A New Efficient Explicit Deferred Correction Framework:Analysis and Applications to Hyperbolic PDEs and Adaptivity

The deferred correction(DeC)is an iterative procedure,characterized by increasing the accuracy at each iteration,which can be used to design numerical methods for systems of ODEs.The main advantage of such framework is the automatic way of getting arbitrarily high order methods,which can be put in the Runge-Kutta(RK)form.The drawback is the larger computational cost with respect to the most used RK methods.To reduce such cost,in an explicit setting,we propose an efcient modifcation:we introduce interpolation processes between the DeC iterations,decreasing the computational cost associated to the low order ones.We provide the Butcher tableaux of the new modifed methods and we study their stability,showing that in some cases the computational advantage does not afect the stability.The fexibility of the novel modifcation allows nontrivial applications to PDEs and construction of adaptive methods.The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

Techniques,Tricks,and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs

GPU computing is expected to play an integral part in all modern Exascale supercomputers.It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers.It is,therefore,very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity.Not every algorithm that is used in the space-time update of the solution of hyperbolic PDEs will take well to GPUs.However,we identify a small core of algorithms that take exceptionally well to GPU computing.Based on an analysis of available options,we have been able to identify weighted essentially non-oscillatory(WENO)algorithms for spatial reconstruction along with arbitrary derivative(ADER)algorithms for time extension followed by a corrector step as the winning three-part algorithmic combination.Even when a winning subset of algorithms has been identified,it is not clear that they will port seamlessly to GPUs.The low data throughput between CPU and GPU,as well as the very small cache sizes on modern GPUs,implies that we have to think through all aspects of the task of porting an application to GPUs.For that reason,this paper identifies the techniques and tricks needed for making a successful port of this very useful class of higher order algorithms to GPUs.Application codes face a further challenge—the GPU results need to be practically indistinguishable from the CPU results—in order for the legacy knowledge bases embedded in these applications codes to be preserved during the port of GPUs.This requirement often makes a complete code rewrite impossible.For that reason,it is safest to use an approach based on OpenACC directives,so that most of the code remains intact(as long as it was originally well-written).This paper is intended to be a one-stop shop for anyone seeking to make an OpenACC-based port of a higher order Godunov scheme to GPUs.We focus on three broad and high-impact areas where higher order Godunov schemes are used.The first area is computational fluid dynamics(CFD).The second is computational magnetohydrodynamics(MHD)which has an involution constraint that has to be mimetically preserved.The third is computational electrodynamics(CED)which has involution constraints and also extremely stiff source terms.Together,these three diverse uses of higher order Godunov methodology,cover many of the most important applications areas.In all three cases,we show that the optimal use of algorithms,techniques,and tricks,along with the use of OpenACC,yields superlative speedups on GPUs.As a bonus,we find a most remarkable and desirable result:some higher order schemes,with their larger operations count per zone,show better speedup than lower order schemes on GPUs.In other words,the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes.Several avenues for future improvement have also been identified.A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs.It is found that GPUs offer a substantial performance benefit over comparable number of CPUs,especially when all the methods designed in this paper are used.

A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.

PDES策略在冷连轧动态系统仿真中的应用研究

并行离散事件仿真(PDES)策略是当今仿真界研究的热点,它是解决复杂系统问题以及分布式仿真交互管理的有效工具之一。冷连轧动态仿真是由多台计算机组成的分布式协同仿真系统,该论文在处理其中的离散事件管理以及系统运行的逻辑管理上不仅使用到并行离散事件仿真策略,而且在某些应用上改进或调整了PDES仿真算法,取得了预期的控制效果。

产生和求解PDEs对称确定方程组的微分特征列集算法理论及其应用

基于作者给出的吴－微分特征列集算法理论〔１，６〕，给出了计算微分方程（组）对称的关键一步产生和求解确定方程组的特征列集算法．利用计算机代数系统Ｍａｔｈｅｍａｔｉｃａ１．２ｆｏｒＤｏｓ〔９〕具体实现了本文提出的算法．

一维泊松方程的结构优化及对PDEs的精确解

关于PDEs的精确解问题,在实际求解中较为困难。本文以一维泊松方程为例,结合Lie对称方法和Wu方法分析方程的古典对称,计算单参数的无穷小向量,并设计方程的最优结构,通过两个定义得出一维泊松方程的最优换位和伴随算子,由此优化方程结构。以此为基础计算一维泊松方程最优结构单个元素对PDEs的精确解,在两种古典对称情况下,分别求出对PDEs的两个新解。此结果对PDEs精确解问题是一次有新意的尝试。

PDES时间推进中保守算法的研究与实现

介绍了PDES时间推进中的保守算法,给出了保守策略,分析了死锁的产生原因,并给出了解决死锁的方法,最后对算法进行了实现。

PDES时间推进中乐观算法的研究与实现

介绍了PDES时间推进中的乐观算法,给出了乐观策略,从本地控制机制、消息回滚机制和全局控制机制三个方面分析了时间弯曲机制,最后对算法进行了实现。

新一代数据交换标准PDES／STEP及其应用简介

本文概要介绍并分析了新一代面向产品定义数据的数据交换标准ＰＤＥＳ／ＳＴＥＰ的基本原理和特点，对其在ＣＡＤ／ＣＡＭ信息集成中的应用也作了简要讨论。

一个面向PDES/STEP的工程数据模型

产品定义交换/产品模型数据交换标准PDES/STEP是一个在IGES、PDDI标准基础上发展起来的、以特征实体造型为基础的国际标准(建议草案)。该标准的主要应用领域是CIM和CAD/CAM系统中的数据交换。本文提出了一个以语义数据模型为基础、并能描述PDES概念模式的数据模型。根据该工程数据模型,将研制一个适合于CIM和CAD/CAM应用的工程数据库管理系统。以PDES数据模型为基础的工程数据库系统不但具有广泛的通用性,而且便于在其上建立PDES/STEP的解释和生成系统。

