We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at t=0t = 0 which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.
我们为非线性声学中的一个重要方程 Westervelt 方程开发了一种数值方法，该方程的衰减形式由一类非局部时间算子表示。基于梯形法则和 A 稳定卷积正交的时间半离散化方法得到了阐述和分析。连续方程的存在性和正则性分析为半离散系统的稳定性和误差分析提供了信息。误差分析包括考虑 t = 0 t = 0 处的奇异性，通过在数值方案中使用修正来解决这一问题。大量的数值实验证实了这一理论。
{"title":"Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping","authors":"Katherine Baker, Lehel Banjai, Mariya Ptashnyk","doi":"10.1090/mcom/3945","DOIUrl":"https://doi.org/10.1090/mcom/3945","url":null,"abstract":"<p>We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">t = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"147 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931313","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}
The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H1(H−1)H^1(H^{-1}), L2(L2)L^2(L^2), and the L2(H1)L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H1(H−
理查兹方程常用于模拟水和空气在土壤中的流动，是多孔介质中多相流的关健方程。它是一个非线性平流-反应-扩散方程，表现出抛物线-双曲和抛物线-椭圆两种退行性。在本研究中，我们为完全退化的理查兹方程的数值近似提供了可靠、完全可计算和局部时空高效的后验误差边界。为了显示全局可靠性，我们为时间积分的 H 1 ( H - 1 ) H^1(H^{-1}) 、L 2 ( L 2 ) L^2(L^2) 和 L 2 ( H 1 ) L^2(H^1) 误差分别导出了非局部时间误差估计。最后一个误差采用了最大原则和退化估计器。然后，在标准的 H 1 ( H - 1 ) ∩ L 2 ( H 1 ) H^1(H^{-1})cap L^2(H^1) 规范中得到全局和局部时空效率误差边界。当不存在非线性时，所采用的可靠性规范和效率规范是一致的。此外，还识别并区分了空间离散化、时间离散化、正交、线性化和数据振荡等误差因素。这些估计值在考虑使用非精确求解器进行迭代线性化时也是有效的。对具有精确解的非退化和退化情况，以及现实情况和基准情况进行了数值测试。结果表明，估计器能正确识别误差，误差最大可达 1 倍。
{"title":"A posteriori error estimates for the Richards equation","authors":"K. Mitra, M. Vohralík","doi":"10.1090/mcom/3932","DOIUrl":"https://doi.org/10.1090/mcom/3932","url":null,"abstract":"<p>The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">H^1(H^{-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared left-parenthesis upper L squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^2(L^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L^2(H^1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis intersection upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"47 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942574","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 is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.
{"title":"Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations","authors":"Constantin Carle, Marlis Hochbruck","doi":"10.1090/mcom/3952","DOIUrl":"https://doi.org/10.1090/mcom/3952","url":null,"abstract":"<p>This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"38 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931379","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 work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.
{"title":"Quinary forms and paramodular forms","authors":"N. Dummigan, A. Pacetti, G. Rama, G. Tornaría","doi":"10.1090/mcom/3815","DOIUrl":"https://doi.org/10.1090/mcom/3815","url":null,"abstract":"<p>We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"43 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931393","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 study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.
