Explicit numerical schemes for unsteady free‐surface flows with shocks

Abstract

Three second-order accurate, explicit finite-difference schemes (MacCormack, Lambda, and Gabutti) are introduced and compared for the analysis of unsteady, free-surface flows having shocks or bores. The details of these schemes, their shock-capturing capabilities, stability restrictions, boundary conditions, and use of artificial viscosity to dampen the numerical oscillations near the shock are presented. Computed results are compared with the analytical solution to demonstrate their validity. A comparison of the computer time required for reproducing the shocks with a similar accuracy shows that the Preissmann's implicit scheme takes four to eight times the CPU time taken by these explicit schemes. Because the latter are easier to program, it makes them attractive for real-life applications. Warming, R. F., and X. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Cornput. Phys., 14, 159-179, 1974.

Solute Transport Modeling in Overland Flow Applied to Fertigation

“…It allows the simulation of hydraulic flows involving shocks traveling along a fixed grid (no shock tracking is necessary). For details about this method see Fennema and Chaudhry (1986) and García-Navarro and Savirón (1992). Being second order accurate in space and time, it offers good resolution and has great conceptual simplicity.…”

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 1: theoretical development

“…As a result, several numerical techniques have been developed in order to solve the Saint-Venant equations deterministically in their full form, without major simplifications. The most frequently used of these techniques are finite-difference methods (Abbott and Ionescu, 1967;Fread, 1973;Beam and Warming, 1976;Fennema and Chaudhry, 1986;Garcia and Kahawita, 1986;Venutelli, 2002), which solve the governing equations explicitly or implicitly along a fixed or adaptive x-t grid (Szymkiewicz, 2010). Finite-element methods are also available for solving such unsteady flow equations (Cooley and Moin, 1976;Szymkiewicz, 1991Szymkiewicz, , 1995Hicks and Steffler, 1995), though they are usually considered to be more effective for two-and three-dimensional flow problems (Szymkiewicz, 1991).…”

Ensemble modeling of stochastic unsteady open-channel flow in terms of its time-space evolutionary probability distribution: theoretical development

“…Finite-difference methods are the most frequently used numerical techniques to solve the unsteady flow equations (Abbott and Ionescu, 1967;Fread, 1973;Beam and Warming, 1976;Fennema and Chaudhry, 1986;Garcia and Kahawita, 15 1986;Venutelli, 2002). In such methods, the derivatives of the governing equations are approximated by a finite-difference formulation which is then substituted into the partial differential forms of the equations, thus transforming the governing equations into difference equations that are solved along a fixed rectangular x-t grid (Gates and AlZahrani, 1996a).…”