Adaptive modeling for free-surface flows

Abstract

Abstract. This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such as a channel network or a river delta, by means of a suitable "combination" of different mathematical models. In this framework a wide spectrum of space and time scales is involved due to the presence of physical phenomena of different nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solve throughout the whole domain the most complex model (with solution u fine ) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional F, we aim to approximate, within a prescribed tolerance τ , the value F(u fine ) by means of the quantity F(u adapted ), u adapted being the so-called adapted solution solving the simpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving from the simplified setting where only two hydrodynamic models, fine and coarse, are considered, we provide an efficient tool able to automatically select the regions of the domain where the coarse model rather than the fine one are to be solved, while guaranteeing |F(u fine ) − F(u adapted )| below the tolerance τ . This goal is achieved via a suitable a posteriori modeling error analysis developed in the framework of a goal-oriented theory. We extend the dual-based approach provided in [Braack and Ern, Multiscale Model Sim. 1 (2003) 221-238], for steady equations to the case of a generic time-dependent problem. Then this analysis is specialized to the case we are interested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dual problems. Finally, in the last part of the paper a widespread numerical validation is carried out.Mathematics Subject Classification. 65J15, 65M15, 65M60.

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

“…Other similar techniques belong to the class of heterogeneous multiscale methods and micro-macro decompositions. See for instance [1,22,49,51,52] and also [10,11,14,25,29,36,48,55] for related works on coupling models with different scales. In particular, let us mention the work by Degond et al [24] (and references therein, in particular [23]) where similar ideas are used to design adaptive methods in a kinetic-fluid context.…”

Model Adaptation Enriched with an Anisotropic Mesh Spacing for Nonlinear Equations: Application to Environmental and CFD Problems

“…In [3] some theoretical assumptions are supplied to justify the dropping of the two residuals ρ(u α )(e z ) and ρ * (u α )(z α , e u ) and of the remainder R in (2.6), while in [30] these hypotheses are numerically corroborated in the unsteady shallow water setting.…”

“…Concerning the model adaptation, the idea is to devise an adapted model which is derived from the monolithic model by dropping the terms which are more computationally expensive [3,4,30]: to fix ideas, in the backward-facing step configuration, we expect that the nonlinear term in the momentum equation of the Navier-Stokes system can be neglected in some parts of the domain. Which actual parts, however, cannot be determined a priori; only an a posteriori adaptation method can predict where the nonlinear term can be actually dropped.…”

Goal-oriented model adaptivity for viscous incompressible flows

“…This framework has been applied to, e.g. heterogeneous materials [18], flow problems [22], atomistic-tocontinuum problems [2], free-surface flows [20], turbulence modeling [13] and flows in arteries or rivers [9].…”