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日期：2024-04-12 04:15

Coursework 2: Godunov-Type Scheme for Free Surface Flow
1 Important Notes
 This final coursework is worth 40% of the total module mark
 The deadline for submission is on Friday 12 April 2024 at 14:00
2 Tasks
 Develop a Godunov-type model for solving the following 1D shallow water equations:

 Test different aspects of your numerical model by applying it to simulate all test cases
listed at the end of this document.
 You should produce figures, tables, etc. to visualize and interpret your results.
 Attach your code setup for test case 2: Tidal Wave over a Varying Bed
3 Submission/Report
You should write a short academic essay no more than 7 pages to clearly introduce your
model and present the results. You may use the following template:
A Godunov-Type Model for Free Surface Flow
Shannon Leaky
School of Civil Engineering & Geosciences
Newcastle University
Introduction
Herein, you should give a background for computational hydraulics, introduce different
numerical methods (finite difference, finite element, finite volume, etc.) and explain why you choose
a finite volume Godunov-type scheme (e.g. Toro 2001) to construct your model. A brief literature
review may be necessary.
Godunov-Type Shallow Flow Model
In this section, you should introduce the governing equations, i.e. the 1D shallow water
equations, and your numerical scheme.
Results and Discussion
You should present your results for all test cases using figures, tables, etc. Detailed discussion
should be provided to interpret the results. The analytical solutions which are provided should be
used to validate your model.
Conclusions
Draw brief conclusions here.
References
Toro EF (2001) Shock-capturing methods for free-surface shallow flows, John Wiley & Sons, Chichester.
Appendix
Attach your code set up for test case 2. The appendix will not be counted into the page limit
4 Test Cases
Test 1: Still water test
The bed elevation of the frictionless 1D channel is described by

where L = 14,000 m is the length of the channel.
 Uniform computational grid: 50 cells;
 Initial conditions:

 Boundary conditions: transmissive / reflective;
 Output results (water surface and velocity profiles) at t = 5000 s.
Test 2: Tidal Wave over a Varying Bed
In the same channel as Test 1, the analytical solutions of a tidal flow are given by

q = 0 throughout the channel
 Boundary conditions: transmissive/open at both ends
 Output results (water surface and velocity profiles) at t = 5 s.
Test 4: Tidal Wave over Steps
A tidal wave flow occurs in a 1500m long frictionless channel with two vertical steps with the bed
profile defined by

An asymptotic analytical solution of the flow is provided by

 Uniform computational grid: 200 cells;
 Initial conditions:

 Boundary conditions: upstream
th ),0(
; and downstream reflective;
 Output results at t = 10,800s and t = 32,400s.