Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python from Coursera

Interested in learning how to solve partial differential equations with numerical methods and how to turn them into python codes? This course provides you with a basic introduction how to apply methods like the finite-difference method, the pseudospectral method, the linear and spectral element method to the 1D (or 2D) scalar wave equation. The mathematical derivation of the computational algorithm is accompanied by python codes embedded in Jupyter notebooks. In a unique setup you can see how the mathematical equations are transformed to a computer code and the results visualized. The emphasis is on illustrating the fundamental mathematical ingredients of the various numerical methods (e.g., Taylor series, Fourier series, differentiation, function interpolation, numerical integration) and how they compare. You will be provided with strategies how to ensure your solutions are correct, for example benchmarking with analytical solutions or convergence tests. The mathematical aspects are complemented by a basic introduction to wave physics, discretization, meshes, parallel programming, computing models.

The course targets anyone who aims at developing or using numerical methods applied to partial differential equations and is seeking a practical introduction at a basic level. The methodologies discussed are widely used in natural sciences, engineering, as well as economics and other fields.

What's inside

Syllabus

Week 01 - Discrete World, Wave Physics, Computers

The use of numerical methods to solve partial differential equations is motivated giving examples form Earth sciences. Concepts of discretization in space and time are introduced and the necessity to sample fields with sufficient accuracy is motivated (i.e. number of grid points per wavelength). Computational meshes are discussed and their power and restrictions to model complex geometries illustrated. The basics of parallel computers and parallel programming are discussed and their impact on realistic simulations. The specific partial differential equation used in this course to illustrate various numerical methods is presented: the acoustic wave equation. Some physical aspects of this equation are illustrated that are relevant to understand its solutions. Finally Jupyter notebooks are introduced that are used with Python programs to illustrate the implementation of the numerical methods.

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Provides basic understanding of partial differential equations and teaches numerical methods to solve them

Applies methods to wave equation, making it relevant to physics, engineering, and other fields

Taught by recognized instructor Heiner Igel, known for work in geophysics

Uses Python codes for implementation, providing a practical approach

Emphasizes mathematical and computational foundations, giving learners a strong base

Introduces parallel programming and computing models, preparing learners for real-world simulations

Reviews summary

Practical numerical methods with python

Learners say this course provides a well-balanced mix of theory and practice in numerical methods, with an emphasis on wave propagation. It offers a range of methods, including finite difference, finite element, and spectral element, all of which are supported by Python code for implementation. The course is well-structured, with engaging assignments and clear explanations from the instructor. Overall, learners found it to be a highly enjoyable and valuable learning experience.

The assignments are engaging and help reinforce the concepts learned in the lectures.

"The assignments and the codes (included inside the course content) gives you a way to judge your progress day by day."

"The algorithms are easy to understand and the introduction of the Jupiter Notebooks is something I enjoyed a lot."

"The simulations are beautiful too."

The course is well-structured, with clear and concise explanations.

"The course was a great introduction to FEM-FDM, and more."

"The maths behind the code implementations are handled by the instructor in such a way that it was neither too long and boring nor not sufficient, but right at the point."

"I believe, when I study many body simulations, I will not get an headache understanding the context and the codes."

The instructor provides clear explanations of the concepts covered in the course.

"The video lectures are very well made and great effort is made to present concepts in an understandable way."

"The existence of the Jupyter notebooks greatly compliment the course's objectives."

"There is a lot of content and the author gives initiative to examine the given code with the follow along videos."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python with these activities:

Review Partial Differential Equations (PDEs)

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Start the course with a clear understanding of the fundamental concepts and terminology related to Partial Differential Equations.

Browse courses on Partial Differential Equations

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Revisit the definitions and classifications of linear and nonlinear PDEs
Review methods for solving basic PDEs, such as separation of variables and the method of characteristics

Explore Online Tutorials on Numerical Methods

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Supplement your understanding of the numerical methods used in this course by seeking out additional hands-on resources.

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Identify online tutorials that cover the specific numerical methods covered in the course
Work through the tutorials, practicing the implementation of the methods

Join a Study Group for Numerical Methods

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Connect with peers, discuss concepts, and solve problems collaboratively to enhance your understanding.

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Reach out to classmates or fellow students to form a study group
Schedule regular meetings to discuss course material, work on assignments together, and provide support

Six other activities

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Solve Practice Problems on Numerical Methods

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Improve your problem-solving skills and reinforce your understanding of the numerical techniques by engaging in deliberate practice.

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Find textbooks or online resources with practice problems on numerical methods
Attempt to solve the problems independently
Check your solutions against the provided answers or consult with instructors/peers for guidance

Develop a Computational Model Using Python

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Apply your knowledge of numerical methods and Python programming to create a functional computational model.

