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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).
Plane one-dimensional relativistic electron oscillations
Published in E.V. Chizhonkov, Mathematical Aspects of Modelling Oscillations and Wake Waves in Plasma, 2019
The situation with the equations in the Eulerian variables (3.1) is much more complicated. They have a non-conservative form, so the use of traditional algorithms [69] presents known difficulties. In particular, as mentioned above, most of the methods are based on an exact or approximate solution of the Riemann problem (the Cauchy problem with discontinuous initial data) [69]. In the case of plasma oscillations, the very formulation of the Riemann problem is devoid of physical meaning: the discontinuous function of the electric field is an attribute of the breaking effect; at this moment, the use of the classical electrodynamics model ceases.
Numerical Methods for Inviscid Flow Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The main challenge in constructing methods for solving the Euler equations is to find ways of estimating the flux terms at the control-volume faces. Several flux splitting schemes were reviewed in the previous section and were interpreted as schemes that transport particles according to the characteristic information (van Leer, 1990). In contrast, the changes in the flux quantities at the cell interface using flux-difference splitting (FDS) have been interpreted as being caused by a series of waves. The wave interpretation is derived from the characteristic field of the Euler equations. The problem of computing the cell-face fluxes for a control volume is viewed as a series of 1-D Riemann problems along the direction normal to the control-volume faces. One way of determining these fluxes is to solve the Riemann problem using the Godunov method as outlined in Section 4.4.8 for the 1-D Burgers equation. Of course, the solution in the present case would be for a generalized problem with arbitrary initial states. The original Godunov method has been substantially improved by employing a variety of techniques to accelerate the solution of the nonlinear wave problem (Gottlieb and Groth, 1988). Because some of the details of the exact solution, obtained at considerable cost, are lost in the cell-averaged representation of the data, the solution of the full Riemann problem is usually replaced by methods referred to as approximate Riemann solvers. The Roe method (Roe, 1980) and the Osher scheme (Osher, 1984) are well known examples of these schemes. Owing to its simplicity, the Roe scheme and its many variations have evolved as one of the most popular of the FDS schemes. In the next section, the Roe scheme will be discussed as applied to the Euler equations. This technique is another way of calculating the flux values at the control-volume boundaries in the finite-volume approach.
Estimation for heterogeneous traffic using enhanced particle filters
Published in Transportmetrica A: Transport Science, 2022
A numerical scheme is used to approximate the solution to the PDE (1) based on the Godunov scheme (Godunov 1959), which requires solving a Riemann problem at every interface between each pair of consecutive and discretised road segments at each time step. On scalar models, the approach leads to the well known cell transmission model (CTM) (Daganzo 1994, 1995). The discretised creeping model reads as follows: where represents the density of class j in the ith cell at time k. The terms and are the numerical fluxes of class j via the upstream and downstream boundaries of cell i at time k.
A hybrid flux splitting method for compressible flow
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
The one-dimensional Riemann problem is commonly used to study the performance of numerical schemes. In the test, the computational domain contains two constant states and at initial time. Here, this one-dimensional test is implemented in the domain . The initial conditions are
A localised dynamic closure model for Euler turbulence
Published in International Journal of Computational Fluid Dynamics, 2018
The final Riemann solver we considered is Advection Upstream Splitting Method (AUSM) which was introduced by Liou and Steffen (1993). The AUSM has been employed in a wide range of problems because of its features like accurate capturing of shocks and contact discontinuities as well as entropy-satisfying solutions. The main idea behind the AUSM scheme is to split the inviscid flux into a convective flux part and a pressure flux part. The convective flow quantities are determined by a cell interface advection Mach number. There are many types of the AUSM solver which are developed to obtain a more accurate and robust result against shocks in all-speed regimes (Kitamura and Hashimoto 2016; Liou 2001; Matsuyama 2014; Zhang et al. 2017). Here, we use the AUSM as a low-diffusion solver to yield the interfacial numerical flux as where the mass flux is given by in which the directional convective Mach number ( in x-direction) is given as The following split formula is used for the pressure flux: There is another type of Riemann solver named The Harten–Lax–van Leer (HLL) approximate Riemann solver which assumes that the lower and upper bounds on the characteristic speeds can be used in the solution of the Riemann problem involving the right and left states (Davis 1988; Harten, Lax, and van Leer 1983; Toro 2013). Shortcomings of HLL scheme lead to a modified HLL scheme called HLLC scheme whereby the missing contact and shear waves in the Euler equations are restored (Toro 2009). Force scheme and Marquina scheme are also some examples of Riemann solver used in shock capturing schemes (Donat, Font, and Marquina 1998; Donat and Marquina 1996; Dumbser et al. 2010; San and Kara 2015; Stecca, Siviglia, and Toro 2010). However, we have not included these solvers into our study since the selected Riemann solvers may represent the lower and upper dissipation bounds within our ILES framework.