About this Course

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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.

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Shareable Certificate

Earn a Certificate upon completion

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Coursera Labs

Includes hands on learning projects.

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Intermediate Level

Basic knowledge of calculus and analysis, series, partial differential equations, and linear algebra.

Approx. 35 hours to complete

English

What you will learn

How to solve a partial differential equation using the finite-difference, the pseudospectral, or the linear (spectral) finite-element method.
Understanding the limits of explicit space-time simulations due to the stability criterion and spatial and temporal sampling requirements.
Strategies how to plan and setup sophisticated simulation tasks.
Strategies how to avoid errors in simulation results.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Coursera Labs

Includes hands on learning projects.

Learn more about Coursera Labs

Intermediate Level

Basic knowledge of calculus and analysis, series, partial differential equations, and linear algebra.

Approx. 35 hours to complete

English

Instructor

Instructor rating

4.86/5 (115 Ratings)

Heiner Igel

Prof. Dr.

Earth and Environmental Sciences

20,499 Learners

1 Course

Syllabus - What you will learn from this course

Content Rating97%(1,730 ratings)

Week1

Week 1

3 hours to complete

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.

3 hours to complete

6 videos (Total 63 min), 1 reading, 1 quiz

Week2

Week 2

4 hours to complete

Week 02 The Finite-Difference Method - Taylor Operators

4 hours to complete

8 videos (Total 41 min)

Week3

Week 3

3 hours to complete

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.

3 hours to complete

9 videos (Total 50 min)

Week4

Week 4

7 hours to complete

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.

7 hours to complete

10 videos (Total 83 min)

10 videos

W4V1 Acoustic Waves 2D – Analytical Solutions8mW4V2 Acoustic Waves 2D – Finite-Difference Algorithm6mW4V3 Python: Acoustic Waves 2D8mW4V4 Acoustic Waves 2D – von Neumann Analysis5mW4V5 Acoustic Waves 2D – Waves in a Fault Zone8mW4V6 Python: Waves in a Fault Zone9mW4V7 Elastic Wave Equation – Staggered Grids16mW4V8 Python: Staggered Grids5mW4V9 Improving numerical accuracy11mW4V10 Wrap up3m

1 practice exercise

Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids45m

Week5

Week 5

6 hours to complete

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.

6 hours to complete

9 videos (Total 62 min)

9 videos

W5V1 Function Interpolation – Trigonometric basis functions5mW5V2 Fourier Series - Examples5mW5V3 Discrete Fourier Series5mW5V4 The Fourier Transform - Derivative6mW5V5 Solving the 1D/2D Wave Equation with Python11mW5V6 Convolutional Operators6mW5V7 Chebyshev Polynomials - Derivatives8mW5V8 Chebyshev Method – 1D Elastic Wave Equation7mW5V9 Summary3m

1 practice exercise

Pseudospectral method45m

Week6

Week 6

2 hours to complete

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.

2 hours to complete

5 videos (Total 42 min)

Week7

Week 7

3 hours to complete

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.

3 hours to complete

7 videos (Total 56 min)

Week8

Week 8

4 hours to complete

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

4 hours to complete

7 videos (Total 51 min)

Week9

Week 9

4 hours to complete

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

4 hours to complete

7 videos (Total 50 min)

Reviews

4.7

117 reviews

5 stars
80.25%
4 stars
15.98%
3 stars
1.88%
2 stars
1.56%
1 star
0.31%

TOP REVIEWS FROM COMPUTERS, WAVES, SIMULATIONS: A PRACTICAL INTRODUCTION TO NUMERICAL METHODS USING PYTHON

by VPMay 1, 2022

This is an outstanding course that covers many subjects briefly. I have learnt so much with the methodology the lecturer have applied for the course.

by ATDec 1, 2021

Great experience. Really came to know about the theory of simulation techniques coupled with the introductory knowledge of python language.

by FDFeb 24, 2019

I already know that I will learn a lot even though I am an undergrad. ( FTD from Colorado School of Mines)

by YHApr 8, 2019

This is a great course for intro to numerical course with additional bonus on python code, although a little bit too fast pace.

View all reviews

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