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

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

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  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 bed friction wind shear

  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 Jingous’s 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)

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