课程信息

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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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中级

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

完成时间大约为35 小时

英语（English）

字幕：法语（French）, （欧洲人讲的）葡萄牙语, 俄语（Russian）, 英语（English）, 西班牙语（Spanish）

您将学到的内容有

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.

可灵活调整截止日期

根据您的日程表重置截止日期。

可分享的证书

完成后获得证书

100% 在线

立即开始，按照自己的计划学习。

中级

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

完成时间大约为35 小时

英语（English）

字幕：法语（French）, （欧洲人讲的）葡萄牙语, 俄语（Russian）, 英语（English）, 西班牙语（Spanish）

授课教师

授课教师评分

4.86/5 （106 个评分）

Heiner Igel

Prof. Dr.

Earth and Environmental Sciences

17,506 个学生

1 门课程

授课大纲 - 您将从这门课程中学到什么

内容评分97%（1,522 个评分）

周

1

周 1

完成时间为 3 小时

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.

完成时间为 3 小时

6 个视频（总计 63 分钟）, 1 个阅读材料, 1 个测验

周

2

周 2

完成时间为 4 小时

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.

完成时间为 4 小时

8 个视频（总计 41 分钟）

周

3

周 3

完成时间为 3 小时

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.

完成时间为 3 小时

9 个视频（总计 50 分钟）

周

4

周 4

完成时间为 7 小时

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.

完成时间为 7 小时

10 个视频（总计 83 分钟）

10 个视频

W4V1 Acoustic Waves 2D – Analytical Solutions8分钟

W4V2 Acoustic Waves 2D – Finite-Difference Algorithm6分钟

W4V3 Python: Acoustic Waves 2D8分钟

W4V4 Acoustic Waves 2D – von Neumann Analysis5分钟

W4V5 Acoustic Waves 2D – Waves in a Fault Zone8分钟

W4V6 Python: Waves in a Fault Zone9分钟

W4V7 Elastic Wave Equation – Staggered Grids16分钟

W4V8 Python: Staggered Grids5分钟

W4V9 Improving numerical accuracy11分钟

W4V10 Wrap up3分钟

1 个练习

Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids45分钟

周

5

周 5

完成时间为 6 小时

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.

完成时间为 6 小时

9 个视频（总计 62 分钟）

9 个视频

W5V1 Function Interpolation – Trigonometric basis functions5分钟

W5V2 Fourier Series - Examples5分钟

W5V3 Discrete Fourier Series5分钟

W5V4 The Fourier Transform - Derivative6分钟

W5V5 Solving the 1D/2D Wave Equation with Python11分钟

W5V6 Convolutional Operators6分钟

W5V7 Chebyshev Polynomials - Derivatives8分钟

W5V8 Chebyshev Method – 1D Elastic Wave Equation7分钟

W5V9 Summary3分钟

1 个练习

Pseudospectral method45分钟

周

6

周 6

完成时间为 2 小时

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.

完成时间为 2 小时

5 个视频（总计 42 分钟）

周

7

周 7

完成时间为 3 小时

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.

完成时间为 3 小时

7 个视频（总计 56 分钟）

周

8

周 8

完成时间为 4 小时

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.

完成时间为 4 小时

7 个视频（总计 51 分钟）

周

9

周 9

完成时间为 4 小时

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.

完成时间为 4 小时

7 个视频（总计 50 分钟）

审阅

4.7

110 条评论

5 stars
80.20%
4 stars
15.77%
3 stars
2.01%
2 stars
1.67%
1 star
0.33%

来自COMPUTERS, WAVES, SIMULATIONS: A PRACTICAL INTRODUCTION TO NUMERICAL METHODS USING PYTHON的热门评论

由 RM 提供Jul 11, 2020

This is an excellent course as I have found. The instructor has taught us many important concepts including the detailed codes. I would love to join further courses from Prof. Igel.

由 AS 提供Oct 21, 2020

Although the mathematical formulation & codes are hard to grasp, I really enjoy and feel challenged by this course. Thank you!

由 AT 提供Dec 1, 2021

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

由 AC 提供Aug 22, 2020

An excellent course towards gaining knowledge in the domain of computational physics. Must recommended

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