computational physics problem solving with python solutions

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Using Python to Solve Computational Physics Problems

Introduction

Laplace equation is a simple second-order partial differential equation. It is also a simplest example of elliptic partial differential equation. This equation is very important in science, especially in physics, because it describes behaviour of electric and gravitation potential, and also heat conduction. In thermodynamics (heat conduction), we call Laplace equation as steady-state heat equation or heat conduction equation.

In this article, we will solve the Laplace equation using numerical approach rather than analytical/calculus approach. When we say numerical approach, we refer to discretization. Discretization is a process to "transform" the continuous form of differential equation into a discrete form of differential equation; it also means that with discretization, we can transform the calculus problem into matrix algebra problem, which is favored by programming.

Here, we want to solve a simple heat conduction problem using finite difference method. We will use Python Programming Language, Numpy (numerical library for Python), and Matplotlib (library for plotting and visualizing data using Python) as the tools. We'll also see that we can write less code and do more with Python.

In computational physics, we "always" use programming to solve the problem, because computer program can calculate large and complex calculation "quickly". Computational physics can be represented as this diagram.

There are so many programming languages that are used today to solve many numerical problems, Matlab for example. But here, we will use Python, the "easy to learn" programming language, and of course, it's free. It also has powerful numerical, scientific, and data visualization library such as Numpy, Scipy, and Matplotlib. Python also provides parallel execution and we can run it in computer clusters.

Back to Laplace equation, we will solve a simple 2-D heat conduction problem using Python in the next section. Here, I assume the readers have basic knowledge of finite difference method, so I do not write the details behind finite difference method, details of discretization error, stability, consistency, convergence, and fastest/optimum iterating algorithm. We will skip many steps of computational formula here.

Instead of solving the problem with the numerical-analytical validation, we only demonstrate how to solve the problem using Python, Numpy, and Matplotlib, and of course, with a little bit of simplistic sense of computational physics, so the source code here makes sense to general readers who don't specialize in computational physics.

Preparation

To produce the result below, I use this environment:

• OS : Linux Ubuntu 14.04 LTS
• Python : Python 2.7
• Numpy : Numpy 1.10.4
• Matplotlib : Matplotlib 1.5.1

If you are running Ubuntu, you can use pip to install Numpy and Matplotib, or you can run this command in your terminal.

and use this command to install Matplotlib:

Note that Python is already installed in Ubuntu 14.04. To try Python, just type Python in your Terminal and press Enter.

You can also use Python, Numpy and Matplotlib in Windows OS, but I prefer to use Ubuntu instead.

Using the Code

This is the Laplace equation in 2-D cartesian coordinates (for heat equation):

Where T is temperature, x is x-dimension, and y is y-dimension. x and y are functions of position in Cartesian coordinates. If you are interested to see the analytical solution of the equation above, you can look it up  here .

Here, we only need to solve 2-D form of the Laplace equation. The problem to solve is shown below:

What we will do is find the steady state temperature inside the 2-D plat (which also means the solution of Laplace equation) above with the given boundary conditions (temperature of the edge of the plat). Next, we will discretize the region of the plat, and divide it into meshgrid, and then we discretize the Laplace equation above with finite difference method. This is the discretized region of the plat.

We set Δ x = Δ y = 1 cm , and then make the grid as shown below:

Note that the green nodes are the nodes that we want to know the temperature there (the solution), and the white nodes are the boundary conditions (known temperature). Here is the discrete form of Laplace Equation above.

In our case, the final discrete equation is shown below.

Now, we are ready to solve the equation above. To solve this, we use "guess value" of interior grid (green nodes), here we set it to 30 degree Celsius (or we can set it 35 or other value), because we don't know the value inside the grid (of course, those are the values that we want to know). Then, we will iterate the equation until the difference between value before iteration and the value until iteration is "small enough", we call it convergence. In the process of iterating, the temperature value in the interior grid will adjust itself, it's "selfcorrecting", so when we set a guess value closer to its actual solution, the faster we get the "actual" solution.