HLA时间管理与PDES仿真算法研究

HLA的接口规范中定义了六类服务 ,其中时间管理服务是联邦成员间正确交互的基础。本文将对HLA/RTI时间管理 ,以及PDES的两种主要算法———保守算法和乐观算法进行研究 ,并对它们进行比较和分析。这两种算法也是作者已经实现的RTI时间管理中使用的算法。

THE ACCELERATED SEARCH-EXTENSION METHOD FOR COMPUTING MULTIPLE SOLUTIONS OF SEMILINEAR PDEs

In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.

计算机辅助工艺设计基于PDES/STEP标准的产品数据准备过程

本文主要论述计算机辅助工艺设计产品数据的准备过程，其基础是建立在PDES／STEP的原始标准上，虽然PDES／STEP目前还不太完善，正在发展中，但它仍能满足工程应用。本文对计算机辅助工艺设计（CAPP）的PDES／STEP参考模型、应用协议、EXPRESS语言以及面向对象数据库（ROSE）等逐一进行介绍。

PDES设备安全评价点巡检系统在漳村煤矿的应用

分析了目前企业中设备状态监测与故障诊断技术现状与存在的问题,介绍了PDES设备安全评价点巡检系统的系统组成和关键技术,以及该系统在漳村煤矿的应用情况,并提出系统推广应用必要性。

Three-Variable Shifted Jacobi Polynomials Approach for Numerically Solving Three-Dimensional Multi-Term Fractional-Order PDEs with Variable Coefficients

In this paper,the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of threedimensional multi-term fractional-order PDEs with variable coefficients.The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem.The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by threevariable shifted Jacobi polynomials are compared with the exact solutions.Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm.Lastly,several numerical examples are presented to test the superiority and efficiency of the proposed method.

AN IMPROVED SEARCH-EXTENTION METHOD FOR SOLVING SEMILINEAR PDES

This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.

ON THE BOREL SUMMABILITY OF FORMAL SOLUTIONS FOR SOME FIRST ORDER SINGULAR PDES WITH IRREGULAR SINGULARITY

In this paper the authors consider the summability of formal solutions for some first order singular PDEs with irregular singularity. They prove that in this case the formal solutions will be divergent, but except a enumerable directions, the formal solutions are Borel summable.

Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large scale.We employ an agent-based model to describe the microscopic dynamics of each individual in a flock,and use a fractional partial differential equation(fPDE)to model the evolution of macroscopic quantities of interest.The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model.Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics,we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations.We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking dynamics.In particular,a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual,while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities.The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally.They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization.We show in one-and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method.The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models,and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields

This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms.Such PDEs are referred to as curl-free or curl-preserving,respectively.They arise very frequently in equations for hyperelasticity and compressible multiphase flow,in certain formulations of general relativity and in the numerical solution of Schrödinger’s equation.Experience has shown that if nothing special is done to account for the curl-preserving vector field,it can blow up in a finite amount of simulation time.In this paper,we catalogue a class of DG-like schemes for such PDEs.To retain the globally curl-free or curl-preserving constraints,the components of the vector field,as well as their higher moments,must be collocated at the edges of the mesh.They are updated using potentials collocated at the vertices of the mesh.The resulting schemes:(i)do not blow up even after very long integration times,(ii)do not need any special cleaning treatment,(iii)can oper-ate with large explicit timesteps,(iv)do not require the solution of an elliptic system and(v)can be extended to higher orders using DG-like methods.The methods rely on a spe-cial curl-preserving reconstruction and they also rely on multidimensional upwinding.The Galerkin projection,highly crucial to the design of a DG method,is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the verti-ces of the mesh with the help of a multidimensional Riemann solver.A von Neumann sta-bility analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work.The stability analysis confirms that with the increasing order of accuracy,our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation.We also show that PNPM-like methods,which only evolve the lower moments while reconstructing the higher moments,retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity.The quadratic energy preservation of these methods is also shown to be excellent,especially at higher orders.The methods are also shown to be curl-preserving over long integration times.

具有PDEs抑制作用的部分中药及其活性成分研究进展

PDEs是拥有多基因、多功能的同工酶,其生化作用是水解细胞内第二信使cAMP或cGMP,参与多种疾病的病理生理过程,如神经退行性疾病、炎症、心血管疾病、勃起功能障碍等的调节。PDEs作为新的药物治疗靶点,目前受到了广泛的关注。临床上常用的PDEs抑制剂存在着对不同亚型的选择性作用差等不足,其临床应用受到一定的限制。近年来研究发现,中药淫羊藿、蛇床子、连翘等以及中药中的主要活性成分ICA、ICSⅡ、Res等对PDEs有显著的抑制作用。本文对PDEs具有抑制作用的部分中药及其活性成分进行简要综述,为寻找选择性更好的天然PDEs抑制剂提供思路。