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.

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

Ludwig-Maximilians-Universität München (LMU)## About this Course

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

## 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.

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

## Syllabus - What you will learn from this course

**3 hours to complete**

### Week 01 - Discrete World, Wave Physics, Computers

**3 hours to complete**

**4 hours to complete**

### Week 02 The Finite-Difference Method - Taylor Operators

**4 hours to complete**

**3 hours to complete**

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

**3 hours to complete**

**7 hours to complete**

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

**7 hours to complete**

## Reviews

- 5 stars80.25%
- 4 stars15.98%
- 3 stars1.88%
- 2 stars1.56%
- 1 star0.31%

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

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.

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

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

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

## Frequently Asked Questions

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