{"title":"Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows","authors":"Zhaohui Fu, Tao Tang, Jiang Yang","doi":"10.1090/mcom/3950","DOIUrl":"https://doi.org/10.1090/mcom/3950","url":null,"abstract":"<p>This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"38 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931340","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 note we present a construction of an infinite family of diagonal quintic threefolds defined over Qmathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B=(B0,B1,B2,B3)B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer bb (depending on BB), the equation B0X05+B1X15+B2X2
在本注释中，我们提出了一个定义在 Q mathbb {Q} 上的对角五元三次方的无穷族的构造，每个对角五元三次方都包含无穷多个有理点。作为应用，我们证明存在无穷多个四元数 B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3})，对于一个合适的选定整数 b b (取决于 B B )、方程 B 0 X 0 5 + B 1 X 1 5 + B 2 X 2 5 + B 3 X 3 5 = b B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b 有无穷多个正整数解。
{"title":"Construction of diagonal quintic threefolds with infinitely many rational points","authors":"Maciej Ulas","doi":"10.1090/mcom/3953","DOIUrl":"https://doi.org/10.1090/mcom/3953","url":null,"abstract":"<p>In this note we present a construction of an infinite family of diagonal quintic threefolds defined over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B equals left-parenthesis upper B 0 comma upper B 1 comma upper B 2 comma upper B 3 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B=(B_{0}, B_{1}, B_{2}, B_{3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of co-prime integers such that for a suitable chosen integer <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b\"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=\"application/x-tex\">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (depending on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), the equation <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 0 upper X 0 Superscript 5 Baseline plus upper B 1 upper X 1 Superscript 5 Baseline plus upper B 2 upper X 2 Superscript 5 Baseline plus upper B 3 upper X 3 Superscript 5 Baseline equals b\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"147 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931203","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 contrast with the diffusion equation which smoothens the initial data to C∞C^infty for t>0t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.
扩散方程在 t > 0 t>0 时（远离域的角落/边缘）将初始数据平滑为 C ∞ C^infty，而亚扩散方程只表现出有限的空间规则性。因此，在用非光滑初始数据求解亚扩散方程时，一般不能期望空间上的高阶精度。本文为非光滑初始数据的亚扩散方程的高阶有限元近似求解构建了一种新的求解分割方法。该方法通过将解拆分为两部分，即与时间相关的平稳部分和与时间无关的非平稳部分，然后通过不同的策略对两部分进行逼近。与时间相关的平滑部分采用空间高阶有限元法和时间卷积正交法进行逼近，而稳定的非平滑部分可采用较小的网格尺寸或其他可获得高阶精度的方法进行逼近。本文举了几个例子来说明如何精确逼近稳定非光滑部分，包括片状光滑初始数据、Dirac-Delta 点初始数据和集中在界面上的 Dirac 量。该论证可直接扩展到具有非光滑源数据的子扩散方程。为了支持理论分析并说明所提出的高阶分裂有限元方法的性能，我们进行了大量的数值实验。
{"title":"High-order splitting finite element methods for the subdiffusion equation with limited smoothing property","authors":"Buyang Li, Zongze Yang, Zhi Zhou","doi":"10.1090/mcom/3944","DOIUrl":"https://doi.org/10.1090/mcom/3944","url":null,"abstract":"<p>In contrast with the diffusion equation which smoothens the initial data to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">t>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"23 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931235","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}
Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove
We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section.
We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.
{"title":"Effective homology and periods of complex projective hypersurfaces","authors":"Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove","doi":"10.1090/mcom/3947","DOIUrl":"https://doi.org/10.1090/mcom/3947","url":null,"abstract":"<p>We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section.</p> <p>We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"64 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931320","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}
<p>Classical continued fractions can be introduced in the field of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic numbers, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic continued fractions offer novel perspectives on number representation and approximation. While numerous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q Subscript p"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">mathbb {Q}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a given prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we propos
