Numerical methods for partial differential equationsNumerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values.
Parallel computingParallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: bit-level, instruction-level, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.
Message Passing InterfaceMessage Passing Interface (MPI) is a standardized and portable message-passing standard designed to function on parallel computing architectures. The MPI standard defines the syntax and semantics of library routines that are useful to a wide range of users writing portable message-passing programs in C, C++, and Fortran. There are several open-source MPI implementations, which fostered the development of a parallel software industry, and encouraged development of portable and scalable large-scale parallel applications.
Parallel algorithmIn computer science, a parallel algorithm, as opposed to a traditional serial algorithm, is an algorithm which can do multiple operations in a given time. It has been a tradition of computer science to describe serial algorithms in abstract machine models, often the one known as random-access machine. Similarly, many computer science researchers have used a so-called parallel random-access machine (PRAM) as a parallel abstract machine (shared-memory).
Numerical analysisNumerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.
SupercomputerA supercomputer is a computer with a high level of performance as compared to a general-purpose computer. The performance of a supercomputer is commonly measured in floating-point operations per second (FLOPS) instead of million instructions per second (MIPS). Since 2017, there have existed supercomputers which can perform over 1017 FLOPS (a hundred quadrillion FLOPS, 100 petaFLOPS or 100 PFLOPS). For comparison, a desktop computer has performance in the range of hundreds of gigaFLOPS (1011) to tens of teraFLOPS (1013).
Numerical stabilityIn the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.
Embarrassingly parallelIn parallel computing, an embarrassingly parallel workload or problem (also called embarrassingly parallelizable, perfectly parallel, delightfully parallel or pleasingly parallel) is one where little or no effort is needed to separate the problem into a number of parallel tasks. This is often the case where there is little or no dependency or need for communication between those parallel tasks, or for results between them. Thus, these are different from distributed computing problems that need communication between tasks, especially communication of intermediate results.
Roadrunner (supercomputer)Roadrunner was a supercomputer built by IBM for the Los Alamos National Laboratory in New Mexico, USA. The US$100-million Roadrunner was designed for a peak performance of 1.7 petaflops. It achieved 1.026 petaflops on May 25, 2008, to become the world's first TOP500 LINPACK sustained 1.0 petaflops system. In November 2008, it reached a top performance of 1.456 petaFLOPS, retaining its top spot in the TOP500 list. It was also the fourth-most energy-efficient supercomputer in the world on the Supermicro Green500 list, with an operational rate of 444.
Finite difference methodIn numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
