Numerical stability of implicit four-point scheme applied to inverse linear flow routing

Szymkiewicz, Romuald

doi:10.1016/0022-1694(95)02785-8

Cited by 20 publications

(23 citation statements)

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“…Finally, the best estimate of the source signal is reverse-computed with the optimal α. Bruen and Dooge (2007) studied, by Fourier analysis, the stability of reverse routing with the de St. Venant equations, discretised via the four-point box scheme of Preissman, similar to the one used by Szymkiewicz (1993Szymkiewicz ( , 1996. They considered three inputs, a short-duration square pulse, a single sinusoid and a train of 10 such sinusoids, with 1 h and 10 h periods of both the pulse and the sinusoid.…”

Section: Reverse Routing Of Imperfect Data With Optimisationmentioning

confidence: 99%

Reverse flood routing with the inverted Muskingum storage routing scheme

Koussis

Mazi

Lykoudis

et al. 2012

Nat. Hazards Earth Syst. Sci.

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Abstract. This work treats reverse flood routing aiming at signal identification: inflows are inferred from observed outflows by orienting the Muskingum scheme against the wave propagation direction. Routing against the wave propagation is an ill-posed, inverse problem (small errors amplify, leading to large spurious responses); therefore, the reverse solution must be smoothness-constrained towards stability and uniqueness (regularised). Theoretical constrains on the coefficients of the reverse routing scheme assist in error control, but optimal grids are derived by numerical experimentation. Exact solutions of the convection-diffusion equation, for a single and a composite wave, are reverse-routed and in both instances the wave is backtracked well for a range of grid parameters. In the arduous test of a square pulse, the result is comparable to those of more complex methods. Seeding outflow data with random errors enhances instability; to cope with the spurious oscillations, the reversed solution is conditioned by smoothing via low-pass filtering or optimisation. Good-quality inflow hydrographs are recovered with either smoothing treatment, yet the computationally demanding optimisation is superior. Finally, the reverse Muskingum routing method is compared to a reverse-solution method of the St. Venant equations of flood wave motion and is found to perform equally well, at a fraction of the computing effort. This study leads us to conclude that the efficiently attained good inflow identification rests on the simplicity of the Muskingum reverse routing scheme that endows it with numerical robustness. Character of the reverse routing problemThe forward calculation of the propagation of a flood wave in an open channel, known as flood routing, is a problem of applied hydrology that has been studied extensively. The relevant methods of solution, all within the framework of one-dimensional free-surface flow, span the spectrum from numerical solutions of the hydraulic equations of Barré de Saint-Venant to storage routing models of the diffusion-wave type (Koussis, 2009), and have utility in such applications as flood warning, river training and urban storm drainage design. On occasion, however, flood related questions are posed in the reverse sense, such as, e.g., in signal identification (hydrologic forensics): "Which inflow created the outflow observed at cross-section X, or the observed flood profile along reach Y?" We may be also interested in operating a reservoir (optimal outflow control) to minimise downstream flood damage (Szöllósi -Nagy, 1987). Bruen and Dooge (2007) point out that reliable solution techniques of the latter problem would be valuable in handling of urban flash flooding.Yet reverse routing is an inverse problem and as such not well posed. A problem is well-posed when its solution exists, is unique and stable (Bronstein and Semendjajew, 1964), that is, small changes in the initial condition (forcing) cause small changes in the response. The reverse routing solution clearly exists, but must be co...

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Section: Reverse Routing Of Imperfect Data With Optimisationmentioning

confidence: 99%

Reverse flood routing with the inverted Muskingum storage routing scheme

Koussis

Mazi

Lykoudis

et al. 2012

Nat. Hazards Earth Syst. Sci.

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“…The Saint-Venant equations are applied to the hydrodynamic simulation [7,8] . For a river with floating ice covers, the one-dimensional continuity equation for flow is…”

Section: Hydrodynamic Modelmentioning

confidence: 99%

Modeling of nitrobenzene in the river with ice process in high-latitude regions

Sun

Zeng

Chen

2009

Sci. China Ser. D-Earth Sci.

