Comment on “An Efficient and Stable Hydrodynamic Model With Novel Source Term Discretization Schemes for Overland Flow and Flood Simulations” by Xilin Xia et al.

Abstract

Xia et al. (2017) proposed a novel, fully implicit method for the discretization of the bed friction terms for solving the shallow‐water equations. The friction terms contain h−7∕3 (h denotes water depth), which may be extremely large, introducing machine error when h approaches zero. To address this problem, Xia et al. (2017) introduces auxiliary variables (their equations (37) and (38)) so that h−4∕3 rather than h−7∕3 is calculated and solves a transformed equation (their equation (39)). The introduced auxiliary variables require extra storage. We implemented an analysis on the magnitude of the friction terms to find that these terms on the whole do not exceed the machine floating‐point number precision, and thus we proposed a simple‐to‐implement technique by splitting h−7∕3 into different parts of the friction terms to avoid introducing machine error. This technique does not need extra storage or to solve a transformed equation and thus is more efficient for simulations. We also showed that the surface reconstruction method proposed by Xia et al. (2017) may lead to predictions with spurious wiggles because the reconstructed Riemann states may misrepresent the water gravitational effect.

Reply to Comment by Lu et al. on “An Efficient and Stable Hydrodynamic Model With Novel Source Term Discretization Schemes for Overland Flow and Flood Simulations”

“…The alternative method by Lu et al (2017) For all of the simulations, the numerical errors predicted by SRM are smaller than those resulting from HR except for the slightly bigger SRM L 1 error for the 1 m simulation. Particularly, the SRM L 1 and L 2 errors, which reflect the overall solution accuracy, are all much smaller than the corresponding HR errors on coarser grids (10 and 100 m).…”

“…This document addresses the comments raised by Lu et al (2017). Lu et al (2017) proposed an alternative numerical treatment for implementing the fully implicit friction discretization in Xia et al (2017). The method by Lu et al (2017) is also effective, but not necessarily easier to implement or more efficient.…”

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This document addresses the comments raised by Lu et al (2017). Lu et al (2017) proposed an alternative numerical treatment for implementing the fully implicit friction discretization in Xia et al (2017).

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“…Lu et al (2017) proposed an alternative numerical treatment for implementing the fully implicit friction discretization in Xia et al (2017). The method by Lu et al (2017) is also effective, but not necessarily easier to implement or more efficient. The numerical wiggles observed by Lu et al (2017) do not affect the overall solution accuracy of the surface reconstruction method (SRM).…”

A Two‐Layer Hydrostatic‐Reconstruction Method for High‐Resolution Solving of the Two‐Layer Shallow‐Water Equations over Uneven Bed Topography

“…A face value reconstruction algorithm may affect a numerical scheme to correctly predict stationary flow at rest (e.g., C-property [1]), as well as to determine quantitative accuracy and model robustness in capturing spatially large gradients. It is found that a singlevalued definition of bed elevation at the cell-interfaces is convenient and of vital importance in devising a C-property satisfying numerical scheme [2][3][4][5]. In solving the 1LSWEs, the hydrostatic-reconstruction method (HRM) proposed by Audusse et al [3] is widely adopted; in this method, singlevalued inter-cell bed elevations are dynamically calculated based on the reconstructed inter-cell water surface elevation and water depth values.…”