On the diffusive wave approximation of the shallow water equations

Alonso, Ricardo; Santillana, Mauricio; Dawson, Clint

doi:10.1017/s0956792508007675

Cited by 39 publications

(49 citation statements)

References 28 publications

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“…Even though some of the material of this section has been presented previously in [1], we present it here for completeness. Models for surface water flows are derived from the incompressible, 3-D Navier-Stokes equations, which consist of momentum equations for the three velocity components and a continuity equation.…”

Section: Motivationmentioning

confidence: 99%

“…The work presented in this paper is a natural continuation of the study presented in [1]. The outline of this paper is as follows.…”

Section: Introductionmentioning

confidence: 97%

“…However, relevant results and techniques coming from nonlinear diffusion equations and other doubly nonlinear diffusion equations [3,10,16,21,24,28] can be applied to the analysis of the DSW equation when topographic effects are ignored (i.e., when z = 0). An extensive survey of these results and techniques can be found in [1]. Difficulties arise when topographic effects are considered (i.e., when z = 0) since the mathematical techniques developed to study nonlinear diffusion equations, such as the PME, the p-Laplacian, and other doubly nonlinear degenerate equations, do not extend directly for this case.…”

Section: Introductionmentioning

confidence: 99%

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A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations

Santillana

Dawson

2009

Comput Geosci

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In 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.

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Section: Motivationmentioning

confidence: 99%

“…The work presented in this paper is a natural continuation of the study presented in [1]. The outline of this paper is as follows.…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations

Santillana

Dawson

2009

Comput Geosci

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“…The state of the body of water is encoded by two functions, the velocity function and the water level function. In almost all instances of flash floods, the diffusive wave approximation to the shallow water equations (DSW, [6]) can be used. The DSW equations are a simplification of the shallow water equations, used to model flows where the vertical momentum is small relative to the horizontal.…”

Section: Previous Workmentioning

confidence: 99%

“…An illustration of the results of the preprocessing scheme is shown in Figure 4 Time(Days) 6 Figure 4. Comparison between raw PIR sensor data (blue) and reconstructed PIR data (green) at the end of the preprocessing stage.…”

Section: B Preprocessing Of Temperature Measurementsmentioning

confidence: 99%

A Novel Dual Traffic/Flash Flood Monitoring System Using Passive Infrared/Ultrasonic Sensors

Mousa

Oudat

Claudel³

2015

2015 IEEE 12th International Conference on Mobile Ad Hoc and Sensor Systems

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Exa-Dune—Flexible PDE Solvers, Numerical Methods and Applications

Bastian¹,

Altenbernd²,

Dreier³

et al. 2020

Software for Exascale Computing - SPPEXA 2016-2019

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In the EXA-DUNE project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project was to add flexible, applicationoriented resilience capabilities into the framework. Continuous improvement of the underlying hardware-oriented numerical methods have included GPU-based sparse approximate inverses, matrix-free sum-factorisation for high-order discontinuous Galerkin discretisations as well as partially matrix-free preconditioners. On top of that, additional scalability is facilitated by exploiting massive coarse grained parallelism offered by multiscale and uncertainty quantification methods where we have focused on the adaptive choice of the coarse/fine scale and the overlap region as well as the combination of local reduced basis multiscale methods and the multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in a land-surface model including subsurface flow and surface runoff.

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On the diffusive wave approximation of the shallow water equations

Cited by 39 publications

References 28 publications

A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations

A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations

A Novel Dual Traffic/Flash Flood Monitoring System Using Passive Infrared/Ultrasonic Sensors

Exa-Dune—Flexible PDE Solvers, Numerical Methods and Applications

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