Semi‐Lagrangian semi‐implicit time‐splitting scheme for the shallow water equations

Time‐splitting technique applied in the context of the semi‐Lagrangian semi‐implicit method allows the use of extended time steps mainly based on physical considerations and reduces the number of numerical operations at each time step such that it is approximately proportional to the number of the points of spatial grid. To control time growth of the additional truncation errors, the standard stabilizing correction method is modified with no penalty for accuracy and efficiency of the algorithm. A linear analysis shows that constructed scheme is stable for time steps up to 2h. Numerical integrations with actual atmospheric fields of pressure and wind confirm computational efficiency, extended stability and accuracy of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

An unstructured grid,three‐dimensional model based on the shallow water equations

A semi‐implicit finite difference model based on the three‐dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi‐implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.     

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order semi‐implicit time‐integrators for a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Semi‐Lagrangian advection on a spherical geodesic grid

A simple and efficient numerical method for solving the advection equation on the spherical surface is presented. To overcome the well‐known ‘pole problem’ related to the polar singularity of spherical coordinates, the space discretization is performed on a geodesic grid derived by a uniform triangulation of the sphere; the time discretization uses a semi‐Lagrangian approach. These two choices, efficiently combined in a substepping procedure, allow us to easily determine the departure points of the characteristic lines, avoiding any computationally expensive tree‐search. Moreover, suitable interpolation procedures on such geodesic grid are presented and compared. The performance of the method in terms of accuracy and efficiency is assessed on two standard test cases: solid‐body rotation and a deformation flow. Copyright © 2007 John Wiley & Sons, Ltd.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐order compact finite difference schemes for the vorticity–divergence representation of the spherical shallow water equations

This work is devoted to the application of the super compact finite difference method (SCFDM) and the combined compact finite difference method (CCFDM) for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence, and height. The fourth‐order compact, the sixth‐order and eighth‐order SCFDM, and the sixth‐order and eighth‐order CCFDM schemes are used for the spatial differencing. To advance the solution in time, a semi‐implicit Runge–Kutta method is used. In addition, to control the nonlinear instability, an eighth‐order compact spatial filter is employed. For the numerical solution of the elliptic equations in the problem, a direct hybrid method, which consists of a high‐order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate, is utilized. The accuracy and convergence rate for all methods are verified against exact analytical solutions. Qualitative and quantitative assessments of the results for an unstable barotropic mid‐latitude zonal jet employed as an initial condition are addressed. It is revealed that the sixth‐order and eighth‐order CCFDMs and SCFDMs lead to a remarkable improvement of the solution over the fourth‐order compact method. Copyright © 2015 John Wiley & Sons, Ltd.     

An upstream flux‐splitting finite‐volume scheme for 2D shallow water equations

An upstream flux‐splitting finite‐volume (UFF) scheme is proposed for the solutions of the 2D shallow water equations. In the framework of the finite‐volume method, the artificially upstream flux vector splitting method is employed to establish the numerical flux function for the local Riemann problem. Based on this algorithm, an UFF scheme without Jacobian matrix operation is developed. The proposed scheme satisfying entropy condition is extended to be second‐order‐accurate using the MUSCL approach. The proposed UFF scheme and its second‐order extension are verified through the simulations of four shallow water problems, including the 1D idealized dam breaking, the oblique hydraulic jump, the circular dam breaking, and the dam‐break experiment with 45° bend channel. Meanwhile, the numerical performance of the UFF scheme is compared with those of three well‐known upwind schemes, namely the Osher, Roe, and HLL schemes. It is demonstrated that the proposed scheme performs remarkably well for shallow water flows. The simulated results also show that the UFF scheme has superior overall numerical performances among the schemes tested. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

