The goal of high performance computing is to pursue the ultimate computational performance. This paper summaries the key technologies that need to be developed in the three phases of high performance computing: hardware engineering, software engineering and performance engineering, and focuses on the performance portability challenges in achieving efficient adaptation between complex application loads and diverse heterogeneous sys -tems under the trend of E-class computing. Finally, the three main technologies involved in performance engineering are discussed in detail, which are pattern-driven performance modeling approach, input-aware intelligent tuning engine, and unified abstraction of software and hardware code generation.

The oil- and gas-bearing formations are widely distributed in China, but the geological structure is complex and the natural energy is usually not sufficient. Numerical simulation methods and software are important tools for reservoir engineers to analyze and manage reservoirs and are among the main tools for determining the remaining oil distribution and enhancing oil/gas recovery ratios in the late stage of reservoir development. The fine geological models increase the spatial resolution, leading to a huge number of grid cells and costly simulation runs, which present many new challenges to numerical algorithms. In this paper, we briefly introduce a simplified composition model, its mathematical model, discretization methods, solution methods, parallel computation, and implementation, with emphasis on several preconditioning and decoupling methods now adopted by commercial and industrial software.

As the nonlocal model can describe the singularity and discontinuity mechanism of some important physical phenomena, it has been widely used in many fields in recent years and has played a strong role in promoting the development of related disciplines. As an important nonlocal model, anomalous diffusion model is used to describe anomalous diffusion phenomena. The nonlocality and multi-scale characteristics of nonlocal models not only promote the discovery of new mathematical theories, but also provide a new perspective for the existing discrete and local continuous models and their connections. Although there have been many achievements, there is a broad space for the development of multi-scale nonlocal and anomalous diffusion models, whether from the perspective of mathematical methods, basic theories or numerical methods. It is a research focus to further develop and improve the basic mathematical theories and methods, and develop new efficient numerical schemes under the condition of real solution regularity, especially the numerical schemes with stability, convergence and asymptotic compatibility. In the past few years, the author of this paper has been committed to the research of mathematical theory and numerical methods of nonlocal models, and has made some meaningful research results in the design of artificial boundary conditions, nonlocal maximum principle and asymptotically compatible numerical schemes. In the numerical analysis of anomalous diffusion equations, the fast algorithm of Caputo derivative and the discrete fractional Grönwall inequality are developed, and the idea of error convolution structure is proposed to represent the local consistent error, which provides some basic analysis frameworks for the optimal error estimation of a class of widely-used variable-step-size numerical schemes. There is still a long way to completely solve various mathematical problems in nonlocal and anomalous diffusion models, which needs further study. It is hoped that this paper can play a role in promoting the in-depth development of basic theories and algorithms of multi-scale nonlocal and anomalous diffusion models.

Chemotaxis is the central mechanism for bacteria to find food and flee from poisons. In multicellular organisms, chemotaxis is critical for the development and health. Along with the understanding of more biological details of chemotaxis, the mathematical models become more complete and complex. We review three models for E.coli chemotaxis at different scales: the Keller-Segel model proposed in the 70’s last century, the velocity jump model given in the 90s and the pathway-based kinetic model proposed at the beginning of this century. We review the biological background of the proposed models, the modeling ideas, some main analytical results about the features of their solutions, the numerical methods and discuss about the connections and differences between models at different scales.

Sparse matrix algorithms are one of the most challenging topics in the supercomputing area. This paper from three aspects, i.e., high scalability, high performance and high practicability, provides an overview of the most important research work published in the last 30 years. Then a large amount of experimental results from evaluating over ten sparse BLAS algorithms on three GPUs are analyzed, and the difficulties of getting the three achievements are discussed. Finally, a serial of challenges of the next generation sparse matrix computations are proposed.

There exist plenty of nonlinear partial differential equations in scientific research and practical engineer, which leads to the importance of numerical methods for nonlinear problems. This review paper introduces a method to build the Aubin-Nitsche estimate based on a low dimensional finite element space defined on a coarse mesh. Based on this new understanding of Aubin-Nitsche technique, an augmented subspace method is designed for semilienar elliptic equations and eigenvalue problems. The corresponding convergence analysis and estimates of computational work are provided to validate the efficiency of the presented method. Especially, the augmented subspace method gives absolutely and nonlinear independently optimal computational work for polynomial types of nonlinear partial differential equations and eigenvalue problems. This paper also shows that the augmented subspace method provides a framework to design efficient nonlinear solvers for general nonlinear equations.

Algebraic multigrid(AMG) is one of the most efficient methods for solving large-scale linear systems arising from the discretized PDEs, it is widely used in numerical simulations for scientific and engineering applications. As the supercomputers become more and more powerful, both application features and machine architectures become more and more complicated, which lead to great challenges in scalability and robustness for AMG. This paper reviews the state-of-the art and challenges of AMG algorithms, including the parallel scalability, algorithmic scalability and float-point performance optimization for extreme-scale applications, especially for the upcoming exascale era.

Interface tracking (IT) is one of the most fundamental problems in the study of multiphase flows. In current IT methods, geometric and topological problems are avoided by converting them to numerically solving partial differential equations. In contrast, we tackle geometric and topological problems with tools in geometry and topology. This review paper is a brief summary of the MARS theory and associated numerical algorithms, including (1) the Yin space as a model of continua, (2) Boolean algebra on the Yin space, (3) homological analysis on topological changes of a flow phase, (4) the theory of donating regions for classifying fluxing points and calculating Lagrangian flux in the context of scalar conservation laws, (5) numerical analysis on the convergence rates of VOF methods, (6) the cubic MARS methods for fourth-order interface tracking, and (7) the HFES method for estimating curvature and normal vectors with fourth-and higher-order accuracy. Results of classical benchmark tests show that the cubic MARS method and the HFES method are advantageous over current IT methods in terms of both accuracy and efficiency.

