A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde

Einstein vs. Bergson

On 6 April 1922, Einstein met Bergson to debate the nature of time: is the time the physicist calculates the same time the philosopher reflects on? Einstein claimed that only scientific time is real, while Bergson argued that scientific time always presupposes a living and perceiving subject. On that day, nearly 100 years ago, conflict was inevitable. Is it still inevitable today? How many kinds of time are there

Einstein vs. Bergson

On 6 April 1922, Einstein met Bergson to debate the nature of time: is the time the physicist calculates the same time the philosopher reflects on? Einstein claimed that only scientific time is real, while Bergson argued that scientific time always presupposes a living and perceiving subject. On that day, nearly 100 years ago, conflict was inevitable. Is it still inevitable today? How many kinds of time are there

Using MapReduce Streaming for Distributed Life Simulation on the Cloud

Distributed software simulations are indispensable in the study of large-scale life models but often require the use of technically complex lower-level distributed computing frameworks, such as MPI. We propose to overcome the complexity challenge by applying the emerging MapReduce (MR) model to distributed life simulations and by running such simulations on the cloud. Technically, we design optimized MR streaming algorithms for discrete and continuous versions of Conway’s life according to a general MR streaming pattern. We chose life because it is simple enough as a testbed for MR’s applicability to a-life simulations and general enough to make our results applicable to various lattice-based a-life models. We implement and empirically evaluate our algorithms’ performance on Amazon’s Elastic MR cloud. Our experiments demonstrate that a single MR optimization technique called strip partitioning can reduce the execution time of continuous life simulations by 64%. To the best of our knowledge, we are the first to propose and evaluate MR streaming algorithms for lattice-based simulations. Our algorithms can serve as prototypes in the development of novel MR simulation algorithms for large-scale lattice-based a-life models.https://digitalcommons.chapman.edu/scs_books/1014/thumbnail.jp