## Research |

**Computational Fluid Dynamics: Shallow water modeling**

As a consequence of our changing climate, large efforts have been made to understand the social risks of storm surges (hypothesized to increase in frequency in warmer climate scenarios) and sea level rise in coastal areas. Of particular interest is the role that wetlands and coastal marshes play in storm surges and flooding events.
For example, coastal marshes and swamps act as a buffer zone between the Gulf of Mexico and inhabited inland areas in Louisiana, where an estimated 60-75 % of residents live within 50 miles of the coast (1993) and where, between 1899 and 1995, over a dozen major hurricanes (class 3-5) have hit (with the two most recent hits being the category 5 hurricanes Katrina and Rita in 2005). Understanding the role of these rich biological ecosystems in our changing climate requires the development of appropriate mathematical models. In my research, I have studied analytically and numerically an effective equation often referred to in the literature as the diffusive wave approximation of the shallow water system of equations (DSW), used to simulate overland flow in wetlands and open channels. This equation is obtained by approximating the depth averaged continuity equations by empirical laws such as Manning’s or Chezy’s formulas and then combining the resulting expression with the free surface boundary condition. I have studied the properties of approximate (weak) solutions to the DSW using the continuous and discontinuous (LDG) Galerkin method, developing error estimates and implementing a 2-D code aimed at simulating water flow on experimental settings as well as real environments. I have also investigated inverse modeling approaches to estimate friction coefficients using the DSW as a physical model. |
Dam break numerical simulation |

## Related publications

A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equationsSantillana M, Dawson C. Computational Geosciences. 2010;14 (1) :31–53. AbstractIn this paper, we study the properties of approximate solutions to a doubly nonlinear and degenerate diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW), using a numerical approach based on the Galerkin finite element method. This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases the porous medium equation and the p-Laplacian. Diverse numerical schemes have been implemented to approximately solve the DSW equation and have been successfully applied as suitable models to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out in order to study the properties of approximate solutions. In this study, we propose a numerical approach as a means to understand some properties of solutions to the DSW equation and, thus, to provide conditions for which the use of the DSW equation may be inappropriate from both the physical and the mathematical points of view, within the context of shallow water modeling. For analysis purposes, we propose a numerical method based on the Galerkin method and we obtain a priori error estimates between the approximate solutions and weak solutions to the DSW equation under physically consistent assumptions. We also present some numerical experiments that provide relevant information about the accuracy of the proposed numerical method to solve the DSW equation and the applicability of the DSW equation as a model to simulate observed quantities in an experimental setting. |

On the diffusive wave approximation of the shallow water equationsAlonso R, Santillana M, Dawson C. European Journal of Applied Mathematics. 2008;19 (05) :575–606. AbstractWe discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, in the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. |

A local discontinuous Galerkin method for a doubly nonlinear diffusion equation arising in shallow water modelingSantillana M, Dawson C. Computer Methods in Applied Mechanics and Engineering. 2010;199 (23) :1424–1436. AbstractIn this paper, we study a local discontinuous Galerkin (LDG) method to approximate solutions of a doubly nonlinear diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW). This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the parabolic p-Laplacian. Continuous in time a priori error estimates are established between the approximate solutions obtained using the proposed LDG method and weak solutions to the DSW equation under physically consistent assumptions. The results of numerical experiments in 2D are presented to verify the numerical accuracy of the method, and to show the qualitative properties of water flow captured by the DSW equation, when used as a model to simulate an idealized dam break problem with vegetation. |

Gradient-based estimation of Manning’s friction coefficient from noisy dataCalo VM, Collier N, Gehre M, Jin B, Radwan H, Santillana M. Journal of Computational and Applied Mathematics. 2013;238 :1–13. AbstractWe study the numerical recovery of Manning's roughness coefficient for the diffusive wave approximation of the shallow water equation. We describe a conjugate gradient method for the numerical inversion. Numerical results for one-dimensional model are presented to illustrate the feasibility of the approach. Also we provide a proof of the differentiability of the weak form with respect to the coefficient as well as the continuity and boundedness of the linearized operator under reasonable assumptions using the maximal parabolic regularity theory. |

Convergence rates for diffusive shallow water equations (DSW) using higher order polynomialsRadwan HG, Vignal P, Collier N, Dalcin L, Santillana M, Calo VM. Journal of the Serbian Society for Computational Mechanics/Vol. 2012;6 (1) :160–168. AbstractIn this paper, we describe the diffusive shallow water equation (DSW) and discuss a numerical strategy to solve it using the generalized-?? method as a method for temporal discretization. This method provides a good norm estimate of the error and guarantees an optimal convergence rate for the spatial discretization. We also discuss the effect of higher polynomial orders on the convergence rates, focusing on the nonlinear DSW problem. Our numerical experiments show that optional convergence rates can be obtained for polynomial orders 1 through 4. |

Analysis and numerical simulation of the diffusive wave approximation of the shallow water equationsSantillana M. ProQuest; 2008. AbstractIn this dissertation, the quantitative and qualitative aspects of modeling shallow water flow driven mainly by gravitational forces and dominated by shear stress, using an effective equation often referred to in the literature as the diffusive wave approximation of the shallow water equations (DSW) are presented. These flow conditions arise for example in overland flow and water flow in vegetated areas such as wetlands. The DSWequation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the time evolution of the p-Laplacian. It has been successfully applied as a suitable model to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out addressing, for example, conditions for which weak solutions may exist, and conditions for which a numerical scheme can be successful in approximating them. This thesis represents a first step in that direction. The outline of the thesis is as follows. First, a survey of relevant results coming from the studies of doubly nonlinear diffusion equations that can be applied to the DSWequation when topographic effects are ignored, is presented. Furthermore, an original proof of existence of weak solutions using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions is shown. Some regularity results about weak solutions are presented as well. Second, a numerical approach is proposed as a means to understand some properties of solutions to the DSW equation, when topographic effects are considered, and conditions for which the continuous and discontinuous Galerkin methods will succeed in approximating these weak solutions are established. |