We are ready for the source code. In order to use Numpy library, we need to import Numpy in our source code, and we also need to import Matplolib.Pyplot module to plot our solution. So the first step is to import necessary modules.

and then, we set the initial variables into our Python source code:

What we will do next is to set the "plot window" and meshgrid . Here is the code:

np.meshgrid() creates the mesh grid for us (we use this to plot the solution), the first parameter is for the x-dimension, and the second parameter is for the y-dimension. We use np.arange() to arrange a 1-D array with element value that starts from some value to some value, in our case, it's from 0 to lenX and from 0 to lenY . Then we set the region: we define 2-D array, define the size and fill the array with guess value, then we set the boundary condition, look at the syntax of filling the array element for boundary condition above here.

Then we are ready to apply our final equation into Python code below. We iterate the equation using for loop.

You should be aware of the indentation of the code above, Python does not use bracket but it uses white space or indentation. Well, the main logic is finished. Next, we write code to plot the solution, using Matplotlib.

That's all, This is the complete code .

It's pretty short,  huh ? Okay, you can copy-paste and save the source code, name it findif.py . To execute the Python source code, open your Terminal, and go to the directory where you locate the source code, type:

and press Enter . Then the plot window will appear.

You can try to change the boundary condition's value, for example, you change the value of right edge temperature to 30 degree Celcius ( Tright = 30 ), then the result will look like this:

Points of Interest

Python is an "easy to learn" and dynamically typed programming language, and it provides (open source) powerful library for computational physics or other scientific discipline. Since Python is an interpreted language, it's slow as compared to compiled languages like C or C++, but again, it's easy to learn. We can also write less code and do more with Python, so we don't struggle to program, and we can focus on what we want to solve.

In computational physics, with Numpy and also Scipy (numeric and scientific library for Python), we can solve many complex problems because it provides matrix solver (eigenvalue and eigenvector solver), linear algebra operation, as well as signal processing, Fourier transform, statistics, optimization, etc.

In addition to its use in computational physics, Python is also used in machine learning, even Google's TensorFlow uses Python.

• 21 st March, 2016: Initial version

This article, along with any associated source code and files, is licensed under The Code Project Open License (CPOL)

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Description

Creators/contributors, contents/summary.

• Introduction
• Computing Software Basics
• Errors and Uncertainties in Computations
• Monte Carlo: Randomness, Walks, and Decays
• Differentiation and Integration
• Matrix Computing
• Trial-and-Error Searching and Data Fitting
• Solving Differential Equations: Nonlinear Oscillations
• ODE Applications: Eigenvalues, Scattering, and Projectiles
• High-Performance Hardware and Parallel Computers
• Applied HPC: Optimization, Tuning, and GPU Programming
• Fourier Analysis: Signals and Filters
• Wavelet and Principal Components Analyses: Nonstationary Signals and Data Compression
• Nonlinear Population Dynamics
• Continuous Nonlinear Dynamics
• Fractals and Statistical Growth Models
• Thermodynamic Simulations and Feynman Path Integrals
• Molecular Dynamics Simulations
• PDE Reviewand Electrostatics via Finite Differences and Electrostatics via Finite Differences
• Heat Flow via Time Stepping
• Wave Equations I: Strings and Membranes
• Wave Equations II: QuantumPackets and Electromagnetic
• Electrostatics via Finite Elements
• Shocks Waves and Solitons
• Fluid Dynamics
• Integral Equations of Quantum Mechanics
• Appendix A Codes, Applets, and Animations

PHY432 — Computational Methods in Physics | Learning | Beckstein Lab

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PHY432 — Computational Methods in Physics

The course provides an introduction to using the computer as a tool to solve problems in physics. Students will learn to analyze problems, select appropriate numerical algorithms, implement them using Python , a programming language widely used in scientific computing, and critically evaluate their numerical results. Problems will be drawn from diverse areas of physics.

This class starts in Spring 2022 as PHY 432 . (The class was running as PHY 494 from Spring 2016 to Spring 2021.)

For a brief overview, see this page and the Syllabus .

The topics of the class include applications in electrostatics and heat transfer as well as applications such as writing a molecular dynamics code to simulate liquid argon or simulating the lenses in an transmission electron microscope.

