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Discontinuous Solutions and Methods of Their Computation
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
For the difference scheme (11.17) to make sense, we still need to define a procedure for evaluating the fluxes (Um+1/2p+1/2)2 given the quantities ump. To do that, we can exploit the solution to a special Riemann problem. This approach leads to one of the most popular and successful conservative schemes known as the Godunov scheme.
Application of Numerical Methods to Selected Model 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
While the Godunov scheme treats shocks as discontinuities, the Enquist–Osher scheme replaces shocks by what van Leer (1984) describes as overturned centered compression waves. Discontinuities are excluded owing to the smooth transitions in the phase space integrals. The treatment of sonic transitions leads to a small deviation in the slope of expansions when such points are present. Such a case is shown in Figure 4.61, where the results produced by applying the Enquist–Osher scheme to both a propagating discontinuity and expansion are depicted.
The Methods of Control of Stabilized Detonation Location in a Supersonic Gas Flow in a Plane Channel
Published in Combustion Science and Technology, 2023
Vladimir A. Levin, Tatiana A. Zhuravskaya
A set of Euler gas dynamics equations coupled with detailed chemical kinetics equations has been solved using operator-split manner. The species production rates were integrated as ordinary differential equations using the Gear method. We use the explicit first-order classical Godunov’s scheme described in detail in Godunov et al. (1976) for solving the equations of gas dynamics. The adaptive computational mesh was used for numerical simulation of studied flows with detonation waves. Walls in divergent and convergent parts of the channel were represented by line strings. The size of mesh has been chosen so that further grid refinements do not lead to a change of the flow pattern and so that the flow behind the detonation front (in particular, the flow in the induction zone) is represented correctly. Thus, we use the fine computational grid with cell size 0.02–0.03 mm in numerical calculations. According to Bocharova and Lebedev (2017), usage of the classical first-order Godunov’s scheme and a very fine computational grid makes accurate numerical modeling of the flows under consideration possible and prevents any nonphysical oscillations of solutions. The numerical modeling was performed using the software package developed by the authors. The hybrid MPI/OpenMP parallelization of computations was applied to reduce time expenditures.
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.
Numerical method to simulate detonative combustion of hydrogen-air mixture in a containment
Published in Engineering Applications of Computational Fluid Mechanics, 2019
Upwind schemes utilize characteristic information for robust and accurate calculation, and among them, Roe’s flux difference splitting family and van Leer’s flux vector splitting family are well known. Upwind schemes originated from Godunov’s scheme (Toro, 1997) in which variables are approximated as constant values in a computational cell and accurate calculation is carried out at the cell interface. Godunov’s scheme is accurate but time-consuming due to calculation at the cell interface. Therefore, approximate numerical schemes such as Roe’s and van Leer’s schemes were derived.