Abstract In the large scale scientific and engineering computations the effective parallel method and algorithm must be appropriate to the structure of the parallel computer with massive parallel processors. Some difference schemes of the problems need improving in order to suit the parallel computer. Some difference schemes themselves have the parallel character for direct use and for the naturally introduced numerical method and algorithm. The difference schemes having such a parallel charactar are called the difference schemes with intrinsic parallelism. The general difference schemes with intrinsic parallelism are constructed for the problems of quasilinear parabolic systems, nonlinear parabolic systems and two and three dimensional semilinear parabolic systems. For these schemes, the existence, uniqueness and the convergence of the discrete solutions are discussed and stated. Their stability with intrinsic parallelism is studied in the sense of the continuous dependence of the discrete solutions on the given data of the original problems. The key of these studies is the restriction conditions on the choice of the meshsteps for the space variables and the time variable.

In this paper, we consider a class of system of nonlinear schrödinger equations with wave operator with the following initial-boundary conditions where u(x, t)=(u_1(x, t),…, u_N(x, t))~T,i=(-1)~(1/2). To solve this problem, we apply the finite difference method, give a convergence proof and the results of numerical calculation.

This paper gives the numerical solutions of two kinds of harmonic canonical integral equations over sector with crack and concave angle, together with their error estimates. Since the Poisson integral formula expresses exactly the singularity, canonical boundary element method has not the shortcoming, due to which the accuracy of general FEM will be lowered seriously in the neighbourhood of singular point.

This note provides a practical modification of the Schur-Cohn-Miller theorem aboutthe location of the roots of polynomials of second order with complex coefficients. Anelementary proof of this theorem is given.

In the case of the non enough smooth function we find that not only the ap-proximation of the cubic Hermite interpolation is unsuitable but often the improperinflection points are arose. In this paper, two classes quadratic Hermite interpolation and quadratic splinefunction interpolation be proposed. We find that in the curve or surface fitting, theprecision and convexity will still be holded. In the paper, the error of above methodsis estimated and the results is better.

If operator of a nonlinear boundary value problem is a monotone one then theiterative method of the finite element solution may be given. In doing so,it is notnecessary to solve the system of nonlinear algebraic equations, but to construct thebasis and the dual basis functions of the finite element subspace. Furthermore thisarticle has proved the covergence of approximate solution.

In [1] Prof. Feng Kang has considered the elliptic differential equations on compositemanifold of different dimensions and has given a precise mathematical foundation for compositeelastic structures. According to this theory, we have discussed in, detail the formulation of linkconditions for rigid connection between different parts of structures and their finite elementimplementation. It was pointed out, for instance, that on the link line of two plates or platewith beam only three translational displacement components and a longitudinal rotation ought tobe set continuously and nothing should be required for the other two transversal rotation angles.However, it is usual for some engineering finite element calculations to demand that all thesix displacement components be continuous, which may make the problem unsolvable in somespecial cases because of superfluous continuity. We further proposed a computational scheme for treating the link conditions, which wereprocessed in connection with substructures method for solving the stiffness matrix equation.

In this paper, a finite difference scheme of two-dimensional parabolic equation is constru-cted in a non-rectangular mesh by the integral method. For an arbitrary quadralateral mesh, anine--point scheme is obtained by using certain interpolation formulae. Meanwhile, an economicalscheme is established and some numerical results are given.

As usual, the subtraction has more influence in the loss of the number of effectivedigits. In this paper an abstract probability model is constructed for the statisticalestimation of the accumulation of rounding errors. The formula S_2-S_1=Δm+(1/2)log_2(T_2V_2/T_1V_1)+1/(2N)log_2(C_2/C_1),which deseribes the matching relation among word lengh S, speed V and memory spaceC of the computer is derived, where T denotes the time of the calculation for a certainproblem, Δm≥0, N=1,2,3. By taking one computer as reference, the above formula canbe used to consider the relationship for a new computer. Some other formulae of thematching relations are obtained.

The Boltzmann equation for neutron transport in configuration space (Ⅰ) is exp-ressed in an integral form of conservation (Ⅱ) in suitable phase space. Based onthis principle together with cellular subdivision and piece-wise linear approximationa conservative difference scheme is established and is applied to the eigenvalue pro-blem for axisymmetric case. The conservativeness assures the accuracy of the method.For the determination of the principal eigenvalue λ_0, and its corresponding eigenfunc-tion of (Ⅲ) a method of artificial criticality is suggested, i.e., an artificial eigen-value k(λ) depending on the parameter λ is introduced (Ⅳ) and λ = λ_0, is obtainedby adjusting λ so that k(λ) = 1. The numerical computation of the system of dif-ference equations is carried out along the direction of the characteristics, thus givesan advantage in computing simplicity and an enormous saving in storage. Thiswork was done in early 1960's and it seems to be worth while to publish. it heresince it still contains some novel points even at present.

This paper gives the affine invariant Convergence theorm concerning differentialcontinuation and Newton iterative proccesses, it also gives the formulae of optimalefficiency of differential-continuation. These formulae are more efficient than someother methods. Last, some numerical exemples are given.

In this paper a simple graphic method is given for physical solution of the initialvalued problem u~+,for x>0, u(x,0)= u~_,for x≤0of the quasilinear equation u/t+ψ(u)/x =0Stability of some difference schemes with high order of accuracy is considered.Several numerical examples are given. Using difference schemes with high order ofaccuracy, weak solutions which have "random" property in some sense are obtainedinstead of physical solutions.