## Free-surface hydrodynamics (Advanced Numerical Methods for)

**Prof. Vincenzo Casulli, Prof. Michael Dumbser**

Timetable 2012-2013

January -February 2013 |
Hours |
Room |

Jan. 21 - Feb. 1 (Casulli) | 9:00-13:00 | D1 |

Jan. 21 - Feb. 1 (excluded Sat. & Sunday) (Dumbser) | 15:00-17:00 | PC-OVEST |

Final assessment: to be decided | to be decided |

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**Programme**

** **** **

**1. Mathematical Models **

1.1 The Navier-Stokes Equations

1.2 A Three-Dimensional Hydrostatic Model

1.3 The Vertically Averaged Model (2Dxy)

1.4 The Laterally Averaged Model (2Dxz)

1.5 The Open Channel Equations (1D)

**2. Eulerian-Lagrangian Methods **

2.1 Advection-Diffusion Equations

2.2 Explicit Central Difference Method

2.3 Implicit Central Difference Method

2.4 Explicit Upwind Method

2.5 Implicit Upwind Method

2.6 Eulerian-Lagrangian Methods

2.7 Semi-Implicit Methods

2.8 The Conjugate Gradient Method

**3. Numerical Methods for the 1D Model**

3.1 Characteristic Analysis

3.2 A Semi-Implicit Finite Difference Method

3.3 An Equation for the Free Surface

**4. Numerical Methods for the 2Dxz Model **

4.1 A Semi-Implicit Finite Difference Method

4.2 Derivation of the Free Surface Equation

4.3 A Particular Case: The 1D Model

**5. Numerical Methods for the 2Dxy Model **

5.1 Characteristic Analysis

5.2 A Semi-Implicit Finite Difference Method

5.3 An Equation for the Free Surface

5.4 A Particular Case: The 1D Model

**6. Numerical Methods for the 3D Model **

6.1 Extensions of the 2D Methods

6.2 A Semi-Implicit Finite Difference Method

6.3 An Equation for the Free Surface

6.4 A Particular Case: The 2Dxy Model

6.5 A Particular Case: The 2Dxz Model

** **

**7. Further Extensions **

7.1 The θ-Method

7.1.1 A Semi-Implicit Finite Difference Method

7.1.2 Solution Algorithm

7.2 Numerical Modelling on Unstructured Grids

7.2.1 Orthogonal Unstructured Grid

7.2.2 A Semi-Implicit Discretization

7.2.3 Solution Algorithm

7.3 Semi-Implicit Subgrid Modelling

7.3.1 Semi-Implicit Finite Volume Discretization

7.3.2 Solution Algorithm

**Laboratory Exercise**

With MATLAB, the participants will implement a semi-implicit finite difference scheme using an Eulerian-Lagrangian approach for the convection terms for the open channel equations (1D) as well as for the vertically averaged model (2Dxy).

A new rigorous treatment for wetting and drying, which is a very frequent problem in civil and environmental engineering, is also part of the laboratory exercises.

### Assessment

The test consists of a presentation on a topic chosen by the student among the subjects of the course. Students who have passed the final assessment will be acknowledged 5 ECTS.

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References

Lecture notes from the instructors.

IMPORTANT NOTICE: all external students, scholars and professionals interested in the course can participate upon payment of the fees as specified in the file downloadable from the box below.