Shallow Water Equations Model Research Articles

Application of 1D model for overland flow simulations on 2D complex domains

High computational demand limits the applications of two-dimensional (2D) shallow water equation models for high resolution overland flow simulations, while one-dimensional (1D) models can achieve higher computing efficiency. This study applied a 1D hydrodynamic model to surrogate 2D models for overland flow simulations on complex 2D domains. With one dimension reduced, the surrogate model simulation would have acceptable accuracy with much higher computing efficiency. The surrogating is fulfilled through mimicking 2D models in mesh generations, so that the 1D channel network is generated in such a way that it geometrically covers the whole domain without overlapping and intersection, and hydrologically follows the steepest slopes. Several benchmark cases on 2D domains in both laboratory and field scales with complex geometry, where no 1D models have been ever applied, are used to compare the 1D and 2D model simulations. The comparisons demonstrate that the 1D model does have potentials to efficiently simulate overland flow on 2D complex domains with accuracy comparable to 2D models.

Evaluating hydrodynamics and implications to sediment transport for tidal restoration at Swan Cove Pool, Virginia

Coastal restoration projects are significantly important in coastal ecosystems as wetland losses accelerate. This study investigates tidal hydrodynamics and the potential impacts of a waterway opening on an existing roadway to restore and revitalise salt marshes in a small estuarine system in Virginia, USA. A depth-integrated, discontinuous Galerkin shallow-water equations model (DG-SWEM) is applied for astronomic tide simulation. The model employs a high-resolution unstructured mesh with a minimum element size of less than one meter and resolves complex tidal flows in the entire barrier island system, including the existing culvert gate and canal system. Compared to the existing system, the water exchange increased dramatically (flushing time also dramatically decreased) under the opened scenarios regardless of the opening width (22.9-, 30.5-, and 38.1-m width). By increasing the opening width, peak velocity through the proposed opening decreased 30–40%, and the maximum shear stress was reduced by more than half. The high-resolution model represented complex tidal flows, including eddies, and assisted in striking a balance of water-exchange capability, opening stability, and minimising their potential erosions for the opening design. Besides, erosion and sediment transport potentials for suspended sediments were estimated using bed shear stress and a Lagrangian particle tracking module. Such proxy modelling approach allows for the impact assessment of civil engineering and ecological waterworks in complex and highly damped tidal flow areas and is readily transferrable to other like systems (e.g. causeway construction, causeway cutting, biota passageways, and inlet modification).

On the data assimilation of initial distribution for 2-dimensionalshallow-water equation model

Abstract The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system, which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.

Thetis-SWAN: A Python-interfaced wave–current interactions coupled system

Wave–Current Interactions (WCI) emerge in nearshore coastal areas, prompting the development of coupled modelling systems to simulate these phenomena. We present a new multi-scale parallelised Python-interfaced WCI coupled system adopting a component-based approach enabling model-component integration without inhibiting their respective development. The underlying principles emphasise model equitability, flexibility and language interoperability. The hybrid model comprises the spectral wave model SWAN and the 2-D shallow-water equation model, Thetis. The coupling is performed through the Basic Model Interface. The coupled WCI model is the first to employ a Python interface, while maintaining the efficiency of different lower-level compiled programming languages, Fortran for SWAN and C for Thetis. We discuss the system implementation, architecture, and underlying physics considered. The coastal waters of Duck, NC, serve as a practical demonstration in simulating WCI. We then elaborate on the rationale for the coupled system design to inform the development of coupled modelling frameworks for environmental systems.

