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

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.

Week 02 The Finite-Difference Method - Taylor Operators

In Week 2 we introduce the basic definitions of the finite-difference method. We learn how to use Taylor series to estimate the error of the finite-difference approximations to derivatives and how to increase the accuracy of the approximations using longer operators. We also learn how to implement numerical derivatives using Python.

Week 03 The Finite-Difference Method - 1D Wave Equation - von Neumann Analysis

We develop the finite-difference algorithm to the acoustic wave equation in 1D, discuss boundary conditions and how to initialize a simulation example. We look at solutions using the Python implementation and observe numerical artifacts. We analytically derive one of the most important results of numerical analysis – the CFL criterion which leads to a conditionally stable algorithm for explicit finite-difference schemes.

Week 04 The Finite-Difference Method in 2D - Numerical Anisotropy, Heterogeneous Media

We develop the solution to the 2D acoustic wave equation, compare with analytical solutions and demonstrate the phenomenon of numerical (non-physical) anisotropy. We extend the von Neumann Analysis to 2D and derive numerical anisotropy analytically. We learn how to initialize a realistic physical problem and illustrate that 2D solution are already quite powerful to understand complex wave phenomena. We introduced the 1D elastic wave equation and show the concept of staggered-grid schemes with the coupled first-order velocity-stress formulation.

Week 05 The Pseudospectral Method, Function Interpolation

We start with the problem of function interpolation leading to the concept of Fourier series. We move to the discrete Fourier series and highlight their exact interpolation properties on regular spatial grids. We introduce the derivative of functions using discrete Fourier transforms and use it to solve the 1D and 2D acoustic wave equation. The necessity to simulate waves in limited areas leads us to the definition of Chebyshev polynomials and their uses as basis functions for function interpolation. We develop the concept of differentiation matrices and discuss a solution scheme for the elastic wave equation using Chebyshev polynomials.

Week 06 The Linear Finite-Element Method - Static Elasticity

We introduce the concept of finite elements and develop the weak form of the wave equation. We discuss the Galerkin principle and derive a finite-element algorithm for the static elasticity problem based upon linear basis functions. We also discuss how to implement boundary conditions. The finite-difference based relaxation method is derived for the same equation and the solution compared to the finite-element algorithm.

Week 07 The Linear Finite-Element Method - Dynamic Elasticity

We extend the finite-element solution to the elastic wave equation and compare the solution scheme to the finite-difference method. To allow direct comparison we formulate the finite-difference solution in matrix-vector form and demonstrate the similarity of the linear finite-element method and the finite-difference approach. We introduce the concept of h-adaptivity, the space-dependence of the element size for heterogeneous media.

Week 08 The Spectral-Element Method - Lagrange Interpolation, Numerical Integration

We introduce the fundamentals of the spectral-element method developing a solution scheme for the 1D elastic wave equation. Lagrange polynomials are discussed as the basis functions of choice. The concept of Gauss-Lobatto-Legendre numerical integration is introduced and shown that it leads to a diagonal mass matrix making its inversion trivial.

Week 09 The Spectral Element Method - 1D Elastic Wave Equation, Convergence Test

We finalize the derivation of the spectral-element solution to the elastic wave equation. We show how to calculate the required derivatives of the Lagrange polynomials making use of Legendre polynomials. We show how to perform the assembly step leading to the final solution system for the elastic wave equation. We demonstrate the numerical solution for homogenous and heterogeneous media.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

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

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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

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.

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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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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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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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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.

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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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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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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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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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