Wave equation models currently discretize the generalized wave continuity equation with a three‐time‐level scheme centered at k and the momentum equation with a two‐time‐level scheme centered at k+1/2; non‐linear terms are evaluated explicitly. However in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This paper examines an implicit treatment of the non‐linear terms using an iterative time‐marching algorithm. Depending on the domain, results from one‐dimensional experiments show up to a tenfold increase in stability and temporal accuracy. The sensitivity of stability to variations in the G‐parameter (a numerical weighting parameter in the generalized wave continuity equation) was examined; results show that the greatest increase in stability occurs with G/τ=2–50. In the one‐dimensional experiments, three different types of node spacing techniques—constant, variable, and LTEA (Localized Truncation Error Analysis)—were examined; stability is positively correlated to the uniformity of the node spacing. Lastly, a scaling analysis demonstrates that the magnitudes of the non‐linear terms are positively correlated to those that most influence stability, particularly the term containing the G‐parameter. It is evident that the new algorithm improves stability and temporal accuracy in a cost‐effective manner. Copyright © 2001 John Wiley & Sons, Ltd.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

A semi‐implicit numerical method for the free‐surface Navier–Stokes equations

Semi‐implicit methods are known for being the basis of simple, efficient, accurate, and stable numerical algorithms for simulating a large variety of geophysical free‐surface flows. Geophysical flows are typically characterized by having a small vertical scale as compared with their horizontal extents. Hence, the hydrostatic approximation often applies, and the free surface can be conveniently represented by a single‐valued function of the horizontal coordinates. In the present investigation, semi‐implicit methods are extended to complex free‐surface flows that are governed by the full incompressible Navier–Stokes equations and are delimited by solid boundaries and arbitrarily shaped free‐surfaces. The primary dependent variables are the velocity components and the pressure. Finite difference equations for momentum, and a finite volume discretization for continuity, are derived in such a fashion that, after simple manipulation, the resulting pressure equation yields a well‐posed piecewise linear system from which both the pressure and the fluid volume within each computational cell are naturally derived. This system is efficiently solved by a nested Newton type iterative scheme, and the resulting fluid volumes are assured to be nonnegative and bounded from above by the available cell volumes. The time step size is not restricted by stability conditions dictated by surface wave speed, but can be freely chosen just to achieve the desired accuracy. Several examples illustrate the model applicability to a large range of complex free‐surface flows and demonstrate the effectiveness of the proposed algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

An accurate and efficient semi‐implicit method for section‐averaged free‐surface flow modelling

An accurate, efficient and robust numerical method for the solution of the section‐averaged De St. Venant equations of open channel flow is presented and discussed. The method consists in a semi‐implicit, finite‐volume discretization of the continuity equation capable to deal with arbitrary cross‐section geometry and in a semi‐implicit, finite‐difference discretization of the momentum equation. By using a proper semi‐Lagrangian discretization of the momentum equation, a highly efficient scheme that is particularly suitable for subcritical regimes is derived. Accurate solutions are obtained in all regimes, except in presence of strong unsteady shocks as in dam‐break cases. By using a suitable upwind, Eulerian discretization of the same equation, instead, a scheme capable of describing accurately also unsteady shocks can be obtained, although this scheme requires to comply with a more restrictive stability condition. The formulation of the two approaches allows a unified implementation and an easy switch between the two. The code is verified in a wide range of idealized test cases, highlighting its accuracy and efficiency characteristics, especially for long time range simulations of subcritical river flow. Finally, a model validation on field data is presented, concerning simulations of a flooding event of the Adige river. Copyright © 2009 John Wiley & Sons, Ltd.     

A CIP/multi‐moment finite volume method for shallow water equations with source terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume‐integrated average (VIA) for each mesh cell, the surface‐integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi‐Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux‐based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non‐oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