Students worked on Final Projects in 2019 as diverse as path integral Monte Carlo, simulations of celestial bodies such as Jupiter’s Trojan asteroids or the solar system, electrons in J J Thompson’s classical experiment, molecular dynamics of organic molecules or rock salt crystals, or the collective behavior of self driving cars. The Final Projects in 2016 included navigation in the solar system, real time digital soundeffect processing, and agent-based modeling such as generating a winning strategy for Black Jack.

Time and requirements

All times and dates to be confirmed.

• Spring 2022: listed as PHY 432
• TTh 1:30pm — 2:45pm
• PHY 252 is a pre- or co-requisite.
• Programming experience as taught in PHY 202 (online) or equivalent required.

If possible, bring your own laptop (Linux, MacOS, Windows).

Course Description

The course provides a practical introduction to using the computer as a tool to solve problems in physics. Students will learn how to work in a scientific computing environment, to analyze a physical problem, select appropriate numerical algorithms to solve the problem, and to implement them. The course will introduce the students to computer graphics and object oriented design. Students will work in teams, critically evaluate their approaches and results, and present them in a professional manner to their peers. The instructor will introduce problems and guide students to their solution.

This is a three-credit hour course. It will be taught in a computer laboratory/studio setting. The emphasis is on practical work , with the instructor initially introducing the problem, and the students then pursuing pre-structured programming exercises and projects. Assessment will primarily focus on projects, including group projects, in which students solve a problem as a small team and present their work as a short report or as a presentation.

In order to facilitate the interactive setting, the capacity will be capped at 24 students. The class is designed to work equally well for remote and in-person students.

PHY 252 is a co-requisite.

Basic programming experience in Python is required; the Online class PHY 202 (by the same instructor) is recommended and can be taken in the preceding Spring, Summer, or Fall semester.

Learning outcomes

Students will learn to program computers in order to solve physical problems. In particular, they will be able to write object oriented code in the open source Python programming language, which is widely used in science and engineering and in some of the biggest tech companies such as Google.

Students will also learn how to solve problems in teams and to communicate their work clearly and effectively.

By the completion of the course, students should be able to quickly apply their knowledge to problems they encounter in other courses and experimental and theoretical research projects.

List of topics

• Working in a scientific computing environment: basic Linux and command line.
• Version control with git
• Programming in Python (including object oriented programming and use of the numpy library).
• Plotting and visualization with matplotlib .
• Numerics fundamentals: numbers, errors
• Differentiation and Integration
• Ordinary differential equations ( ODE ), integration of equations of motion
• Linear algebra (vectors, matrices, eigenvalue problems, SVD )
• Root finding and optimization
• Partial differential equations ( PDE ) e.g. Poisson’s equation, diffusion equation, wave equation
• Monte Carlo methods (importance sampling, Ising model)

For mode details see the course modules overview .

Books and Resources

Recommended text books.

• Computational Modeling and Visualization of Physical Systems with Python , Jay Wang (2016) Wiley- VCH . ISBN : 978-1-119-17918-4
• Computational Physics: Problem Solving with Python, 3rd Edition , Rubin H. Landau, Manuel J. Páez, Cristian C. Bordeianu. (2015) Wiley- VCH . ISBN : 978-3-527-41315-7

Additional books and online resources

• Effective Computation in Physics. Field Guide to Research with Python , Anthony Scopatz and Kathryn D. Huff. (2015) O’Reilly
• A Survey of Computational Physics , Rubin Landau, Manuel J. Páez, and Cristian C. Bordeianu. (2011) Princeton University Press. Free online ComPADRE edition and PDF
• Software Carpentry (especially the lessons on the Unix Shell , Version Control with Git , and Programming with Python )

Course website

• Course page py4phy.github.io/PHY432
• PHY432 GitHub site

For a number of lessons in Spring 2020, videos were created, which you can look at to get a feel for the class.

Discuss: “PHY432 — Computational Methods in Physics”

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Computational Problems for Physics: With Guided Solutions Using Python (Series in Computational Physics) 1st Edition, Kindle Edition

Our future scientists and professionals must be conversant in computational techniques. In order to facilitate integration of computer methods into existing physics courses, this textbook offers a large number of worked examples and problems with fully guided solutions in Python as well as other languages (Mathematica, Java, C, Fortran, and Maple). It’s also intended as a self-study guide for learning how to use computer methods in physics. The authors include an introductory chapter on numerical tools and indication of computational and physics difficulty level for each problem. Readers also benefit from the following features:

• Detailed explanations and solutions in various coding languages.

