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Shallow Water Flow Modeling
Published in Saeid Eslamian, Faezeh Eslamian, Flood Handbook, 2022
Franziska Tügel, Ilhan Özgen-Xian, Franz Simons, Aziz Hassan, Reinhard Hinkelmann
The preferred method for such robust shallow water flow models is a second-order accurate finite-volume method (FV) (see, e.g., the models reported in Simons et al., 2014; Liang et al., 2016; Ginting and Mundani, 2019; and many others. The FV method solves the integral form of the PDEs in each computational cell. A flux function is defined across each cell interface, which drives the change of the solution in time. Hence, the solution is locally conserved in each cell, and since the integral form of the PDEs is used, discontinuous solutions are allowed (Toro, 2001; Hinkelmann, 2005). In the FV method, defining the flux function is non-trivial. There are many methods in the literature to calculate the flux function (LeVeque, 2002). A particularly established way is to solve a local Riemann problem across the cell edge. The Riemann problem is defined as an initial-value problem with piecewise constant values separated by a single discontinuity. Solving this problem exactly or approximately to define the flux function across the cell interface gives the so-called Godunov-type method (Godunov, 1959). The order of accuracy of the FV method is usually increased by means of a total variation diminishing (TVD) method, which avoids spurious oscillations. A common scheme applied in the literature is the TVD-MUSCL scheme (van Leer, 1979).
Implicit and explicit TVD methods for discontinuous open channel flows
Published in R. A. Falconer, K. Shiono, R. G. S. Matthew, Hydraulic and Environmental Modelling: Estuarine and River Waters, 2019
Finally Figure 5 shows the convergence rate for both schemes. The ratio of the right hand side of Eq. 20 to the same quantity at the first time step is plotted in logarithmic scale for every time step. The explicit algorithm converges in an oscillating manner and much slower than the implicit one (notice the different scaling in the number of time steps of Figure 6a and b). Although the computational cost per time step is much cheaper for the former, TVD schemes always require the evaluation of complicated nonlinear expressions every time step and this is very expensive. Actually for this test case the overall computational cost of the implicit scheme was about one fifth that of the explicit one.
Simulation of the effects of dilution gas for the formation of CJ plane during the oblique detonation
Published in Numerical Heat Transfer, Part A: Applications, 2023
MUSCL (Monotonic Up-Stream-Centered Schemes) [47] schemes are also known as Godunov-type schemes and can be described as a projection-evolution scheme consisting of a numerical (projection) and a physical (evolution) stage. It is necessary to evaluate the flow variables at the cell interface. The MUSCL scheme is a well-known high-order and TVD approach. When a solution contains discontinuous surfaces or large gradient differences, the numerical dissipation of MUSCL is useful. Van Leer first proposed the MUSCL method in 1979. It has more numerical dissipation and smears out flow structures partially. Solving multi-component flows between interfaces may be fatal due to this error.
Vorticity Confinement Applied to Accurate Prediction of Convection of Wing Tip Vortices and Induced Drag
Published in International Journal of Computational Fluid Dynamics, 2021
Alex Povitsky, Kristopher C. Pierson
TVD schemes work by reducing second-order schemes to first-order schemes in the presence of oscillations. These oscillations cause values of the limiter function, , to tend to zero in Equation (9). The second-order scheme was created through linear interpolation of the flux terms shown in Equation (9). For a pure second-order scheme, , while for a first order scheme. where and f is TVD limiter.
Combined Vorticity Confinement and TVD Approaches for Accurate Vortex Modelling
Published in International Journal of Computational Fluid Dynamics, 2020
Alex Povitsky, Kristopher C. Pierson
TVD schemes operate by reducing second-order schemes to first-order schemes in the presence of oscillations. These oscillations cause values of the limiter function,, to tend to zero in Equation (8). When the vortex is over-confined, the local maximum of velocity becomes larger than its physical value. This leads to formation of a non-physical vortex flowfield as time increases. Switch to first-order scheme should reduce the local maximum.