A parametric finite‐difference method for shallow sea waves

This paper presents a parametric finite‐difference scheme concerning the numerical solution of the one‐dimensional Boussinesq‐type set of equations, as they were introduced by Peregrine (J. Fluid Mech. 1967; 27 (4)) in the case of waves relatively long with small amplitudes in water of varying depth. The proposed method, which can be considered as a generalization of the Crank‐Nickolson method, aims to investigate alternative approaches in order to improve the accuracy of analogous methods known from bibliography. The resulting linear finite‐difference scheme, which is analysed for stability using the Fourier method, has been applied successfully to a problem used by Beji and Battjes (Coastal Eng. 1994; 23 : 1–16), giving numerical results which are in good agreement with the corresponding results given by MIKE 21 BW (User Guide. In: MIKE 21, Wave Modelling, User Guide. 2002; 271–392) developed by DHI Software. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi‐Lagrangian level set method for incompressible Navier–Stokes equations with free surface

In this paper, we formulate a level set method in the framework of finite elements‐semi‐Lagrangian methods to compute the solution of the incompressible Navier–Stokes equations with free surface. In our formulation, we use a quasi‐monotone semi‐Lagrangian scheme, which is both unconditionally stable and essentially non oscillatory, to compute the advective terms in the Navier–Stokes equations, the transport equation and the equation of the reinitialization stage for the level set function. The method we propose is quite robust and flexible with regard to the mesh and the geometry of the domain, as well as the magnitude of the Reynolds number. We illustrate the performance of the method in several examples, which range from a benchmark problem to test the volume conservation property of the method to the flow past a NACA0012 foil at high Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

A flux‐splitting solver for shallow water equations with source terms

This paper introduces a stable flux‐splitting solver for one‐dimensional (1D) shallow water equations. This solver is specifically designed to satisfy a strengthened consistency condition for stationary solutions that ensures the stability and accuracy of the scheme. It applies to channels with variable depth and width, including terms modelling friction at bottom and vertical walls. Some numerical tests by comparison to both analytical solutions and experimental measurements show the good performances of the scheme. Copyright © 2003 John Wiley & Sons, Ltd.     

A mass‐conservative staggered immersed boundary model for solving the shallow water equations on complex geometries

In this work, an approach is proposed for solving the 3D shallow water equations with embedded boundaries that are not aligned with the underlying horizontal Cartesian grid. A hybrid cut‐cell/ghost‐cell method is used together with a direction‐splitting implicit solver: Ghost cells are used for the momentum equations in order to prescribe the correct boundary condition at the immersed boundary, while cut cells are used in the continuity equation in order to conserve mass. The resulting scheme is robust, does not suffer any time step limitation for small cut cells, and conserves fluid mass up to machine precision. Moreover, the solver displays a second‐order spatial accuracy, both globally and locally. Comparisons with analytical solutions and reference numerical solutions on curvilinear grids confirm the quality of the method. Copyright © 2015 John Wiley & Sons, Ltd.     

A two‐dimensional numerical scheme of dry/wet fronts for the Saint‐Venant system of shallow water equations

We propose a new two‐dimensional numerical scheme to solve the Saint‐Venant system of shallow water equations in the presence of partially flooded cells. Our method is well balanced, positivity preserving, and handles dry states. The latter is ensured by using the draining time step technique in the time integration process, which guarantees non‐negative water depths. Unlike previous schemes, our technique does not generate high velocities at the dry/wet boundaries, which are responsible for small time step sizes and slow simulation runs. We prove that the new scheme preserves ‘lake at rest’ steady states and guarantees the positivity of the computed fluid depth in the partially flooded cells. We test the new scheme, along with another recent scheme from the literature, against the analytical solution for a parabolic basin and show the improved simulation performance of the new scheme for two real‐world scenarios. Copyright © 2014 John Wiley & Sons, Ltd.     

An improved technique for solving two‐dimensional shallow water problems

A numerical model for solving the 2D shallow water equations is proposed herewith. This model is based on a finite volume technique in a generalized co‐ordinate system, coupled with a semi‐implicit splitting algorithm in which a Helmholtz equation is used for the surface elevation. Several benchmark problems have proven the good accuracy of this method in complex geometries. Nevertheless, several numerical perturbations were noted in the surface elevation. After finding the origin, a new numerical technique is suggested, to avoid these perturbations. Several severe tests are proposed to validate this technique. Copyright © 1999 John Wiley & Sons, Ltd.