Introduction The goal of this article is to coalesce a discussion around best practices for scholarly research that utilizes computational methods, by providing a formalized set of best practice recommendations to guide computational scientists and other stakeholders wishing to disseminate reproducible research, facilitate innovation by enabling data and code re-use, and enable broader communication of the output of computational scientific research. Developing Best Practices A typical computational scientist today is being inundated with new software tools to help with research [ 16 ], new requirements for publication [ 17 ], and evolving standards as his or her field responds to the changing nature and increasing quantity of available data [ 18 ].

Best Practices indicate making the data and code maximally available and open for re-use. One way to make this legally possible is through the use of open licensing [ 20 ] and the Reproducible Research Standard [ 21 ]. This document assumes you have the legal right to make the data and code publicly available, or can obtain permission from the data and code owners.

Best practices indicate negotiating open licensing for data and code with collaborators prior to beginning the research project. Provenance, workflow tracking, and publishing environments are important tools that help enable reproducibility and re-use by others, while minimizing the burden on the researcher. For example, using a version control system such as git or mercurial throughout the project simplifies making the code available at the time of publication.

For an example of work that follows these processes, refer to [ 22 ]. Data must be available and accessible.

Availability and accessibility can be broken down into three sub-discussions. Version Control for Data: At minimum, provide a version for datasets you generate or collect. If you did not generate or collect the data yourself, provide a link and citation to the source of each dataset you incorporated, including which version of the data you used if the data source does not provide version information, provide the exact time and date you accessed the data. As of yet there are no standards or conventions being widely practiced, but this is a very active topic.

An additional best practice would be to include a DOI digital objet identifier and a hash for bit-level identification of the data [ 22a ]. Raw Data Availability: Results should be reproduced from the earliest digital data in the experiment, whether that is raw data coming from instruments or observations, or data as accessed from a secondary source.

The goal is that all data manipulations be made transparent, beginning with the initial version of the data with which the researcher started working. Meta-data should accompany the raw data. Meta-data should be machine and human-readable and use standard terminology [ 23 ]. External and Redundant Storage: In the simplest case, there are no external data files, for example in some simulations. In the most complex case, data are massive, distributed, and possibly updated in real time.

The intermediary cases involve data files that can be readily downloaded and accessed by the user. Going roughly from the simplest to the most challenging cases: Simulated Data: In the case of simulated data, sharing the code that generated the data are enough if the code executes reasonably quickly.

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When a simulation takes an extended amount of time to regenerate the simulated data, pre-calculated data should be provided along with the code used to generate them. If you are able to store your datasets at your institution and link to them from your institutional webpage, that is a good step.

It will help your citation count, help others find your data, and help verification of your work. But it is insufficient. You must make your datasets available at an external repository dedicated to providing access to scientific datasets in perpetuity. These datasets should be versioned as discussed previously to enable citation to the particular version that will permit verification of the findings based on it in the paper.

The first is access, since uploading and download very large files is very time-consuming if not prohibitive. If you created the dataset yourself, you may have to make a one-time upload.

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If you did not create the dataset yourself, it is likely sufficient to cite the version of the third party data that you accessed and provide the computer code you used to manipulate the data. Large data are very likely to come with its own infrastructure. Infrastructure for large data are becoming available for researchers beyond these groups of domain-specific data repositories.

Both Globus Online and HUBzero provide different types of computational environments for non-domain-specific scientific research and their own methods for data availability. Many of these infrastructure efforts provide suggested citations and versioning for data, and this is just as crucial as it is in the small data case. Streaming Data: These data seem like the most challenging case but are actually likely to fall into one of the above categories.

Published results must be obtained on some amount of fixed data, and this particular dataset can be readily shared as above. In these cases it is likely scientifically relevant to validate models on future streams of data, but that is left to the domain of new, potentially publishable research that will share its data when published. There are exceptions to this principle, including confidential data and proprietary data. Workarounds should be attempted and may exist for confidential data [ 24 ] and proprietary data [ 25 ]. Code and methods must be available and accessible.

Input values should be included with code and scripts that generated the results, along with random number generator seeds if randomization is used. Version control should be utilized for code development, facilitating re-use by others. This discussion can be broken into subdiscussions. There are many advantages to using version control for the code you and your collaborators write during a project, and releasing the code to the wider world using version control is important.

Doing so permits others to know precisely which version of the code generated what results, allows others to make modifications and feed them back into the system without disrupting the original code, and perhaps most importantly permits a community to develop around the research questions, complete with mature functionality for bug tracking and fixes, new code developments, centralized code dissemination, and collaboration. Here is an example of scientific code associated with a published paper, available on GitHub.

Vagrant and Docker are two technologies to consider for use. For example, the BrainScaleS Brain-inspired multiscale computation in neuromorphic hybrid systems project provides a Docker image [ 26 ] for the neural network simulators nest, neuron, brian, with PyNN and music. A researcher who uses this technology stack can include a Dockerfile with their repository.

In the wider software community, projects such as Jenkins, Travis CI, and drone. Outputs from these runs can also be shared.

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Authors should provide some code samples with test parameters and data sets that demonstrate the codes use. This book conveys the fundamentals of mathematical models, numerical methods and algorithms. Opening with a tutorial on mathematical models and analysis, it proceeds to introduce the most important classes of numerical methods, with finite element, finite difference and spectral methods as central tools. The concluding section describes applications in physics and engineering, including wave propagation, heat conduction and fluid dynamics. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

Explanations of the most basic and fundamental mathematical and numerical principles that makes a thorough understanding of advanced methods easy A unified approach demonstrating the power of mathematics when applied to different types of problems in mathematical and numerical analysis An introductory tutorial for those who have forgotten their calculus and linear algebra, or never had much of it Presentation of important applications in physics and engineering, representing the most significant types of mathematical models Numerous illustrations for ease of understanding see more benefits.

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## Fundamentals of Scientific Computing

Your points will be added to your account once your order is shipped. Click on the cover image above to read some pages of this book! The book of nature is written in the language of mathematics -- Galileo Galilei. How is it possible to predict weather patterns for tomorrow, with access solely to today's weather data? And how is it possible to predict the aerodynamic behavior of an aircraft that has yet to be built?

## Numerical Methods Syllabus

The answer is computer simulations based on mathematical models - sets of equations - that describe the underlying physical properties. However, these equations are usually much too complicated to solve, either by the smartest mathematician or the largest supercomputer. This problem is overcome by constructing an approximation: a numerical model with a simpler structure can be translated into a program that tells the computer how to carry out the simulation.

This book conveys the fundamentals of mathematical models, numerical methods and algorithms. Opening with a tutorial on mathematical models and analysis, it proceeds to introduce the most important classes of numerical methods, with finite element, finite difference and spectral methods as central tools. The concluding section describes applications in physics and engineering, including wave propagation, heat conduction and fluid dynamics. Help Centre.