About this Course
4.8
33 ratings
12 reviews

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Flexible deadlines

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Intermediate Level

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

Approx. 16 hours to complete

Suggested: 9 hours/week...

English

Subtitles: English

What you will learn

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    How to solve a partial differential equation using the finite-difference, the pseudospectral, or the linear (spectral) finite-element method.

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    Understanding the limits of explicit space-time simulations due to the stability criterion and spatial and temporal sampling requirements.

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    Strategies how to plan and setup sophisticated simulation tasks.

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    Strategies how to avoid errors in simulation results.

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Intermediate Level

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

Approx. 16 hours to complete

Suggested: 9 hours/week...

English

Subtitles: English

Syllabus - What you will learn from this course

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. ...
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
2
4 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
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. ...
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
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. ...
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
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. ...
9 videos (Total 62 min), 1 quiz
9 videos
W5V2 Fourier Series - Examples5m
W5V3 Discrete Fourier Series5m
W5V4 The Fourier Transform - Derivative6m
W5V5 Solving the 1D/2D Wave Equation with Python11m
W5V6 Convolutional Operators6m
W5V7 Chebyshev Polynomials - Derivatives8m
W5V8 Chebyshev Method – 1D Elastic Wave Equation7m
W5V9 Summary3m
1 practice exercise
Pseudospectral method45m
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. ...
5 videos (Total 42 min), 1 quiz
5 videos
W6V2 Weak Form - Galerkin Principle7m
W6V3 Solution Scheme9m
W6V4 Boundary Conditions - System Matrices9m
W6V5 Relaxation Method - Python: Static Eleasticity7m
1 practice exercise
Finite-element method - Static problem45m
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. ...
7 videos (Total 56 min), 1 quiz
7 videos
W7V2 Solution Algorithm - 1D Elastic Case12m
W7V3 Differentiation Matrices8m
W7V4 Python: 1D Elastic Wave Equation11m
W7V5 h-adaptivity6m
W7V6 Shape Functions9m
W7V7 Dynamic Elasticity - Summary2m
1 practice exercise
Dynamic elasticity - Finite elements45m
Week
8
4 hours to complete

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. ...
7 videos (Total 51 min), 1 quiz
7 videos
W8V2 Weak Form - Matrix Formulation9m
W8V3 Element Level5m
W8V4 Lagrange Interpolation12m
W8V5 Python:Lagrange Interpolation6m
W8V6 Numerical Integration7m
W8V7 Python Numerical Integration4m
1 practice exercise
Lagrange Interpolation - Numerical Integration45m
Week
9
4 hours to complete

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. ...
7 videos (Total 50 min), 1 quiz
7 videos
W9V2 System of Equations - Element Level6m
W9V3 Global Assembly8m
W9V4 Python: 1D Homogeneous Case13m
W9V5 Python: Heterogeneous Case in 1D8m
W9V6 Convergence Test4m
W9V7 Wrap Up2m
1 practice exercise
Spectral-element method - Convergence test45m
4.8
12 ReviewsChevron Right

Top Reviews

By NLMar 14th 2019

Well thought out. The material is ordered logically and easy to follow. This online course compliments the book from which it is based on.

By YHApr 9th 2019

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

Instructor

Avatar

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

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