An automatic mesh generator for coupled 1D–2D hydrodynamic models

Abstract. Two-dimensional (2D), depth-averaged shallow water equation (SWE) models are routinely used to simulate flooding in coastal areas – areas that often include vast networks of channels and flood-control topographic features and/or structures, such as barrier islands and levees. Adequately resolving these features within the confines of a 2D model can be computationally expensive, which has led to coupling 2D simulation tools to less expensive one-dimensional (1D) models. Under certain 1D–2D coupling approaches, this introduces internal constraints that must be considered in the generation of the 2D computational mesh used. In this paper, we further develop an existing automatic unstructured mesh generation tool for SWE models, ADMESH+, to sequentially (i) identify 1D constraints from the raw input data used in the mesh generation process, namely the digital elevation model (DEM) and land–water delineation data; (ii) distribute grid points along these internal constraints, according to feature curvature and user-prescribed minimum grid spacing; and (iii) integrate these internal constraints into the 2D mesh size function and mesh generation processes. The developed techniques, which include a novel approach for determining the so-called medial axis of a polygon, are described in detail and demonstrated on three test cases, including two inland watersheds with vast networks of channels and a complex estuarine system on the Texas, USA, coast.

Aceh's tsunami wave evolution and its interaction with hybrid protection structure

The 2004 Aceh tsunami tragedy was one of the most catastrophic occurrences, resulting in damage and severe casualties in multiple countries. This study proposes a hybrid coastal protection system made up of mangroves, a sea dike, a trench, or a combination of the three structures to prevent similar devastation in the future. This system is expected to reduce the tsunami wave height, thus lowering their potential damage. The tsunami wave propagation is reproduced using a nonlinear shallow water equation model. To construct a numerical scheme, a staggered grid finite volume method is implemented. This scheme is then validated using several benchmark tests. Once validated, the computational results are compared to experimental data collected at the Laboratory of Coastal Dynamics in Yogyakarta, Indonesia. Both the physical and numerical models use the downscaled Aceh tsunami waves and real bathymetry. Several scenarios of structures combination are presented with the aim to determine the most effective combination. A sensitivity analysis is also conducted to support the results.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

Development of the global tsunami forecasting system considering the dynamic interaction of tide-tsunami around the Korean Peninsula

Tsunamis are extreme natural events that pose a significant threat to coastal communities, making a comprehensive understanding of tsunami propagation mechanisms necessary for forecasting and evacuation purposes. While previous forecasting models have successfully examined several factors influencing tsunami propagation, the impact of the dynamic interaction between tides and tsunamis has yet to be investigated thoroughly. The Yellow Sea is characterized by high tidal elevations and strong tidal currents, which can accelerate the tsunami impacts on the Korean coasts. This study developed a regional tide-tsunami interaction model based on the shallow water equation model to quantitatively investigate the dynamic tide-tsunami interaction and evaluate its influence on tsunami propagation and amplification mechanism. High-resolution numerical tests were conducted for two worst-case tsunami scenarios that occurred in the Korean Peninsula, including the 2010 Chilean tsunami (far-field forecasting) and the 2011 Tohoku tsunami (near-field forecasting). The performance of the numerical model was validated utilizing the observational tide data collected along the Korean coasts. The numerical model effectively reproduces the horizontal distribution of instantaneous free surface displacement and velocity. The results reveal that the dynamic tide-tsunami interaction induced by these tsunamis generally reduces the water level and velocity in the ocean while amplifying these quantities as the tsunamis approach the coastal regions. However, due to the complex and arbitrary features of the topography, the impact of the dynamic tide and tsunami interaction on water elevation and velocity is inconsistent even compared with measurements from the adjacent tidal gauges, which suggests that the dynamic interaction can play an opposite role during the propagation and amplification process. Furthermore, the different arrival times of tsunamis along the Korean coasts are dominated by the corresponding phase of the local tidal currents that develop in each region.

