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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
The MacCormack method (MacCormack, 1969) is a widely used scheme for solving fluid flow equations. It is a variation of the two-step Lax–Wendroff scheme that removes the necessity of computing unknowns at the grid points j+12 and j−12. Because of this feature, the MacCormack method is particularly useful when solving nonlinear PDEs, as is shown in Section 4.4.3. When applied to the linear wave equation, this explicit, predictor–corrector method becomes
Compressible Flow
Published in M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta, Finite Difference Methods in Heat Transfer, 2017
M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta
As discussed earlier, the MacCormack method can result in accurate solutions in smooth regions but has poor stability near shocks, which become apparent in the form of spurious oscillations and overshoots. Although such instabilities can be controlled by artificial viscosity, the choice of the parameter that gives its magnitude is empirical, and its successful use is highly dependent on the user's experience. Methods developed more recently are solution-sensitive and combine different methods based on solution features (e.g., gradients). Such methods are commonly classified as flux-averaged methods or solution-averaged methods. Flux-averaged methods can still be further classified as flux-limited, flux-corrected, and self-adjusting hybrid methods. For linear systems of equations, there is no fundamental distinction between flux averaging and solution averaging, but such is not the case for nonlinear systems (Laney 1998). In this section, a flux-averaged scheme will be applied to the quasi-one-dimensional flow equations. It is the weighted average flux–total variation diminishing (WAF-TVD) scheme that was presented by Toro (1999).
Investigation of overland flow by incorporating different infiltration methods into flood routing equations
Published in Urban Water Journal, 2020
Sezar Gülbaz, Uğur Boyraz, Cevza Melek Kazezyılmaz-Alhan
In this study, conservative form of kinematic and diffusion wave equations is solved while dynamic wave is solved in non-conservative form using MacCormack finite-difference method. The MacCormack scheme is selected because of its simplicity and robustness (Kazezyılmaz-Alhan, Medina, and Rao 2005). Furthermore, MacCormack method is a widely used method in the literature including nonlinear and complicated partial differential equations. Although the discrete forms of non-conservative form of an equation may lead to greater numerical error when compared to the solution of conservative form, this may not be the case for the examples presented in this study since the examples involve only single or double rainfall events. The results of the numerical solutions are compared with the results of the analytical solutions to show this point. Furthermore, the mass balance calculation of example 1 is carried out to check the continuity error. For this purpose, the volume of the rainfall has been compared with the sum of volume of flood and infiltration under the said rainfall event (Table 2). The maximum continuity error among the examples is found as 6%. However, it should be noted that one may need to solve the conservative form of the flood wave equations in case of a real watershed. Further information on solutions of conservative form of the equations can be found in the literature (Gąsiorowski 2015, 2013; Gąsiorowski and Szymkiewicz 2007). Moreover, in case of vast floodplains, the numerical scheme may be selected among the implicit finite-difference methods to avoid any instability.
Numerical experiments on effect of river mouth morphology on tsunami behavior in rivers
Published in Coastal Engineering Journal, 2018
Yuta Mitobe, Hitoshi Tanaka, Kazuya Watanabe, Neetu Tiwari, Yasunori Watanabe
where h is water depth (m), z is bed elevation (m), U = (U, V) is velocity (m/s), g is the gravity acceleration (= 9.8 m/s2), and n is manning coefficient. The manning coefficient was set to be 0.025 in the area under water in the initial condition and 0.03 in the other area (= land area). The governing equations are solved with MacCormack method numerically.