TEAM 4 Impacts and consequences of numerical methods

TEAM 4 Impacts and consequences of numerical methods (finite differences and finite volumes) on hydraulic models (shallow water equations and diffusive wave equation). Supervisors: Pr. K. Brenner and Pr. P. Gilewski 1

1. Introduction Outline 2. Methodology 3. Numerical methods 4. Hydraulic models 5. Results 6. Conclusions 2

1. Introduction o To find the fundamental theories behind the interface of the models. o To find the impacts of the numerical method which is used in different software. o To run the model to see the results of IBER, HEC-RAS and get a flood map o To understand the consequences of the numerical methods when choosing a software. 3

2. Methodology • Theoretical review of the numerical methods used • Modelling: • Shallow water equation VS Diffusive wave equation (HEC-RAS) • Finite differences (HEC-RAS) VS Finite volumes (IBER) • Implicit scheme (HEC-RAS) VS Explicit scheme (IBER) 4

2. Methodology Evaluate combinations of hydraulic model and numerical method in terms of: 1. Flooding extent (model accuracy) 2. Computational time 3. Limitations of approach (e. g. use of structured or unstructured mesh) 5

3. Numerical Methods The Shallow Water Equations - Vertically-averaged mass and momentum conservation laws - Assumes that the horizontal momentum is much greater than the vertical momentum 6

3. Numerical Methods Diffusive wave equation g – acceleration due to gravity Sf – slope of the energy grade line S – bed slope u, h – respectively velocity composants unther X and Y axix - Simplification of the shallow water equation wherein inertial forces are neglected - Considers only pressure, gravity and friction forces 7

3. Numerical Methods Finite Volumes Numerical Methods IBER (unstructured) CITYCAT MIKE 21 FM (unstructured) HEC-RAS (structured) Finite Differences Finite Elements MIKE SHE MIKE 21 (structured) ISIS MIKE 11 TELEMAC 2 D (Not tested) 8

3. Numerical Methods Finite Difference Method s Establish nodal networks s Derive finite difference approximations for the governing equation at both interior and exterior nodal points s Develop a system of simultaneous algebraic nodal equations s Solve the system of equations using numerical schemes 9

3. Numerical Methods Finite Difference Method Dx m, n+1 m-1, n m+1, n Dy m, n-1 x=m. Dx, y=n. Dy m+½, n m-½, n intermediate points 10

3. Numerical Methods Finite Difference Method • Explicit • Use all the information at the previous time step to compute the value at this time step. • Proceed point by point through the domain. • Implicit • Use information from one point at the previous time step to compute the value at all points of this time step. • Solve for all points in domain simultaneously. 11

3. Numerical Methods Finite Difference Method 12

3 -Numerical methods Finite volume method - Always follows the conservation laws (unlike the finite difference) - Differential equations are integrated over the control volume using the divergence theorem 13

3. Numerical Methods Finite Volumes (FVM) The method is based on Integral conservation law rather than Partial differences. Is enforced for small control volumes defined by the computational mesh: 14

3. Numerical Methods Explicit Scheme Implicit Scheme Known quantities of previous time steps n Unknown quantities at time step n+1 Conditionally stable (time steps values limited) Unconditionally stable Allow for long time steps values More easy to adapt to graphics processor unit (less time consuming) More computational effort in each solution step In the last years the use of finite volumes has became More complex to program more popular, solved by the explicit scheme An implicit solution is more stable than an explicit solution, and longer time step can be used. An explicit solution is simpler to program. • Low order scheme is stable but quite dissipative around discontinuities/shocks. • Higher order scheme can be more accurate but shows oscillation in the vicinity of discontinuity. 15

3. Numerical Methods Accuracy and stability (The CFL Condition) • The Courant number CFL condition for depth-averaged 2 D shallow water equations is defined as: 16

3. Numerical Methods Accuracy and stability (The CFL Condition) • Using the stability condition CFL< 1 in above equation, the following stability criterion is obtained for the optimal time step: • • Where: - c = √ g. H is the magnitude of the velocity (whose dimension is length/time). - ∆t is the time step (whose dimension is time). - ∆x is the length interval (whose dimension is length). 17

3. Numerical Methods Method Characteristics Finite Differences Finite Volumes Differential formulation of governing equations. • Do not hold discontinuities. • Do not hold unstructured mesh. • The method is not conservative. • More accurate while refining the grid. • Uses a topologically square of lines for discretization of PDEs*. • Need to convert the given mesh to structured numerical grid internally. Integral formulation of governing equations. • Hold discontinuities • Hold structured and unstructured meshes • Conservative by nature • More accurate while refining the grid. • Can handle almost any PDEs*. • More time consuming than FDM**. • The method has a clear physical interpretation. *PDEs : Partial Differential Equations **FDM : Finite Difference Method 18

3. Numerical Methods Meshing • Its type is dicted by the numerical method: • Finite volumes Unstructured mesh • Finite differences Structured mesh 19

4. Hydraulic models IBER • Iber is a numerical model for simulation of turbulent free surface unsteady flow and environmental processes in river hydraulics. • Iber is a 2 D hydraulic model • Shallow water 2 D equations • Finite Volumes • Implicit scheme 20

4. Hydraulic models IBER • the necessary inputs to perform a simulation with Iber • • Geometry Roughness Initial conditions Boundary conditions • Iber can work with triangular or quadrilateral elements, or with mixed meshes. 21

4. Hydraulic models IBER DEM Mesh Time step lenght 5 m 100 m 3600 S 4 hours 22

4. Hydraulic models IBER 23

4. Hydraulic models HEC-RAS • HEC-RAS is a numerical model that models the hydraulics of water flow through natural rivers and other channels. • HEC-RAS is a 1 D & 2 D hydraulic model • Shallow water equations or Diffusive wave equations • Finite Difference Method (roe scheme) • Implicit scheme 24

4. HEC-RAS Using Shallow Water Equations 25

4. HEC-RAS Using Diffusive Wave Equations 26

4. HEC-RAS shallow water equations & Diffusive Wave Equations 27

4. Conclusions o Numerical methods do have an impact on the models o There is not a better numerical method, it depends on the problem that needs to be solved and the available data o theoretically, explicit scheme is supposed to be more accurate but we couldn’t test it due to the lack of a reference floodplain o Explicit methods need a lot of calculation time 28

References C. B. Vreugdenhil: Numerical Methods for Shallow Water Flow, Boston: Kluwer Academic Publishers (1994) E. J. Kubatko: Development, Implementation, and Verification of hp-Discontinuous Galerkin Models for Shallow Water Hydrodynamics and Transport, Ph. D. Dissertation (2005) S. B. Pope: Turbulent Flows, Cambridge University Press (2000) J. O. Hinze: Turbulence, 2 nd ed. , New York: Mc. Graw-Hill (1975) J. T. Oden: A Short Course on Nonlinear Continuum Mechanics, Course Notes (2006) R. L. Panton: Incompressible Flow, Hoboken, NJ: Wiley (2005) 29

Thank you for your attention ! Questions 30