A Surrogate Model for Shallow Water Equations Solvers with Deep Learning

A Surrogate Model for Shallow Water Equations Solvers with Deep Learning

Coping with geometric discontinuities in porosity-based shallow water models

The use of classic two-dimensional (2D) shallow water equations (SWE) for flooding simulation in complex urban environments is computationally expensive, due to the need of refined meshes for the representation of obstacles and building. Aiming to reduce the computational burden, a class of sub-grid SWE models, where small-scale building features are preserved on relatively coarse meshes by means of macroscale porosity parameters, has been recently introduced in the literature. Among the other porosity-based models, the single porosity (SP) model is relevant because the corresponding one-dimensional (1D) Riemann problem is the building block for the construction of many porosity-based numerical schemes. Like the Riemann problem connected to mathematical models such as the SWE with variable bed elevation and the 1D Euler equations in contracting pipes, the SP Riemann problem may exhibit multiple solutions for certain initial conditions. This ambiguity can be solved by restoring the microscale information of the 2D SWE model that is lost at the SP macroscale. In the present paper, we disambiguate the solutions' multiplicity by systematically comparing the solution of the SP Riemann problem at local porosity discontinuities with the corresponding 2D SWE numerical solutions in contracting channels. An additional result of this comparison is that the SP Riemann problem should incorporate an adequate amount of head loss when strongly supercritical flows past sudden porosity reductions occur. An approximate Riemann solver, able to pick the physically congruent solution among the alternatives and equipped with the required head loss amount, shows promising results when implemented in a 1D single porosity finite volume scheme.

Techniques to embed channels in finite element shallow water equation models

Two new and novel techniques to efficiently represent channels in large-scale finite element shallow water equation models are introduced. The first enables the discontinuous representation of bathymetric depth by allowing nodes along a discontinuity to have two land surface elevations, one representing the bottom of the discontinuity and the other the top of the discontinuity. Elemental integration proceeds using the nodal depth corresponding to whether the element is on the upper or lower side of the discontinuity and enables efficient treatment of steep sided features in a two-dimensional horizontal discretization. The second technique consolidates nodal equations at paired nodes across a channel. Doing this eliminates cross-channel variability and consequently eliminates the across-channel Courant-Friedrichs-Lewy stability constraint on the model time step. Together the two techniques allow channels of various sizes as well as other instances of steep topography to be embedded seamlessly, efficiently and in a fully coupled manner within an otherwise two-dimensional horizontal spatial discretization. This new capability is especially useful in the interface zone subject to both coastal and hydrological influences as it effectively captures bi-directional channelized flow, bi-directional flow between the channel and the floodplain, and coupled flow when the channel and floodplain are fully submerged. This paper describes the methodological development and presents a series of tests that verify the performance of this novel solution approach.

Modeling the effects of large-scale interior headland restoration on tidal hydrodynamics and salinity transport in an open coast, marine-dominant estuary

The effects of large-scale interior headland restoration on tidal hydrodynamics and salinity transport in an open coast, marine dominant estuary (Grand Bay, Alabama, U.S.A) are investigated using a two-dimensional model, the Discontinuous-Galerkin Shallow Water Equations Model (DG-SWEM). Three restoration alternatives are simulated for present-day conditions, as well as under 0.5 m of sea level rise (SLR). Model results show that the restoration alternatives have no impact on tidal range within the estuary but change maximum tidal velocities by ±5 cm/s in the present-day scenarios and by ±7 cm/s in the scenarios with 0.5 m of SLR. Differences in average salinity concentrations for simulated tropical and frontal seasons show increases and decreases on the order of 2 pss in the embayments surrounding the restoration alternatives; differences were larger (on the order of ±4 pss) for the scenarios with 0.5 m of SLR. There were minimal changes in average salinity outside of the estuary and no changes offshore. The size and position of the alternatives played a role in the salinity response as a result of changing the estuarine shoreline geometry and affecting the fetch within the bay. SLR was more impactful in increasing exposure to low salinity values (i.e., less than 5 pss) than the presence of the restoration alternatives. Overall, the modeled results indicate that these large-scale restoration actions have limited and localized impacts on the hydrodynamics and salinity patterns in this open coast estuary. The results also demonstrate the nonlinear response of salinity to SLR, with increases and decreases in the maximum, mean and minimum daily salinity concentrations from present-day conditions. This nonlinear response was a result of changes in the directions of the residual currents, which affected salinity transport.