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To deal with the accidental release of pollutants into frozen rivers, those rivers in high-latitude regions should be treated differently. Pollutants could be stored up in the ice and released when the ice melts in spring, perhaps resulting in secondary pollution in the river. A water quality model of nitrobenzene has been developed in this paper to assess the influence on river water quality due to the freezing process. The model is made up of three modules: thermodynamic module, hydrodynamic module and water quality module. The thermodynamic module considers the complex heat exchange processes between the water body and the atmosphere, the water body and the river bed, the water body and the ice cover, and so on. The growth of the ice cover is simulated in a simplified form whilst taking consideration of heat balance. The hydrodynamic model uses the Saint-Venant equations in non-constant flow and the influence of the ice cover is measured. The degradation, diffusion, absorption and desorption, and the influence of the freezing process are incorporated in the water quality module and the concept of Continuously Stirred Tanked Reactors (CSTRs) model is applied in the model construction. The model has been applied to supporting the management of accidental pollution in the Songhua River. The model was calibrated and validated with monitored data. Regional sensitivity analysis based on a task-based Hornberger-Spear-Young (HSY) algorithm was carried out to examine the model structure and the result showed that the model could describe the system very well. Then the conditioned model was applied to predicting the concentration of nitrobenzene in the water when the ice melted in spring. The predictive result showed that the release of pollutants in the ice could lead to an increase in the concentration of pollutants in the water, but the increase would be very small because there was only a small amount of pollutants stored in the ice. A secondary pollution could therefore be avoided. Also, with necessary modifications, the model established in this study could be applied to the water quality management in rivers that freeze in winter.nitrobenzene, river ice, water quality modeling Rivers in high-latitude regions might freeze in winter and the ice cover would change the hydrodynamics, thermodynamics and the geometry of the flow section. When ice cover is formed, the cross-section area of flow decreases and the wetted perimeter increases, which reduces the water transport capacity of the river. The freezing process also changes water quality in the river as the ice cover could also accumulate pollutants during the ice-up period and then release them when the ice melts. It was also mentioned that ice cover could affect the DO concentration in the water significantly [1] .Driven by engineering and environmental problems, a lot of research into river ice has been carried out in recent years and many studies have been conducted to simulate the river freezing process [2][3][4][5] . Most of the existing models only focus on...

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“…The non-linear open channel flow equations can be discretised in a four-point Preissman scheme (Preissman, 1961) similar to that used by Szymkiewicz (1993Szymkiewicz ( , 1996, and how the value of F at the point P is be approximated, Eqn. (6)…”

Section: Numerical Solutions By Finite Differencesmentioning

confidence: 99%

Harmonic analysis of the stability of reverse routing in channels

Bruen¹,

Dooge²

2007

Hydrol. Earth Syst. Sci.

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Normal downstream routing of a flood flow is a highly stable process for Froude numbers less than 1 and hence the results are reliable. In contrast, reverse routing in an upstream direction, which may be required for flow control, is potentially unstable. This paper reports the results of a study of the practical limits on channel lengths for reverse routing. Harmonic analysis is applied to the full non-linear solution of the St. Venant equations for three different wave patterns and two different wave periods, for a particular channel with a Froude number of 0.5. Reverse routing can be done for prismatic channels longer than 100 km. For long periods (>10 hours) the shape of the upstream hydrograph is recovered well. However, when the wave period is short (< 1hour), the high frequency components of the upstream hydrograph and, thus, its shape, are not recovered. These limits are influenced by the channel morphology and shape of the wave. Further work is needed to determine how these factors interact.

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Numerical stability of implicit four-point scheme applied to inverse linear flow routing

Cited by 20 publications

References 1 publication

Reverse flood routing with the inverted Muskingum storage routing scheme

Reverse flood routing with the inverted Muskingum storage routing scheme

Modeling of nitrobenzene in the river with ice process in high-latitude regions

Harmonic analysis of the stability of reverse routing in channels

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