Godunov-type large time step scheme for shallow water equations with bed-slope source term

Abstract

This paper presents a Godunov-type large time step (LTS) solver of the non-homogeneous shallow water equations (SWEs). Source terms are decomposed into simple characteristic waves in approximate Riemann solvers (ARS) and exact Riemann solvers (ERS), and information is transferred over multiple cells per time step using the LTS method. Benchmark simulations are presented using different solution algorithms (ARS and ERS with and without entropy fixes) for two rarefactions driven by divergent flow, a pair of bores driven by opposing flows, and a dam break over a shelf-like step. In these cases, spurious flow discontinuities and oscillations can occur for Courant–Friedrichs–Lewy number (CFL) > 1 in the absence of an entropy fix. Implementation of a weak-solution LTS entropy fix improves the results, but shock shifting nevertheless occurs in certain cases. The paper also considers steady, frictionless, transcritical flow over a bed hump. In this final case, the model is run for integer CFL ranging from 1 to 10. For CFL ≤ 3, satisfactory results are obtained (without divergence and oscillation) using ARS without an entropy fix. For larger CFL, the results either diverge or exhibit convergent oscillations downstream of the hydraulic jump. Use of an entropy fix designed for implementation in an LTS scheme improves the results for CFL ≤ 5.

Introduction

Advection-dominated shallow environmental flows may exhibit high, even discontinuous, gradients in free surface elevation, flow velocity, and mass transport fluxes. Examples include the hydraulic jump in an open channel that occurs as the flow transitions from supercritical to subcritical conditions, the hydraulic bore caused by a dam break or tidal incursion into a river, and steep-fronted flows that characterize urban flood inundation. Theoretical analysis of such flows requires solving conservation laws for mass and momentum expressed as hyperbolic systems with source and sink terms representing bed gradient, frictional, and other effects (see e.g. Abbott and Minns [1] and Cunge et al. [2]). Since the 1980s, many approximate Riemann solvers (ARS) have been proposed for such hyperbolic systems following the pioneering work of Roe [3], including schemes by Osher and Solomon [4], Harten et al. [5], and Liou and Christopher [6], and summarized in books by Toro [7, 8], LeVeque [9] and Guinot [10], and reviews by Toro and García-Navarro [11], etc. The majority of these schemes are restricted by the Courant–Friedrichs–Lewy (CFL) condition (CFL = cΔt/Δx < 1 where c is the wave celerity, Δt is the time step and Δx is the spatial step), which requires a very small time step that reduces computational efficiency.

In 1982, LeVeque [12] devised a Godunov-type large time step (LTS) scheme that overcame the CFL constraint in a first-order upwind difference solver applied to simulate one-dimensional shock wave propagation. The essence of this work (thus LTS) was to track the influence of a single computational interface through multiple adjacent cells, while the conventional method forbids the interface to pass through even one cell, i.e. CFL < 1. The more cells the solver can track through, the more relaxed the CFL condition could be. Soon afterwards, Harten [13] applied the LTS scheme within a total-variation diminishing (TVD) solver of the Euler equations, and demonstrated its applicability to a shock wave in gas dynamics. In 1988, LeVeque [14] extended his original scheme to two-dimensions and second order accuracy, and presented results for obliquely propagating shock waves. Even though the scheme is highly efficient computationally, it has not been as widely implemented as classical schemes. In 2006, Murillo et al. [15] applied the LTS scheme to solving the shallow water equations and demonstrated its effectiveness in simulating various benchmark cases of steady and transient flows including dam breaks and a hydraulic jump. Meanwhile, Qian and Lee used LTS to solve the Euler equations in aerodynamics for inviscid flow past airfoils [16] and later extended the method to TVD schemes [17]. Recently, Xu et al. [18] devised a method to suppress spurious oscillations in a LTS scheme for an exact Riemann solvers (ERS) of the shallow water equations (SWEs), and accurately reproduced dam break flows for CFL up to 25. Morales-Hernández et al. [19] developed a two-dimensional LTS scheme for the SWEs that can handle wetting and drying, which was applied to a circular dam break, a dam break over an adverse slope, a tsunami interacting with a beach, and a flood event induced by dam failure in the Ebro River. Lindqvist et al. [20] compared the performance of several LTS-TVD schemes and examined numerical entropy fixes for application to generalized transonic rarefactions in nonlinear systems. The foregoing work relates only to the use of Godunov-type LTS solvers in homogenous hyperbolic partial differential equations.

In practice, there are many cases involving additional source terms, such as ones related to wind and bed friction stresses, Coriolis acceleration from the rotation of the Earth, bed porosity as a momentum source term, etc. It is well established that variable depth as a source term has implications for the correct balancing of the scheme, such as the dam break over a bed hump considered by LeVeque [21]. In general, the conservation properties of a Godunov-type shallow flow solver can be jeopardized by incorrect treatment of source terms in the shallow water equations related to flow geometry. According to García-Navarro and Vázquez-Cendón [22] problems can be caused to the conservation properties of the scheme through careless treatment of source terms related to the flow geometry. Bed and wind friction stresses are commonly introduced as source terms in the shallow water equations. In such cases, the equations can become computationally stiff, and are handled using implicit schemes [23], and fractional step and f-wave methods [24].

Herein, we extend the principle of LTS to SWEs with the bed-slope source term. Recalling the principle of LTS, we focus on incorporating source terms into a Riemann solver such that the Riemann solver can accurately track the influence of the source term (together with interfacial waves) from one interface through multiple adjacent cells. The major challenge consists of two problems. The first problem is to find or develop a proper Riemann solver for SWEs with source terms and the second is to examine whether such a Riemann solver can incorporate the LTS method. In general, Godunov-type schemes for shallow water equations with the bed-slope source term can be discretized into a series of local step Riemann problems (SRPs). Each SRP is solved using either an ARS or ERS. Murillo and García-Navarro [25] used an augmented Riemann solver to solve the SRPs as multiple states for the shallow water equations with source terms. Based on different conservation principles (mass or momentum), multiple ERSs for local SRPs were proposed for shallow flow over a non-uniform bed [26, 27]. Bernetti et al. [28] treated the channel bed as an additional equation, and the enlarged hyperbolic system thus enabled both mass and momentum conservations to be considered at the same time. Based on the enlarged equation set, Bernetti et al. derived a new ERS. However, this ERS assumes some prior knowledge of the wave structure, and so its generalization is challenging. Rosatti et al. [29] further showed that the wave generated at the bed discontinuity is in fact a contact wave, across which Riemann invariants do not hold, rendering the foregoing ERSs questionable, especially under the condition of a negative step bed. To date, previous studies have not provided a satisfactory ERS for shallow flow on the non-uniform bed. With the aim to enable LTS for SWEs with bed-slope source term, we firstly develop a new ERS, based on the governing equation set derived by Rosatti et al. [29]., but not requiring prior knowledge of wave structures, and then we derive LTS algorithms from the new ERS and an existing ARS for SWEs with the bed-slope source term so that we can evaluate their performance under multiple benchmark scenarios.

The paper is structured as follows. Section 2 presents the governing shallow water equations with a bed-slope source term in hyperbolic matrix-vector form. Section 3 describes the ARS and ERS used to evaluate the local step Riemann problems, and gives details of the LTS scheme employed to accelerate the solver, and entropy fixes used to minimize spurious oscillations and discontinuities in the solutions. Section 4 presents results obtained for several unsteady, transcritical flows over a step, and steady, transcritical flow over a bed hump. Section 5 lists the main findings.