A hybrid 1D-2D Lagrangian solver with moving coupling to simulate dam-break flow

In dam-break problems, two-dimensional (2D) models on the vertical plane appropriately capture the non-hydrostatic pressure distribution but with a high computational cost. This study presents a hybrid Lagrangian solver of one-dimensional (1D) shallow water flow and a 2D large-eddy simulation with a sub-particle scale to simulate dam-break flows over frictional beds. The proposed 1D-2D hybrid scheme is a useful trade-off between the accuracy of the 2D models on the vertical plane and the computational efficiency of 1D shallow water equations models. The configuration of sub-domains is established so that the 1D solves the regions with dominantly hydrostatic pressure, while the 2D solver accounts for the non-hydrostaticity. A moving coupling method is introduced based on two-layer shallow water equations with a uniform velocity distribution that constructs two sub-domains, including inner and outer zones in the section coupling model (SCM), or upper and lower layers within the layer coupling model (LCM). Then, the region with high vertical accelerations is substituted by the 2D model in the SCM or mixed SCM-LCM in which the overlapping zone and boundary interface move with time, based on the 1D flow feature. Also, a new method is proposed to deal with irregular boundary interfaces in the LCM while satisfying the momentum transfer. The present hybrid model is advantageous as it meets mass conservation, and there is no need for typical inflow/outflow boundary conditions at the coupling interfaces for simulating dam-break waves. Moving particle simulation (MPS) is utilized to solve the Lagrangian equations, which is advantageous in dealing with free-surface problems with large deformation. The model validation reveals that the hybrid results of water depths are in very good agreement with observed data and those obtained from the 2D model for selected dam-break flows with the ratio of initial depths downstream to upstream up to 0.4. In this respect, the average root-mean-square-error (RMSE) is computed at 8.09 mm in the SCM, 8.16 mm in the mixed SCM-LCM, and 7.48 mm in the pure 2D model, while both hybrid models save the computational time by 55% compared to the 2D model. Moreover, the mixed SCM-LCM preserves the nonlinearity of the wavefront over long times, whereas the SCM converts the wavefront into a shockwave considering the same number of fluid particles.

On Tsunami Waves Induced by Atmospheric Pressure Shock Waves After the 2022 Hunga Tonga‐Hunga Ha'apai Volcano Eruption

AbstractEmploying a linear shallow water equation (LSWE) model in the spherical coordinates, this paper investigates the tsunami waves generated by the atmospheric pressure shock waves due to the explosion of the submarine volcano Hunga Tonga–Hunga Ha'apai on 15 January 2022. Using the selected 59 atmospheric pressure records in the Pacific Ocean, an empirical atmospheric pressure model is first constructed. Applying the atmospheric pressure model and realistic bathymetric data in the LSWE model, tsunami generation and propagation are simulated in the Pacific Ocean. The numerical results show clearly the co‐existence of the leading locked waves, propagating with the speed of the atmospheric pressure waves (∼1,100 km/hr), and the trailing free waves, propagating with long gravity ocean wave celerity (∼750 km/hr). During the event, tsunamis were reported by 41 Deep‐ocean Assessment and Reporting of Tsunamis (DART) buoys in the Pacific Ocean, which require corrections because of the occurrence of atmospheric pressure waves. The numerically simulated tsunami arrival time and the amplitudes of the wave crest and trough of the leading locked waves compare reasonably well with the corrected DART measurements. The comparisons for the trailing waves are less satisfactory, since free waves could also have been generated by other tsunami generation mechanisms, which have not been considered in the present model, and by the scattering of locked waves over changing bathymetry. In this regard, the numerical results show clearly that the deep Tonga trench (∼10 km) amplifies the trailing waves in the Southeast part of the Pacific Ocean via the Proudman resonance condition.

