Об этом курсе
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Гибкие сроки

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Промежуточный уровень

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

Прибл. 17 часа на выполнение

Предполагаемая нагрузка: 8 hours/week...

Английский

Субтитры: Английский

Чему вы научитесь

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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% онлайн

Начните сейчас и учитесь по собственному графику.

Гибкие сроки

Назначьте сроки сдачи в соответствии со своим графиком.

Промежуточный уровень

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

Прибл. 17 часа на выполнение

Предполагаемая нагрузка: 8 hours/week...

Английский

Субтитры: Английский

Программа курса: что вы изучите

Неделя
1
3 ч. на завершение

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 видео ((всего 63 мин.)), 1 материал для самостоятельного изучения, 1 тест
6 видео
W1V2 Spatial scales and meshing12мин
W1V3 Waves in a discrete world6мин
W1V4 Parallel Simulations10мин
W1V5 A bit of wave physics16мин
W1V6 Python and Jupyter notebooks10мин
1 материал для самостоятельного изучения
Jupiter Notebooks and Python10мин
1 практическое упражнение
Discretization, Waves, Computers45мин
Неделя
2
4 ч. на завершение

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 видео ((всего 41 мин.)), 1 тест
8 видео
W2V2 Definitions3мин
W2V3 Taylor Series5мин
W2V4 Python: First Derivative10мин
W2V5 Operators5мин
W2V6 High Order3мин
W2V7 Python: High Order7мин
W2V8 Summary1мин
1 практическое упражнение
Taylor Series and Finite Differences20мин
Неделя
3
3 ч. на завершение

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.

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9 видео ((всего 50 мин.)), 1 тест
9 видео
W3V2 Algorithm4мин
W3V3 Boundaries, Sources4мин
W3V4 Initialization4мин
W3V5 Python: Waves in 1D5мин
W3V6 Analytical Solutions4мин
W3V7 Python: Waves in 1D3мин
W3V8 Von Neumann Analysis19мин
W3V9 Summary1мин
1 практическое упражнение
Acoustic Wave Equation with Finite Differences in 1D - CFL criterion
Неделя
4
7 ч. на завершение

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 видео ((всего 83 мин.)), 1 тест
10 видео
W4V2 Acoustic Waves 2D – Finite-Difference Algorithm6мин
W4V3 Python: Acoustic Waves 2D8мин
W4V4 Acoustic Waves 2D – von Neumann Analysis5мин
W4V5 Acoustic Waves 2D – Waves in a Fault Zone8мин
W4V6 Python: Waves in a Fault Zone9мин
W4V7 Elastic Wave Equation – Staggered Grids16мин
W4V8 Python: Staggered Grids5мин
W4V9 Improving numerical accuracy11мин
W4V10 Wrap up3мин
1 практическое упражнение
Acoustic Wave Equation in 2D - Numerical Anisotropy - Staggered Grids45мин
Неделя
5
6 ч. на завершение

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 видео ((всего 62 мин.)), 1 тест
9 видео
W5V2 Fourier Series - Examples5мин
W5V3 Discrete Fourier Series5мин
W5V4 The Fourier Transform - Derivative6мин
W5V5 Solving the 1D/2D Wave Equation with Python11мин
W5V6 Convolutional Operators6мин
W5V7 Chebyshev Polynomials - Derivatives8мин
W5V8 Chebyshev Method – 1D Elastic Wave Equation7мин
W5V9 Summary3мин
1 практическое упражнение
Pseudospectral method45мин
Неделя
6
2 ч. на завершение

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.

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5 видео ((всего 42 мин.)), 1 тест
5 видео
W6V2 Weak Form - Galerkin Principle7мин
W6V3 Solution Scheme9мин
W6V4 Boundary Conditions - System Matrices9мин
W6V5 Relaxation Method - Python: Static Eleasticity7мин
1 практическое упражнение
Finite-element method - Static problem45мин
Неделя
7
3 ч. на завершение

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.

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7 видео ((всего 56 мин.)), 1 тест
7 видео
W7V2 Solution Algorithm - 1D Elastic Case12мин
W7V3 Differentiation Matrices8мин
W7V4 Python: 1D Elastic Wave Equation11мин
W7V5 h-adaptivity6мин
W7V6 Shape Functions9мин
W7V7 Dynamic Elasticity - Summary2мин
1 практическое упражнение
Dynamic elasticity - Finite elements45мин
Неделя
8
4 ч. на завершение

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.

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7 видео ((всего 51 мин.)), 1 тест
7 видео
W8V2 Weak Form - Matrix Formulation9мин
W8V3 Element Level5мин
W8V4 Lagrange Interpolation12мин
W8V5 Python:Lagrange Interpolation6мин
W8V6 Numerical Integration7мин
W8V7 Python Numerical Integration4мин
1 практическое упражнение
Lagrange Interpolation - Numerical Integration45мин
Неделя
9
4 ч. на завершение

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.

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7 видео ((всего 50 мин.)), 1 тест
7 видео
W9V2 System of Equations - Element Level6мин
W9V3 Global Assembly8мин
W9V4 Python: 1D Homogeneous Case13мин
W9V5 Python: Heterogeneous Case in 1D8мин
W9V6 Convergence Test4мин
W9V7 Wrap Up2мин
1 практическое упражнение
Spectral-element method - Convergence test45мин
4.8
Рецензии: 14Chevron Right

Лучшие отзывы о курсе Computers, Waves, Simulations: A Practical Introduction to Numerical Methods using Python

автор: 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.

автор: 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.

Преподаватели

Avatar

Heiner Igel

Prof. Dr.
Earth and Environmental Sciences

О Мюнхенский университет Людвига-Максимилиана (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....

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