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.
Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python
Offered By
Ludwig-Maximilians-Universität München (LMU)
About this Course
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.
Syllabus - What you will learn from this course
Week
13 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.
6 videos (Total 63 min), 1 reading, 1 quiz
6 videos
W1V2 Spatial scales and meshing12m
W1V3 Waves in a discrete world6m
W1V4 Parallel Simulations10m
W1V5 A bit of wave physics16m
W1V6 Python and Jupyter notebooks10m
1 reading
Jupiter Notebooks and Python10m
1 practice exercise
Discretization, Waves, Computers45m
Week
24 hours to complete
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.
8 videos (Total 41 min), 1 quiz
8 videos
W2V2 Definitions3m
W2V3 Taylor Series5m
W2V4 Python: First Derivative10m
W2V5 Operators5m
W2V6 High Order3m
W2V7 Python: High Order7m
W2V8 Summary1m
1 practice exercise
Taylor Series and Finite Differences20m
Week
33 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.
9 videos (Total 50 min), 1 quiz
9 videos
W3V2 Algorithm4m
W3V3 Boundaries, Sources4m
W3V4 Initialization4m
W3V5 Python: Waves in 1D5m
W3V6 Analytical Solutions4m
W3V7 Python: Waves in 1D3m
W3V8 Von Neumann Analysis19m
W3V9 Summary1m
1 practice exercise
Acoustic Wave Equation with Finite Differences in 1D - CFL criterion
Week
47 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.
10 videos (Total 83 min), 1 quiz
10 videos
W4V2 Acoustic Waves 2D – Finite-Difference Algorithm6m
W4V3 Python: Acoustic Waves 2D8m
W4V4 Acoustic Waves 2D – von Neumann Analysis5m
W4V5 Acoustic Waves 2D – Waves in a Fault Zone8m
W4V6 Python: Waves in a Fault Zone9m
W4V7 Elastic Wave Equation – Staggered Grids16m
W4V8 Python: Staggered Grids5m
W4V9 Improving numerical accuracy11m
W4V10 Wrap up3m
1 practice exercise
Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids45m
Instructor
Heiner Igel
Prof. Dr.Earth and Environmental Sciences
About Ludwig-Maximilians-Universität München (LMU)
As one of Europe's leading research universities, LMU Munich is committed to the highest international standards of excellence in research and teaching. Building on its 500-year-tradition of scholarship, LMU covers a broad spectrum of disciplines, ranging from the humanities and cultural studies through law, economics and social studies to medicine and the sciences.
Frequently Asked Questions
When will I have access to the lectures and assignments?
Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.
What will I get if I purchase the Certificate?
When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.
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Is financial aid available?
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