Accurate Numerical Treatment of the Source Terms in the Non-linear Shallow Water Equations

1
Accurate Numerical Treatment of the Source Terms
in the Non-linear Shallow Water Equations

• J.G. Zhou, C.G. Mingham, D.M. Causon and D.M.
  Ingram
• Centre for Mathematical Modelling and Flow
  Analysis
• Department of Computing and Mathematics
• Manchester Metropolitan University
• Chester Street, Manchester M1 5GD, U.K.

2
Outline

• Introduction
• Numerics
• Results
• Conclusions

3
Introduction

• Shallow water equations can be a good model for
  many flow situations
• e.g rivers, lakes, estuaries, near shore
• Realistic problems have variable bathymetry
• In conservative Godunov schemes it is difficult
  to balance flux gradients and source terms
  containing depth leading to errors
• Surface Gradient Method (SGM) developed to
  overcome difficulties

4
Surface Gradient Method

• Simpler than competitors
• (e.g Leveque, Vazquez-Cendon)
• Centred Discretisation
• Computationally efficient
• Accurate solutions for wide range of demanding
  problems
• e.g. transcritical flow with bores over bumps
• Solves SWE without source term splitting
• Can be extended to a Cartesian cut cell framework
  (AMAZON-CC)

5
Shallow Water Equations(inviscid)
Conserved quantities
Flux tensor
g acceleration due to gravity, h water depth,
? g h, V u i v j velocity.
6
Source Terms
bed slope
wind shear
bed friction
7
Numerical Scheme

• High resolution, Godunov type
• Conservative
• Finite volume (AMAZON-CC uses Cartesian
• cut cells for automatic boundary fitted mesh)
• Interface flux via MUSCL reconstruction
• Riemann flux by HLL approx Riemann solver
• Surface Gradient Method (SGM) for accurate
• source term discretisation

8
Numerical Scheme
2-stage
1) Predictor
n time level, i,j cell index, m cell side, A
cell area, Lm side vector, F(Um) interface
flux.
discretised source term
9
MUSCL Reconstruction
1-D Cartesian,
10
Numerical Scheme
2) Corrector
Riemann flux from HLL approximate Riemann
solver
11
Surface Gradient Method
Uses h rather than h for reconstruction of f
Applying MUSCL to h gives,
12
Surface Gradient Method
Bathymetry given at cell interfaces. To get
required cell centre values assume piecewise
linear,
Bed slopes approximated by central difference,
Scheme retains conservative property
13
AMAZON-CC
Techniques are easily extended to Cartesian cut
cell grids
AMAZON-CC simulation of a landslide generated
tsunami in a fjord
14
Results
What about a 1-D picture v exact soln
15
Results
Seawall modelled using bed slope (left) and solid
boundary (right)
16
Results
Fig 2 from Jingouss paper wind induced
circulation
17
Results
Fig 4 from Jingou, overtop sea wall
18
Conclusions

• The Surface Gradient Method is a simple way to
  treat source terms within a conservative Godunov
  type scheme
• Results are good for a wide range of demanding
  test cases
• The method can be incorporated into a Cartesian
  cut cell framework
• (AMAZON-CC)