Title: Accurate Numerical Treatment of the Source Terms in the Non-linear Shallow Water Equations
1Accurate 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.
2Outline
- Introduction
- Numerics
- Results
- Conclusions
3Introduction
- 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
4Surface 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)
5Shallow Water Equations(inviscid)
Conserved quantities
Flux tensor
g acceleration due to gravity, h water depth,
? g h, V u i v j velocity.
6Source Terms
bed slope
wind shear
bed friction
7Numerical 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
8Numerical 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
9MUSCL Reconstruction
1-D Cartesian,
10Numerical Scheme
2) Corrector
Riemann flux from HLL approximate Riemann
solver
11Surface Gradient Method
Uses h rather than h for reconstruction of f
Applying MUSCL to h gives,
12Surface 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
13AMAZON-CC
Techniques are easily extended to Cartesian cut
cell grids
AMAZON-CC simulation of a landslide generated
tsunami in a fjord
14Results
What about a 1-D picture v exact soln
15Results
Seawall modelled using bed slope (left) and solid
boundary (right)
16Results
Fig 2 from Jingouss paper wind induced
circulation
17Results
Fig 4 from Jingou, overtop sea wall
18Conclusions
- 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)