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Identify a specific problem or phenomenon that can be modeled using numerical methods
Design and implement a Python program to solve the problem numerically
Validate your model by comparing its results with analytical solutions or experimental data

Become a Mentor for Junior Students

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Reinforce your understanding of course concepts and contribute to the learning community by mentoring students.

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Identify opportunities to volunteer as a mentor or tutor for students taking introductory courses in numerical methods
Provide support and guidance to mentees, helping them understand concepts and solve problems

Create a Comprehensive Course Summary

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Retain and reinforce the knowledge acquired in the course by creating a detailed and organized collection of course materials.

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Regularly review your lecture notes, textbooks, and other course materials
Summarize key concepts, formulas, and algorithms in your own words
Organize your materials logically for easy reference and future review

Attend a Workshop on Advanced Numerical Methods

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Expand your knowledge and skills by participating in a workshop focused on advanced numerical techniques and their applications.

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Research and identify workshops that align with your interests and learning goals
Register for and actively participate in the workshop

Contribute to Open-Source Numerical Methods Projects

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Gain practical experience and contribute to the wider community by participating in open-source projects related to numerical methods.

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Identify open-source projects that are relevant to your interests and skill level
Explore the project's documentation and codebase
Contribute to the project by reporting bugs, suggesting improvements, or adding new features

Career center

Learners who complete Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python will develop knowledge and skills that may be useful to these careers:

Acoustic Engineer

Acoustic Engineers design and analyze the propagation and control of sound. They may work on projects such as architectural acoustics, noise control, or underwater acoustics. This course can help build a foundation for a career in acoustic engineering by providing a thorough understanding of wave physics, numerical methods, and their applications in acoustics.

See salaries and explore the career path for Acoustic Engineer

Geophysicist

Geophysicists study the physical properties of the Earth and its atmosphere. They use a variety of methods to collect and analyze data, including seismic waves, gravity measurements, and electrical signals. This course can help build a foundation for a career in geophysics by providing a practical introduction to numerical methods used to solve partial differential equations, which are essential for modeling and interpreting geophysical data.

See salaries and explore the career path for Geophysicist

Mechanical Engineer

Mechanical Engineers design and develop mechanical systems, including machines, engines, and robots. They use a variety of numerical methods to analyze and optimize their designs. This course can help build a foundation for a career in mechanical engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

See salaries and explore the career path for Mechanical Engineer

Materials Scientist

Materials Scientists study the properties and behavior of materials, including metals, ceramics, and polymers. They use a variety of numerical methods to model and predict the behavior of materials. This course can help build a foundation for a career in materials science by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Aerospace Engineer

Aerospace Engineers design and develop aircraft, spacecraft, and missiles. They use a variety of numerical methods to analyze and optimize their designs. This course can help build a foundation for a career in aerospace engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Computational Physicist

Computational Physicists use computers to model and simulate physical systems. They use a variety of numerical methods to solve complex problems in physics. This course can help build a foundation for a career in computational physics by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

See salaries and explore the career path for Computational Physicist

Data Scientist

Data Scientists use data to solve problems and make predictions. They use a variety of numerical methods to analyze and interpret data. This course can help build a foundation for a career in data science by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Financial Analyst

Financial Analysts use financial data to make investment recommendations. They use a variety of numerical methods to analyze and interpret financial data. This course may be useful for building a foundation for a career in financial analysis by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Software Engineer

Software Engineers design, develop, and maintain software applications. They use a variety of numerical methods to optimize the performance of their software. This course may be useful for building a foundation for a career in software engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Operations Research Analyst

Operations Research Analysts use mathematical and analytical methods to improve the efficiency of operations. They use a variety of numerical methods to solve complex optimization problems. This course may be useful for building a foundation for a career in operations research by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Petroleum Engineer

Petroleum Engineers design and develop methods for extracting oil and gas from the Earth. They use a variety of numerical methods to model and simulate the flow of fluids in porous media. This course may be useful for building a foundation for a career in petroleum engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Civil Engineer

Civil Engineers design and build infrastructure, such as bridges, roads, and buildings. They use a variety of numerical methods to analyze and optimize their designs. This course may be useful for building a foundation for a career in civil engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Electrical Engineer

Electrical Engineers design and develop electrical systems, such as power plants, electrical motors, and electronic devices. They use a variety of numerical methods to analyze and optimize their designs. This course may be useful for building a foundation for a career in electrical engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Chemical Engineer

Chemical Engineers design and develop chemical processes, such as those used to produce pharmaceuticals, plastics, and fuels. They use a variety of numerical methods to model and simulate chemical reactions. This course may be useful for building a foundation for a career in chemical engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

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Biomedical Engineer

Biomedical Engineers design and develop medical devices and systems. They use a variety of numerical methods to model and simulate biological systems. This course may be useful for building a foundation for a career in biomedical engineering by providing a practical introduction to numerical methods, including the finite-difference method and the spectral-element method.

See salaries and explore the career path for Biomedical Engineer