• Problems are ranked based on computational and physics difficulty.

• Basics of numerical methods covered in an introductory chapter.

• Programming guidance via flowcharts and pseudocode.

Rubin Landau is a Distinguished Professor Emeritus in the Department of Physics at Oregon State University in Corvallis and a Fellow of the American Physical Society (Division of Computational Physics).

Manuel Jose Paez-Mejia is a Professor of Physics at Universidad de Antioquia in Medellín, Colombia.

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About the author.

Rubin Landau is a Distinguished Professor Emeritus in the Department of Physics at Oregon State University in Corvallis and a Fellow of the American Physical Society (Division of Computational Physics). His research specialty is computational studies of the scattering of elementary particles from subatomic systems and momentum space quantum mechanics. Landau has taught courses throughout the undergraduate and graduate curricula, and, for over 20 years, in computational physics. He was the founder of the OSU Computational Physics degree program, an Executive Committee member of the APS Division of Computational Physics, and the AAPT Technology Committee. At present Landau is the Education co-editor for AIP/IEEE Computing in Science & Engineering and co-editor of this Taylor & Francis book series on computational physics. He has been a member of the XSEDE advisory committee and has been part of the Education Program at the SuperComputing (SC) conferences for over a decade.

Manuel Jose Paez-Mejia has been a Professor of Physics at Universidad de Antioquia in Medellín, Colombia since January 1969. He has been teaching courses in Modern Physics, Nuclear Physics, Computational Physics, Numerical Methods, Mathematical Physics, and Programming in Fortran, Pascal, and C languages. He has authored scientific papers in nuclear physics and computational physics, as well as texts on the C Language, General Physics, and Computational Physics (coauthored with Rubin Landau and Cristian Bordeianu). In the past, he and Landau conducted pioneering computational investigations of the interactions of mesons and nucleons with few-body nuclei. Professor Paez has led workshop in Computational Physics throughout Latin America, and has been Director of Graduate Studies in Physics at the Universidad de Antioquia.

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Computational Physics: Problem Solving with Python 3rd Edition Solutions

The use of computation and simulation has become an essential part of the scientific process. Being able to transform a theory into an algorithm requires significant theoretical insight detailed physical and mathematical understanding and a working level of competency in programming. This upper-division text provides an unusually broad survey of the topics of modern computational physics from a multidisciplinary computational science point of view. Its philosophy is rooted in learning by doing (assisted by many model programs) with new scientific materials as well as with the Python programming language. Python has become very popular particularly for physics education and large scientific projects. It is probably the easiest programming language to learn for beginners yet is also used for mainstream scientific computing and has packages for excellent graphics and even symbolic manipulations. The text is designed for an upper-level undergraduate or beginning graduate course and provides the reader with the essential knowledge to understand computational tools and mathematical methods well enough to be successful. As part of the teaching of using computers to solve scientific problems the reader is encouraged to work through a sample problem stated at the beginning of each chapter or unit which involves studying the text writing debugging and running programs visualizing the results and the expressing in words what has been done and what can be concluded. Then there are exercises and problems at the end of each chapter for the reader to work on their own (with model programs given for that purpose).Read more

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Title: a python gpu-accelerated solver for the gross-pitaevskii equation and applications to many-body cavity qed.

Abstract: TorchGPE is a general-purpose Python package developed for solving the Gross-Pitaevskii equation (GPE). This solver is designed to integrate wave functions across a spectrum of linear and non-linear potentials. A distinctive aspect of TorchGPE is its modular approach, which allows the incorporation of arbitrary self-consistent and time-dependent potentials, e.g., those relevant in many-body cavity QED models. The package employs a symmetric split-step Fourier propagation method, effective in both real and imaginary time. In our work, we demonstrate a significant improvement in computational efficiency by leveraging GPU computing capabilities. With the integration of the latter technology, TorchGPE achieves a substantial speed-up with respect to conventional CPU-based methods, greatly expanding the scope and potential of research in this field.

