Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography

Navas-Montilla, Adrián; Murillo, J.

doi:10.1016/j.jcp.2017.03.057

Cited by 11 publications

(28 citation statements)

References 41 publications

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“…f. The more complicated bottom profile (see line 2 in Figure 5) helps identifying the best advantages of numerical schemes [22]. We have exact stationary solutions on a stepped bottom for subcritical (Froude number Fr < 1) and supercritical regimes (Fr > 1) [24,39].…”

Section: The Set Of Test Problems Verifying the Numerical Modelsmentioning

confidence: 99%

“…As a rule, numerical schemes of the second-order accuracy give satisfactory results and allow to correctly solve a wide range of tasks for diverse applications [17]. Special attention should be focused on the numerical way of a source term setting, since in the case of discontinuous topography, it may induce an error at the shock wave front [22].…”

Section: Introductionmentioning

confidence: 99%

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A Numerical Simulation of the Shallow Water Flow on a Complex Topography

Хоперсков¹,

Khrapov²

2018

Numerical Simulations in Engineering and Science

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In current chapter, we have thoroughly described a numerical integration scheme of nonstationary 2D equations of shallow water. The scheme combines the smoothed particle hydrodynamics (SPH) and the total variation diminishing (TVD) methods, which are sequentially used at various steps of the combined SPH-TVD algorithm. The method is conservative and well balanced. It provides stable through calculations in presence of nonstationary "water-dry bottom" boundaries on complex irregular bottom topography including the transition of such a boundary between wet and dry bottom through the computational boundary. Multifarious tests demonstrate the effectiveness of the combined SPH-TVD scheme application for a solution of diverse problems of the engineering hydrology.

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Section: The Set Of Test Problems Verifying the Numerical Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

Хоперсков¹,

Khrapov²

2018

Numerical Simulations in Engineering and Science

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“…Up to now, most of these studies have been carried out in the framework of Euler equations, but the growing needs for the computation of complex geophysical flows motivate their application to the SWE. For such system of equations, numerical shockwave anomalies appear in the resolution of hydraulic jumps [16]. Due to the non-linearity and non-monotonicity of the Hugoniot locus [16], the slowly-moving shock anomaly is always present in the finite volume (FV) resolution of hydraulic jumps.…”

Section: Introductionmentioning

confidence: 99%

“…For such system of equations, numerical shockwave anomalies appear in the resolution of hydraulic jumps [16]. Due to the non-linearity and non-monotonicity of the Hugoniot locus [16], the slowly-moving shock anomaly is always present in the finite volume (FV) resolution of hydraulic jumps.…”

Section: Introductionmentioning

confidence: 99%

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Improved Riemann solvers for an accurate resolution of 1D and 2D shock profiles with application to hydraulic jumps

Navas-Montilla

Murillo

2019

Journal of Computational Physics

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From the early stages of CFD, the computation of shocks using Finite Volume methods has been a very challenging task as they often prompt the generation of numerical anomalies. Such anomalies lead to an incorrect and unstable representation of the discrete shock profile that may eventually ruin the whole solution. The two most widespread anomalies are the slowly-moving shock anomaly and the carbuncle, which are deeply addressed in the literature in the framework of homogeneous problems, such as Euler equations. In this work, the presence of the aforementioned anomalies is studied in the framework of the 1D and 2D SWE and novel solvers that effectively reduce both anomalies, even in cases where source terms dominate the solution, are presented. Such solvers are based on the augmented Roe (ARoe) family of Riemann solvers, which account for the source term as an extra wave in the eigenstructure of the system. The novel method proposed here is based on the ARoe solver in combination with: (a) an improved flux extrapolation method based on a previous work, which circumvents the slowly-moving shock anomaly and (b) a contact wave smearing technique that avoids the carbuncle. The resulting method is able to eliminate the slowly-moving shock anomaly for 1D steady cases with source term. When dealing with 2D cases, the novel method proves to handle complex shock structures composed of hydraulic jumps over irregular bathymetries, avoiding the presence of the aforementioned anomalies.

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Implementation of exactly well‐balanced numerical schemes in the event of shockwaves: A 1D approach for the shallow water equations

Akbari

Pirzadeh

2022

Numerical Methods in Fluids

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This article presents a numerical technique that ensures the exact solution of stationary shockwaves, known as the hydraulic jump, for the one‐dimensional shallow water equations with the geometric source term. The mathematical description of the hydraulic jump is given by the Rankine–Hugoniot condition, at the interface of a control volume, where one variable in the system, the discharge, is supposed to be constant while the remaining variables are discontinuous. However, at the discrete level, the solution could be, for instance, a 3‐state jump, consisting of extra (intermediate) states resulting in the so‐called numerical anomaly. The anomaly is due to the nonlinearity of the Hugoniot locus and the fact that, at these points, a naive formulation is unlikely to avoid the inaccurate representation. The concept used here is simple, limit all discontinuous variables, that is, the surface flow data inside the cell that contains a shock with the data of the nearby cells where the shock is traveling to, allowing the solution to be linear. When used with a number of well‐known numerical solvers, namely Lax–Friedrichs, HLL, and Roe schemes, this approach is confirmed to accurately capture the moving and stationary shockwaves in the trans‐critical flow over nonflat beds.

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Overcoming numerical shockwave anomalies using energy balanced numerical schemes. Application to the Shallow Water Equations with discontinuous topography

Cited by 11 publications

References 41 publications

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

A Numerical Simulation of the Shallow Water Flow on a Complex Topography

Improved Riemann solvers for an accurate resolution of 1D and 2D shock profiles with application to hydraulic jumps

Implementation of exactly well‐balanced numerical schemes in the event of shockwaves: A 1D approach for the shallow water equations

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