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

Approx. 35 hours to complete

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Course Summary

Learn how to simulate waves using computers in this hands-on course. Gain an understanding of wave behavior and apply it to real-world scenarios, such as ocean waves and earthquakes.

Key Learning Points

Learn how to simulate waves using MATLAB and Python
Gain a deeper understanding of wave behavior
Apply your knowledge to real-world scenarios

Job Positions & Salaries of people who have taken this course might have

- USA: $80,000
- India: ₹5,00,000
- Spain: €35,000
- USA: $80,000
- India: ₹5,00,000
- Spain: €35,000
- USA: $63,000
- India: ₹4,50,000
- Spain: €30,000
- USA: $80,000
- India: ₹5,00,000
- Spain: €35,000
- USA: $63,000
- India: ₹4,50,000
- Spain: €30,000
- USA: $75,000
- India: ₹6,00,000
- Spain: €40,000

Learning Outcomes

Understand wave behavior and how to simulate it
Gain practical skills in using MATLAB and Python
Apply your knowledge to real-world scenarios

Prerequisites or good to have knowledge before taking this course

Basic knowledge of programming
Basic understanding of physics

Course Difficulty Level

Intermediate

Course Format

Online
Self-paced
Hands-on

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Notable People in This Field

Dr. Michio Kaku
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Description

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.

Knowledge

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.

Outline

Week 01 - Discrete World, Wave Physics, Computers
W1V1 General Introduction
W1V2 Spatial scales and meshing
W1V3 Waves in a discrete world
W1V4 Parallel Simulations
W1V5 A bit of wave physics
W1V6 Python and Jupyter notebooks
Jupiter Notebooks and Python
Discretization, Waves, Computers

Week 02 The Finite-Difference Method - Taylor Operators
W2V1 Introduction
W2V2 Definitions
W2V3 Taylor Series
W2V4 Python: First Derivative
W2V5 Operators
W2V6 High Order
W2V7 Python: High Order
W2V8 Summary
Taylor Series and Finite Differences

Week 03 The Finite-Difference Method - 1D Wave Equation - von Neumann Analysis
W3V1 Wave Equation
W3V2 Algorithm
W3V3 Boundaries, Sources
W3V4 Initialization
W3V5 Python: Waves in 1D
W3V6 Analytical Solutions
W3V7 Python: Waves in 1D
W3V8 Von Neumann Analysis
W3V9 Summary

Week 04 The Finite-Difference Method in 2D - Numerical Anisotropy, Heterogeneous Media
W4V1 Acoustic Waves 2D – Analytical Solutions
W4V2 Acoustic Waves 2D – Finite-Difference Algorithm
W4V3 Python: Acoustic Waves 2D
W4V4 Acoustic Waves 2D – von Neumann Analysis
W4V5 Acoustic Waves 2D – Waves in a Fault Zone
W4V6 Python: Waves in a Fault Zone
W4V7 Elastic Wave Equation – Staggered Grids
W4V8 Python: Staggered Grids
W4V9 Improving numerical accuracy
W4V10 Wrap up
Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids

Week 05 The Pseudospectral Method, Function Interpolation
W5V1 Function Interpolation – Trigonometric basis functions
W5V2 Fourier Series - Examples
W5V3 Discrete Fourier Series
W5V4 The Fourier Transform - Derivative
W5V5 Solving the 1D/2D Wave Equation with Python
W5V6 Convolutional Operators
W5V7 Chebyshev Polynomials - Derivatives
W5V8 Chebyshev Method – 1D Elastic Wave Equation
W5V9 Summary
Pseudospectral method

Week 06 The Linear Finite-Element Method - Static Elasticity
W6V1 Introduction - Static Elasticity
W6V2 Weak Form - Galerkin Principle
W6V3 Solution Scheme
W6V4 Boundary Conditions - System Matrices
W6V5 Relaxation Method - Python: Static Eleasticity
Finite-element method - Static problem

Week 07 The Linear Finite-Element Method - Dynamic Elasticity
W7V1Introduction - Dynamic Elasticity
W7V2 Solution Algorithm - 1D Elastic Case
W7V3 Differentiation Matrices
W7V4 Python: 1D Elastic Wave Equation
W7V5 h-adaptivity
W7V6 Shape Functions
W7V7 Dynamic Elasticity - Summary
Dynamic elasticity - Finite elements

Week 08 The Spectral-Element Method - Lagrange Interpolation, Numerical Integration
W8V1 Introduction
W8V2 Weak Form - Matrix Formulation
W8V3 Element Level
W8V4 Lagrange Interpolation
W8V5 Python:Lagrange Interpolation
W8V6 Numerical Integration
W8V7 Python Numerical Integration
Lagrange Interpolation - Numerical Integration

Week 09 The Spectral Element Method - 1D Elastic Wave Equation, Convergence Test
W9V1 Lagrange Derivative - Legendre Polynomials
W9V2 System of Equations - Element Level
W9V3 Global Assembly
W9V4 Python: 1D Homogeneous Case
W9V5 Python: Heterogeneous Case in 1D
W9V6 Convergence Test
W9V7 Wrap Up
Spectral-element method - Convergence test

Summary of User Reviews

Learn about the principles of computer simulations and wave mechanics in this highly rated course on Coursera. Users found the course to be engaging and informative, with many praising the interactive simulation exercises that reinforced the concepts taught.