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A GPU-Accelerated Mixed-Precision WENO Method for Extremal Black Hole and Gravitational Wave Physics Computations

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Communications on Applied Mathematics and Computation 2023年第1期5卷 97-115页

作者： Scott E.Field sigal gottlieb Zachary J.Grant Leah F.Isherwood Gaurav Khanna University of Massachusetts Dartmouth North DartmouthMA 02747USA Department of Computational and Applied Mathematics Oak Ridge National LaboratoryOak RidgeTN 37830USA University of Rhode Island KingstonRI 02881USA

We develop and use a novel mixed-precision weighted essentially non-oscillatory(WENO)method for solving the Teukolsky equation,which arises when modeling perturbations of Kerr black holes.We show that WENO methods outperform higher-order finite-difference methods,standard in the discretization of the Teukolsky equation,due to the need to add dissipation for stability purposes in the latter.In particular,as the WENO scheme uses no additional dissipation,it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio bina-ries.In the mixed-precision approach,the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of≈3.3.In addition,we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations.Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics.We apply our solver to study the behavior of the Aretakis charge—a conserved quantity,that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.

关键词： Numerical methods-Finite differencing Hyperbolic Partial differential equations Black holesblack holes WENO

Preface to the Focused Issue on WENO Schemes

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Communications on Applied Mathematics and Computation 2023年第1期5卷 1-2页

作者： sigal gottlieb Jan S.Hesthaven Jianxian Qiu Chi-Wang Shu Qiang Zhang Yong-Tao Zhang Department of Mathematics University of Massachusetts DartmouthNorth DartmouthMA 02747USA Ecole Polytechnique Federale de Lausanne(EPFL) CH-1015 LausanneSwitzerland School of Mathematical Sciences Xiamen UniversityXiamen 361005FujianChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA Department of Mathematics Nanjing UniversityNanjing 210093JiangsuChina Department of Applied and Computational Mathematics and Statistics University of Notre DameNotre DameIN 46556USA

The weighted essentially-oscillatory(WENO)schemes are a class of finite volume and finite difference methods for solving convection-dominated problems,mainly hyperbolic conservation laws.The idea comes from the earlier essentially-oscillatory(ENO)schemes,first developed in[1]in finite volume version and in[6]in finite difference version,for solving hyperbolic conservation laws.

关键词： hyperbolic finite essentially

A Strong Stability Preserving Analysis for Explicit Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

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Communications on Applied Mathematics and Computation 2019年第1期1卷 21-59页

作者： Zachary Grant sigal gottlieb David C.Seal Department of Computational and Applied Mathematics Oak Ridge National LaboratoryOak RidgeUSA Department of Mathematics University of Massachusetts DartmouthDartmouthUSA Department of Mathematics U.S.Naval AcademyAnnapolisUSA

High-order strong stability preserving(SSP)time discretizations are often needed to ensure the nonlinear(and sometimes non-inner-product)strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs.Multi-derivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs,and the strong stability properties of these methods are of interest.In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second-derivative formulation.However,many spatial discretizations do not satisfy strong stability properties when coupled with this second-derivative formulation,but rather with a more natural Taylor series formulation.In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation.We call these strong stability preserving Taylor series(SSP-TS)methods.We also prove that the maximal order of SSP-TS methods is p=6,and define an optimization procedure that allows us to find such SSP methods.Several types of these methods are presented and their efficiency compared.Finally,these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods,the need for the SSP property,and the sharpness of the SSP time-step in many cases.

关键词： Strong stability preserving Taylor series.Hyperbolic conservation laws Two derivative Runge Kutta

A Review of David gottlieb’s Work on the Resolution of the Gibbs Phenomenon

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Communications in Computational Physics 2011年第3期9卷 497-519页

作者： sigal gottlieb Jae-Hun Jung Saeja Kim Mathematics Department University of Massachusetts DartmouthNorth DartmouthMA 02747USA Mathematics Department Suny BuffaloBuffaloNew York 14260-2900USA

Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic.The global expansions are contaminated by the presence of a local discontinuity,and the result is that the partial sums are oscillatory and feature non-uniform convergence.This characteristic behavior is called the Gibbs phenomenon.However,David gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

关键词： Gibbs phenomenon post-processing Galerkin approximation collocation approximation spectral methods exponential accuracy

Numerical Issues in the Implementation of High Order Polynomial Multi-Domain Penalty Spectral Galerkin Methods for Hyperbolic Conservation Laws

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Communications in Computational Physics 2009年第2期5卷 600-619页

作者： sigal gottlieb Jae-Hun Jung Department of Mathematics University of Massachusetts at DartmouthNorth DartmouthMA 02747-2300USA Department of Mathematics SUNY at BuffaloBuffaloNY14260-2900USA

In this paper,we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws.This formulation has a penalty parameter which can vary in space and time,allowing for flexibility in the penalty formulation.This flexibility is particularly advantageous for problems with an inhomogeneous mesh.We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter.The penalty parameter has an effect on both the accuracy and stability of the method.We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods.The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used.Finally,we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors.Numerical examples for linear and nonlinear problems are presented.

关键词： High order polynomial Galerkin methods penalty boundary conditions discontinuous Galerkin methods hyperbolic conservation laws round-off errors truncation methods

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时间相关问题的谱方法

国外科技新书评介 2008年第5期 8-9页

作者： Jan S. Hesthaven sigal gottlieb David gottlieb 吴永礼 Brown University University of Massachusetts Dartmouth 中国科学院力学研究所

本书是《剑桥应用和计算数学专著》系列丛书第21册。谱方法是非常适用于以时间相关的偏微分方程来模拟的问题。此方法快速、有效而精确，已为广大的数学家和专业人员所应用。在书中，作者论述了谱方法的基本理论，并通过严格的推理和许... 详细信息

本书是《剑桥应用和计算数学专著》系列丛书第21册。谱方法是非常适用于以时间相关的偏微分方程来模拟的问题。此方法快速、有效而精确，已为广大的数学家和专业人员所应用。在书中，作者论述了谱方法的基本理论，并通过严格的推理和许多实例使读者理解谱方法的技术，此外，还在福里叶展开和正交多项式的基础上，提供详细的处理方法，时域积分的计算技术则用Runge-Kutta法。

关键词：解谱方法时间相关 Runge-Kutta法计算技术偏微分方程正交多项式计算数学数学家