Benchmarking a two-way coupled coastal wave–current hydrodynamics model

Wave–current interaction phenomena are often represented through coupled model frameworks in ocean modelling. However, benchmarking of these models is scarce, revealing a substantial research challenge. We seek to address this through a selection of cases for coupled wave–current interaction modelling. This comprises a series of analytical and experimental test cases spanning three diverse conditions of wave run-up, one scenario of waves opposing a current flow, and a 2-D arrangement of waves propagating over a submerged bar. We simulate these through coupling the spectral wave model, Simulating WAves Nearshore (SWAN), with the coastal hydrodynamics shallow-water equation model, Thetis, using the Basic Model Interface (BMI) structure. By comparing calibrated versus default parameter settings we identify and highlight calibration uncertainties that emerge across a range of potential applications. Calibrated model results exhibit good correlation against experimental and analytical data, alongside benchmarked wave–current model predictions, where available. Specifically, inter-model comparisons show equivalent accuracy. Finally, the coupled model we developed as part of this work showcases its ability to account for wave–current effects, in a manner extensible to other coupled processes through BMI and applicable to more complex geometries.

Water-balanced inlet and outlet boundary conditions of the lattice Boltzmann method for shallow water equations

Conservation of mass or water balance is important for the simulation of shallow water projects. The traditional treatment of discharge and water level boundary conditions ignores it in the lattice Boltzmann model for shallow water equations (LABSWE). In this paper, we introduced the water balance equation at these boundary conditions and proposed a universal treatment for the straight boundary of any direction. Six schemes on the unknown particle distribution functions (PDFs) at boundary lattices for given macroscopic quantities (h,u,v) were compared in terms of accuracy and stability, and the nonequilibrium extrapolation (NEQ) scheme and full link nonequilibrium extrapolation (Ful-NEQ) scheme were finally adopted in the proposed treatment. The numerical tests show that the proposed treatment fully satisfies the specified boundary conditions and conservation of mass. Moreover, the distribution of the velocities at the boundaries is consistent along downstream which can reduce the artificial influence length of the boundary.

New formulation of the two-dimensional steep-slope shallow water equations. Part II: Numerical modeling, validation, and application

Numerical models based on the two-dimensional (2D) shallow water equations (SWE) are commonly used for flood hazard assessment, although the basic assumption of small bottom slopes is not always strictly satisfied, such as in mountain areas. When terrain slopes are large, the steep-slope shallow water equations (SSSWE) are theoretically more suitable because the restrictive hypothesis of small bottom slopes is not introduced in deriving these equations. A new formulation of the 2D SSSWE, in which the water depth is measured in the vertical direction, and the flow velocity is assumed parallel to the bottom surface, is proposed in the companion paper (Part I). The pressure distribution on the vertical is assumed linear (yet non-hydrostatic), and the effect of flow curvature is neglected. In this paper, the new SSSWE are solved with an explicit MUSCL-type second-order accurate finite volume scheme using the centered FORCE method for flux evaluation. The SSSWE model is validated against existing experimental data of one-dimensional (1D) dam-break flows on sloping channels with fixed slopes. The numerical results of the SSSWE and SWE models are compared both in this benchmark test case and in other numerical tests, including a 1D dam-break flow moving on an adverse slope, a 2D dam-break flow spreading on an inclined plane, and a 2D dam-break flow propagating in a sloping parabolic channel. Finally, the two models are applied to the real-field test case of the Cancano dam (Adda River, northern Italy), which is characterized by very steep and irregular topography, especially in the upper portion of the valley. The results show that, on the whole, the SSSWE are more accurate in describing dam-break flows over steep topographies than the conventional SWE and predict less severe flooding with slower wave propagation. The two models are practically equivalent when bottom slopes are relatively small.

Can the 2D shallow water equations model flow intrusion into buildings during urban floods?

