma(mathbb T)}le C tau, $$ where $u^n$ denotes the numerical solution at $t_n=ntau$. Moreover, the mass of the numerical solution $M(u^n)$ verifies $$ left|M(u^n)-M(u_0)right|le Ctau^5. $$ In particular, our scheme dose not cost any additional derivative for the first-order convergence and the numerical solution obeys the almost mass conservation law. Furthermore, if $u_0in H^1(mathbb T)$, we rigorously prove that $$ |u^n-u(t_n)|_{H^1(mathbb T)}le Ctau^{frac12-}, $$ where $C= C(|u_0|_{H^1(mathbb T)})>0$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79556893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.
{"title":"Consistency of finite volume approximations to nonlinear hyperbolic balance laws","authors":"M. Ben-Artzi, Jiequan Li","doi":"10.1090/mcom/3569","DOIUrl":"https://doi.org/10.1090/mcom/3569","url":null,"abstract":"This paper addresses the three concepts of consistency, stability and convergence in the context of compact finite volume schemes for systems of nonlinear hyperbolic conservation laws. The treatment utilizes the framework of “balance laws”. Such laws express the relevant physical conservation laws in the presence of discontinuities. Finite volume approximations employ this viewpoint, and the present paper can be regarded as being in this category. It is first shown that under very mild conditions a weak solution is indeed a solution to the balance law. The schemes considered here allow the computation of several quantities per mesh cell (e.g., slopes) and the notion of consistency must be extended to this framework. Then a suitable convergence theorem is established, generalizing the classical convergence theorem of Lax and Wendroff. Finally, the limit functions are shown to be entropy solutions by using a notion of “Godunov compatibility”, which serves as a substitute to the entropy condition.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79941835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider sums of the form $sum phi(gamma)$, where $phi$ is a given function, and $gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
{"title":"Accurate estimation of sums over zeros of the Riemann zeta-function","authors":"R. Brent, Dave Platt, T. Trudgian","doi":"10.1090/mcom/3652","DOIUrl":"https://doi.org/10.1090/mcom/3652","url":null,"abstract":"We consider sums of the form $sum phi(gamma)$, where $phi$ is a given function, and $gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82942630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.
{"title":"Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering kernel","authors":"Qiwei Sheng, C. Hauck","doi":"10.1090/MCOM/3670","DOIUrl":"https://doi.org/10.1090/MCOM/3670","url":null,"abstract":"We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86701682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.
{"title":"Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography","authors":"F. Bouchut, Xavier Lhébrard","doi":"10.1090/MCOM/3600","DOIUrl":"https://doi.org/10.1090/MCOM/3600","url":null,"abstract":"We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72710881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform convergent expansions of integral transforms","authors":"Jose L. Lopez, Pablo Palacios, P. Pagola","doi":"10.1090/MCOM/3601","DOIUrl":"https://doi.org/10.1090/MCOM/3601","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88665625","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chun Liu, Cheng Wang, S. Wise, Xingye Yue, Shenggao Zhou
In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ ell ^infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.
{"title":"A positivity-preserving, energy stable and convergent numerical scheme for the Poisson-Nernst-Planck system","authors":"Chun Liu, Cheng Wang, S. Wise, Xingye Yue, Shenggao Zhou","doi":"10.1090/MCOM/3642","DOIUrl":"https://doi.org/10.1090/MCOM/3642","url":null,"abstract":"In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility \u0000\u0000 \u0000 \u0000 H\u0000 \u0000 −\u0000 1\u0000 \u0000 \u0000 H^{-1}\u0000 \u0000\u0000 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 and \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of \u0000\u0000 \u0000 0\u0000 0\u0000 \u0000\u0000 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the \u0000\u0000 \u0000 \u0000 ℓ\u0000 ∞\u0000 \u0000 ell ^infty\u0000 \u0000\u0000 bound for \u0000\u0000 \u0000 n\u0000 n\u0000 \u0000\u0000 and \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81078982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^times$ sent to $k^times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerator are given when the extension is not Galois, or more generally when the torus is defined by an etale algebra. We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to tori in $mathrm{GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.
给定一个数字域扩展$K/ K $，中间域$K^+$由相应的素数阶伽罗瓦群$p$的中心元素固定，我们在$K $上构造一个代数环面，其有理点是$K^乘以$的元素，通过范数映射$N_{K/K^+}$传递到$K^乘以$。我们的目标是通过Ono的公式明确地计算出环面的Tamagawa数，该公式将其表示为上同调不变量的比率。当K/ K为伽罗瓦时，给出了这种环面特征格的上同性的较为完整和详细的描述。当扩展不是伽罗瓦时，或者更一般地说，当环面由一个代数定义时，给出包含分子的部分结果。我们还介绍了SAGE为此目的开发的工具，使我们能够构建和计算上同调，并探索这种代数环面的局部-全局原理。特别注意$[K:K^+]=2$和$K$是cm域的情况。这种情况对应于$ mathm {GSp}_{2n}$中的tori，大多数示例将在该设置中。这是由有限域上的阿贝尔变分和双线性形式的哈塞原理的应用所激发的。
{"title":"Explicit Tamagawa numbers for certain algebraic tori over number fields","authors":"Thomas Rüd","doi":"10.1090/mcom/3771","DOIUrl":"https://doi.org/10.1090/mcom/3771","url":null,"abstract":"Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^times$ sent to $k^times$ via the norm map $N_{K/K^+}$. The goal is to compute the Tamagawa number of that torus explicitly via Ono's formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when $K/k$ is Galois. Partial results including the numerator are given when the extension is not Galois, or more generally when the torus is defined by an etale algebra. \u0000We also present tools developed in SAGE for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. \u0000Particular attention is given to the case when $[K:K^+]=2$ and $K$ is a CM-field. This case corresponds to tori in $mathrm{GSp}_{2n}$, and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76409128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Previously, we established lower bounds for the usual measures (trace, length, Mahler measure) of totally positive algebraic integers, i.e., all of whose conjugates are positive real numbers. We used the method of explicit auxiliary functions and we noticed that the house of most of the totally positive polynomials involved in our functions are bounded by 5.8. Thanks to this observation, we are able to use the same method and give upper bounds for the usual measures of totally positive algebraic integers with house bounded by this value. To our knowledge, theses upper bounds are the first results of this kind.
{"title":"Upper bounds for the usual measures of totally positive algebraic integers with house less than 5.8","authors":"V. Flammang","doi":"10.1090/mcom/3580","DOIUrl":"https://doi.org/10.1090/mcom/3580","url":null,"abstract":"Previously, we established lower bounds for the usual measures (trace, length, Mahler measure) of totally positive algebraic integers, i.e., all of whose conjugates are positive real numbers. We used the method of explicit auxiliary functions and we noticed that the house of most of the totally positive polynomials involved in our functions are bounded by 5.8. Thanks to this observation, we are able to use the same method and give upper bounds for the usual measures of totally positive algebraic integers with house bounded by this value. To our knowledge, theses upper bounds are the first results of this kind.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85769196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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