https://github.com/jerrychen97/hydrodynamics

This is a repo for the homework 3 of Computational Physics: solving the Burger equation
https://github.com/jerrychen97/hydrodynamics

Last synced: 13 days ago
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This is a repo for the homework 3 of Computational Physics: solving the Burger equation

• Host: GitHub
• URL: https://github.com/jerrychen97/hydrodynamics
• Owner: JerryChen97
• Created: 2020-03-20T19:09:24.000Z (almost 5 years ago)
• Default Branch: master
• Last Pushed: 2020-03-21T03:22:47.000Z (almost 5 years ago)
• Last Synced: 2024-11-11T14:58:49.717Z (2 months ago)
• Language: Jupyter Notebook
• Homepage:
• Size: 13.2 MB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
# Hydrodynamics

This is a repo for the homework3 of Computational Physics: solving the Burger equation

## Introduction

In this homework we aim to solve the nonlinear partial differential equation



This is also well-known as the inviscid Burgers equation. 

## Theoretical Analysis

By characteristic method, we can get the characteristic equation



and then we know that 



This implies that along the characteristic curve ,  is constant,



along with the solution to the characteristic equation



offering us an implicit solution to the inviscid Burgers equation



Even though usually due to the complexity of , we will not be able to solve the explicit solution, we can still manipulate  to give a parametric curve as the solution.

However, such a solution cannot guarantee itself to be a function, which is necessary to be the real solution.

After a certain point , this characteristic solution will not hold anymore, along with the formation of shock.

## Numerical Calculation

In this homework we calculated the convergence order of the numerical method offered by William at different resolutions:

- Before the formation of shock, we compare the numerical results with the theoretical implicit solution. The original code of Will is edited by me to be able to offer an exact solution inside the class.

- After the formation of shock, we compare the numerical results with each other to approximate the convergence order, i.e.



where  is the lattice space.


## Findings

Not everywhere (w.r.t. ) our minmod method will reach the order of 2.

This is because only at the shock region, where the numerical method shows obvious errors, the difference of errors among different resolutions can be presented well.

![alt text](allerrors.png "Title")

![alt text](pointwise.png "Title")

Other conclusions are quite similar. 

For the details please check the comments in the `burgers.ipynb`.

## Acknowledgement

Thanks to Cristian for elaborate discussion with me regarding all respects of this homework.

## Reference

- Burgers Equation, Mikel Landajuela, http://www.bcamath.org/projects/NUMERIWAVES/Burgers_Equation_M_Landajuela.pdf

- Verifying Numerical Convergence Rate, http://www.csc.kth.se/utbildning/kth/kurser/DN2255/ndiff13/ConvRate.pdf