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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
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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.
Outline • Introduction • Numerics • Results • Conclusions
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
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)
Shallow Water Equations(inviscid) Conserved quantities Flux tensor g: acceleration due to gravity, h: water depth, = g h, V = u i + v j velocity.
Source Terms bed slope bed friction wind shear
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
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
MUSCL Reconstruction 1-D Cartesian,
Numerical Scheme 2) Corrector: : Riemann flux from HLL approximate Riemann solver
Surface Gradient Method Uses h rather than h for reconstruction of f Applying MUSCL to h gives,
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
AMAZON-CC Techniques are easily extended to Cartesian cut cell grids AMAZON-CC simulation of a landslide generated tsunami in a fjord
Results What about a 1-D picture v exact soln
Results Seawall modelled using bed slope (left) and solid boundary (right)
Results Fig 2 from Jingous’s paper wind induced circulation
Results Fig 4 from Jingou, overtop sea wall
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)