课程信息
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中级

中级

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

完成时间(小时)

完成时间大约为28 小时

建议:6 hours/week...
可选语言

英语(English)

字幕:英语(English)

您将学到的内容有

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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% 在线

100% 在线

立即开始,按照自己的计划学习。
可灵活调整截止日期

可灵活调整截止日期

根据您的日程表重置截止日期。
中级

中级

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

完成时间(小时)

完成时间大约为28 小时

建议:6 hours/week...
可选语言

英语(English)

字幕:英语(English)

教学大纲 - 您将从这门课程中学到什么

1
完成时间(小时)
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. ...
Reading
6 个视频 (总计 63 分钟), 1 个阅读材料, 1 个测验
Video6 个视频
W1V2 Spatial scales and meshing12分钟
W1V3 Waves in a discrete world6分钟
W1V4 Parallel Simulations10分钟
W1V5 A bit of wave physics16分钟
W1V6 Python and Jupyter notebooks10分钟
Reading1 个阅读材料
Jupiter Notebooks and Python10分钟
Quiz1 个练习
Discretization, Waves, Computers45分钟
2
完成时间(小时)
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....
Reading
8 个视频 (总计 41 分钟), 1 个测验
Video8 个视频
W2V2 Definitions3分钟
W2V3 Taylor Series5分钟
W2V4 Python: First Derivative10分钟
W2V5 Operators5分钟
W2V6 High Order3分钟
W2V7 Python: High Order7分钟
W2V8 Summary1分钟
Quiz1 个练习
Taylor Series and Finite Differences20分钟
3
完成时间(小时)
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. ...
Reading
9 个视频 (总计 50 分钟), 1 个测验
Video9 个视频
W3V2 Algorithm4分钟
W3V3 Boundaries, Sources4分钟
W3V4 Initialization4分钟
W3V5 Python: Waves in 1D5分钟
W3V6 Analytical Solutions4分钟
W3V7 Python: Waves in 1D3分钟
W3V8 Von Neumann Analysis19分钟
W3V9 Summary1分钟
Quiz1 个练习
Acoustic Wave Equation with Finite Differences in 1D - CFL criterion
4
完成时间(小时)
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. ...
Reading
10 个视频 (总计 83 分钟), 1 个测验
Video10 个视频
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
Quiz1 个练习
Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids45分钟

讲师

Avatar

Heiner Igel

Prof. Dr.
Earth and Environmental Sciences

关于 路德维希马克西米利安慕尼黑大学

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....

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