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Applied Research Associate for Machine Learning on FPGAs

The Omega Group in the Physics Department at BNL focuses on the ATLAS experiment at CERN in Geneva, Switzerland. The Omega Group has major responsibilities in the operation of the Liquid Argon calorimeter, Trigger & Data Acquisition, and software development, and has leading roles in the ATLAS High Luminosity Large-Hadron Collider (HL-LHC) upgrade (Silicon Strip detector, Liquid Argon calorimeter, and Trigger & Data Acquisition). The group also has strong detector R&D programs on silicon detectors and in trigger/data acquisition. BNL serves as the host laboratory for the U.S. ATLAS Operations Program, the U.S. ATLAS HL-LHC Upgrade Project, the ATLAS Tier-1 computing facility, and is an active U.S. ATLAS Center.

The Omega Group in the Physics Department has an opening for a postdoc to join in the development of future collider experiments and in the current upgrade of the ATLAS experiment. The successful candidate will have a strong interest in the LHC scientific program and experience in firmware development with a focus on high level synthesis and machine learning. In this position you will contribute to R&D for machine learning on FPGA in low-latency real-time systems, and towards the development of future detectors and data acquisition systems. This position has a high level of interaction with an international and multicultural scientific community.

Essential Duties and Responsibilities:

• Developing and integrating firmware for low-latency real-time systems with a focus on High Level Synthesis for Machine Learning applications
• R&D for new detector and data acquisition concepts towards future detectors
• Participation in detector and trigger performance as part of the ATLAS scientific program with a strong focus on Electroweak and Higgs physics

Required Knowledge, Skills, and Abilities:

• Ph.D. in experimental particle, nuclear, or applied physics, electrical engineering, computing science, or a related discipline
• Experience with programming FPGAs (VHDL or System Verilog), heterogeneous devices including Systems-on-Chip, GPUs, or HPC machines
• Experience with High Level Synthesis tools for FPGAs
• Experience with machine learning techniques and tools
• Experience with data analysis and knowledge of statistical techniques for data analysis
• Knowledge of mixed-signal circuits, digital signal processing, and modern circuit components
• Experience with laboratory equipment, HDL-based design, simulation, and testing
• Ability to work in a large international collaboration
• Hands-on problem solving skills
• Clear and concise verbal and written communication skills
• Research experience demonstrated through a record of publications in peer-reviewed journals
• Proven ability for disseminating research results by writing manuscripts and giving academic presentations

Preferred Knowledge, Skills, and Abilities:

• Experience with computing and software development for high energy or nuclear physics experiments
• Expertise programming FPGAs, heterogeneous devices, GPUs, or HPC machines
• Experience with detector and electronics testing, construction, commissioning, operations or R&D for high energy or nuclear physics experiments
• Analysis with LHC data based on C/C++ and Python
• Knowledge of EDA tools for circuit simulation, schematics entry, and PCB layout design
• Experience in readout system development of scientific detector systems

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• Brookhaven National Laboratory is committed to providing fair, equitable and competitive compensation. The full salary range for this position is $70200 - $116200 / year. Salary offers will be commensurate with the final candidate’s qualification, education and experience and considered with the internal peer group.

Please express your interest via electronic application including cover letter, CV, and Research Statement as Word documents or PDF in the resume field of the application. If references are requested, the application is complete when all of these documents have been received.

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Brookhaven employees are subject to restrictions related to participation in Foreign Government Talent Recruitment Programs, as defined and detailed in United States Department of Energy Order 486.1A. You will be asked to disclose any such participation at the time of hire for review by Brookhaven. The full text of the Order may be found at: https://www.directives.doe.gov/directives-documents/400-series/0486.1-BOrder-a/@@images/file

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Brookhaven Science Associates is an equal opportunity employer that values inclusion and diversity at our Lab. We are committed to ensuring that all qualified applicants receive consideration for employment and will not be discriminated against on the basis of race, color, religion, sex, sexual orientation, gender identity, national origin, age, status as a veteran, disability or any other federal, state or local protected class. BSA takes affirmative action in support of its policy and to advance in employment individuals who are minorities, women, protected veterans, and individuals with disabilities. We ensure that individuals with disabilities are provided reasonable accommodation to participate in the job application or interview process, to perform essential job functions, and to receive other benefits and privileges of employment. Please contact us to request accommodation.

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