经典的续分数可以引入 p p -adic 数领域，其中 p p -adic 续分数为数的表示和逼近提供了新的视角。虽然研究人员已经提出了许多 p p -adic 续分数展开算法，但一些优秀性质的建立，如经典续分数的拉格朗日定理（该定理表明每个二次无理数都可以周期性展开），仍是一个未知数。在本文中，我们介绍了几种新算法，旨在为给定素数 p p 在 Q p mathbb {Q}_p 中展开代数数。我们给出了有理数展开的部分商数上限，并证明了对于小素数 p p ，我们的算法能生成所有二次无理数的周期性续分展开。实验数据表明，与现有算法相比，我们的算法在二次无理数的周期性展开方面表现出更好的性能。此外，对于更大的素数 p p，我们提出了一种建立 p p -adic 连续分数展开算法的潜在方法。与之前的算法一样，该算法旨在扩展 Q p mathbb {Q}_p 中的代数数，同时为 Q p mathbb {Q}_p 中的所有二次无理数生成周期性扩展。
{"title":"Convergence, finiteness and periodicity of several new algorithms of 𝑝-adic continued fractions","authors":"Zhaonan Wang, Yingpu Deng","doi":"10.1090/mcom/3948","DOIUrl":"https://doi.org/10.1090/mcom/3948","url":null,"abstract":"<p>Classical continued fractions can be introduced in the field of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic numbers, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic continued fractions offer novel perspectives on number representation and approximation. While numerous <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q Subscript p\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a given prime <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we propos","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"64 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931310","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}
Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol Zamfirescu
Inspired by Sheehan’s conjecture that no 44-regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe’s computational enumerative results from [Exp. Math. 27 (2018), no. 4, 426–430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen’s independent dominating set method. This extension allows us to find a second hamiltonian cycle that inherits linearly many edges from the first hamiltonian cycle. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen by showing that for k∈{5,6}k in {5, 6} there exist infinitely many kk-regular hamiltonian graphs having no independent dominating set with respect to a prescribed hamiltonian cycle. Motivated by an observation of Aldred and Thomassen, we prove that for every κ∈{2,3}kappa in { 2, 3 } and any positive integer kk, there are infinitely many non-regular gra
受希恩（Sheehan）关于没有任何 4 4 不规则图恰好包含一个哈密尔顿循环的猜想的启发，我们证明了关于规则图和近似规则图中哈密尔顿循环的结果。我们完全推翻了海索普关于正则哈密顿图中哈密顿循环的最小数目的猜想，从而扩展了扎姆费斯库的一个结果，并修正和补充了海索普在[Exp. Math. 27 (2018)，no. 4，426-430]中的计算枚举结果。此后，我们利用洛瓦兹局部定理（Lovász Local Lemma）扩展了托马森的独立支配集方法。通过这种扩展，我们可以找到第二个哈密顿循环，它从第一个哈密顿循环中继承了线性多条边。关于这种方法的局限性，我们回答了哈克塞尔（Haxell）、西蒙（Seamone）和韦斯特拉特（Verstraete）的一个问题，并通过证明对于 k ∈ { 5 , 6 } k in {5, 6} 存在无限多的 k k -regular 哈密尔顿图，这些图相对于规定的哈密尔顿循环没有独立支配集，解决了托马森问题的第一个开放案例。受 Aldred 和 Thomassen 的观察结果的启发，我们证明了对于每一个 κ ∈ { 2 , 3 }，都有一个独立的支配集。 和任意正整数 k k，存在无限多的连通性 κ kappa 的非规则图，其中包含一个哈密顿循环，并且每个顶点都有 3 3 或 2 k 2k 度。
{"title":"Few hamiltonian cycles in graphs with one or two vertex degrees","authors":"Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol Zamfirescu","doi":"10.1090/mcom/3943","DOIUrl":"https://doi.org/10.1090/mcom/3943","url":null,"abstract":"<p>Inspired by Sheehan’s conjecture that no <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4\"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding=\"application/x-tex\">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe’s computational enumerative results from [Exp. Math. <bold>27</bold> (2018), no. 4, 426–430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen’s independent dominating set method. This extension allows us to find a second hamiltonian cycle that inherits linearly many edges from the first hamiltonian cycle. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen by showing that for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k element-of StartSet 5 comma 6 EndSet\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k in {5, 6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there exist infinitely many <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regular hamiltonian graphs having no independent dominating set with respect to a prescribed hamiltonian cycle. Motivated by an observation of Aldred and Thomassen, we prove that for every <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa element-of StartSet 2 comma 3 EndSet\"> <mml:semantics> <mml:mrow> <mml:mi>κ<!-- κ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">kappa in { 2, 3 }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and any positive integer <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there are infinitely many non-regular gra","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"26 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931236","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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