The multiple flow paths existing in urban environments lead to complex flow fields during urban flooding. Modelling these flow processes with three-dimensional numerical models may be scientifically sound; however, such numerical models are computationally demanding. To ascertain whether urban floods can be modelled with faster tools, this study investigated for the first time the capacity of the 2D shallow water equations (SWE) in modelling the flow patterns within and around urban blocks with openings, i.e., involving flow exchanges between the flows in the streets and within the urban blocks (e.g., through alleys leading to courtyards or through broken windows or doors). Laboratory experiments of idealized urban floods were simulated with two academic 2D SWE models, with their most notable difference being the parameterization of the eddy viscosity. Specifically, the first model had a turbulence closure based on flow depth and friction velocity while the second model had a depth-averaged k-ε turbulence closure. Thirteen urban layouts were considered with steady flow and five with unsteady flow. Both models simulated the flow depths accurately for the steady cases. The discharge distribution in the streets and the flow velocities were predicted with lower accuracy, particularly in layouts with large open spaces. The average deviation of the modelled discharge distribution at the outlets was 2.5% and 7.3% for the first and second model, respectively. For the unsteady cases, only the first model was tested. It predicted well the velocity pattern during the falling limb of a flood wave, while it did not reproduce all recirculation zones in the rising limb. The peak flow depths in the streets and the peak discharges at the outlets were predicted with an average deviation of 6.7% and 8.6%, respectively. Even though some aspects of the flow in an urban setup are 3D, the findings of this study support the modelling of such processes with 2D SWE models.

Bathymetry and friction estimation from transient velocity data for one-dimensional shallow water flows in open channels with varying width

The shallow water equations (SWE) model a variety of geophysical flows. Flows in channels with rectangular cross sections may be modeled with a simplified one-dimensional SWE with varying width. Among other model parameters, information about the bathymetry and friction coefficient is needed for the correct and precise prediction of the flow. Although synthetic values of the model parameters may suffice for testing numerical schemes, approximations of the bathymetry and other parameters may be required for specific applications. Estimations may be obtained by experimental methods, but some of those techniques may be expensive, time consuming, and not always available. In this work, we propose to solve the inverse problem to estimate the bathymetry and the Manning's friction coefficient from transient velocity data. This is done with the aid of a cost functional, which includes the SWE through Lagrange multipliers. We prove that the velocity data determine uniquely the derivative of the bathymetry in a linearized shallow water system. That is, the inverse problem is identifiable. The solution is obtained by solving the constrained optimization problem by a continuous descent method. The direct and the adjoint problems are both solved numerically using a second-order accurate Roe-type upwind scheme. Numerical tests are included to show the merits of the algorithm.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

The Shallow Water equations model is commonly used to reproduce a wide variety of environmental flows. Its applications include river hydraulics and flood forecasting.Therefore, an algorithm to solve the Riemann problem for the Shallow Water equations can be a considerable asset for the development of numerical schemes devoted to the solution of such equations. Nevertheless, for real world applications, it becomes mandatory to consider the influence of the irregular shape of the cross section of natural channels in the solution.This work presents an energy based exact Riemann solver for the Shallow Water equations for channels with an arbitrarily shaped cross section with the presence of a non flat bottom and dry states.The procedure for solving the Riemann problem involves the solution of a system of non linear algebraic equations, for which a CPU time efficient solution strategy based on nested Newton-Raphson methods is presented.The proposed solver can be used to develop Finite Volume or Discontinuous Galerkin schemes, based on the Godunov method. For a non flat bottom, the Shallow Water equations are a non conservative system, once the bottom elevation is included in the state variables. Many available schemes take advantage of this fact to use approximate path-conservative Riemann solvers. In other cases, the term related to the non flat bottom is treated as a source term. The use of this exact solver avoids the necessity of considering the bottom topography as a source term and of reverting to approximate solvers. These strategies can in facts pose numerical problems in situations such as transonic rarefactions and flows approaching abrupt changes in bottom topography.The solver is tested by reproducing the solution of Riemann problems with a Godunov finite volume scheme. An analysis of the performance of the solver shows that it requires less computational time than path-conservative schemes and that it has better stability and convergence properties, especially in situations involving resonant waves, such as transonic flows across a bottom step.