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Automatic differentiation using operator overloading (ADOO) for implicit resolution of hyperbolic single phase and two-phase flow models. (English) Zbl 1453.65049
Summary: Implicit time integration schemes are widely used in computational fluid dynamics to speed-up computations. Indeed, implicit schemes usually allow for less stringent time-step stability constraints than their explicit counterpart. The derivation of an implicit scheme is however a challenging and time-consuming task, increasing substantially with the model equations complexity since this method usually requires fairly accurate evaluation of the spatial scheme’s matrix Jacobian. This article presents a flexible method to overcome the difficulties associated to the computation of the derivatives, based on the forward mode of automatic differentiation using operator overloading (ADOO). Flexibility and simplicity of the method are illustrated through implicit resolution of various flow models of increasing complexity such as the compressible Euler equations, a two-phase flow model in full equilibrium [S. Le Martelot et al., “Towards the direct numerical simulation of nucleate boiling flows”, Int. J. Multiphase Flow 66, 62–78 (2014; doi:10.1016/j.ijmultiphaseflow.2014.06.010)] and a symmetric variant [the second author et al., J. Fluid Mech. 495, 283–321 (2003; Zbl 1080.76062)] of the two-phase flow model of M. R. Baer and J. W. Nunziato [Int. J. Multiphase Flow 12, 861–889 (1986; Zbl 0609.76114)] dealing with mixtures in total disequilibrium.
MSC:
65D25 Numerical differentiation
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
65-04 Software, source code, etc. for problems pertaining to numerical analysis
76T10 Liquid-gas two-phase flows, bubbly flows
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