The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

The paper develops high order accurate Runge-Kutta discontinuous local evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow water equations (SWEs). Instead of using the dimensional splitting method or solving one-dimensional Riemann problem in the direction normal to the cell interface, the RKDLEG methods are built on genuinely multi-dimensional approximate local evolution operator of the locally linearized SWEs on a sphere by considering all bicharacteristic directions. Several numerical experiments are conducted to demonstrate the accuracy and performance of our RKDLEG methods, in comparison to the Runge-Kutta discontinuous Galerkin method with Godunov's flux etc.

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial differential equations and are widely used in the simulation of tsunami wave propagations. Our algorithms are tailored to take advantage of the single instruction multiple data (SIMD) architecture of graphic processing units. The time integration is accelerated by local time stepping based on a multi-rate Adams-Bashforthscheme. A total variational bounded limiter is adopted for nonlinear stability of the numerical scheme. This limiter is coupled with a mass and momentum conserving positivity preserving limiter for the special treatment of a dry or partially wet element in the triangulation. Accuracy, robustness and performance are demonstrated with the aid of test cases. Furthermore, we developed a unified multi-threading model OCCA. The kernels expressed in OCCA model can be cross-compiled with multi-threading models OpenCL, CUDA, and OpenMP. We compare the performance of the OCCA kernels when cross-compiled with these models.

We describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under a pre-balanced formulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.

The Bay of Bengal is surrounded by coastline except to the south, where there is open sea. The coastline bends most sharply along the coast of Bangladesh, and there are many small and large islands in the off shore region there. In order to incorporate the island boundaries and the curved coastline properly, in any numerical scheme it is often necessary to consider a very fine grid resolution along the coastal belts whereas this is unnecessary away from the coasts. However, a very fine resolution involves more memory and more CPU time in the numerical solution process, and invites numerical instability. On the other hand, boundary-fitted curvilinear grids in hydrodynamic models for coastal seas, bays and estuaries not only fit to the coastline but also render the finite difference schemes simpler and more accurate. In this article, the boundary-fitted curvilinear grids for the model represent the complete boundary of the area considered by four curves defined by four functions, and the four boundaries of two of the larger islands are then represented approximately by two general functions. An appropriate independent coordinate transformation maps the curvilinear physical area to a square domain, and each island boundary is transformed to a rectangle within this square domain. The vertically integrated shallow water equations are transformed to the new space domain, and solved by a regular explicit finite difference scheme. The model is applied to compute the water levels due to astronomical tides, and also the water levels due to surges associated with tropical storms that hit the coast of Bangladesh.

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

This paper considers the effect of a hard-wall beach on the downstream side of submerged parallel bars in a breakwater. In previous research, it was assumed that the beach can absorb all of the transmitted wave energy, when an optimal dimension for a submerged parallel bar is obtained and the wave amplitude is reduced as more bars are installed. However, for a hard-wall beach there are waves reflected from the beach that change the long-term wave interaction. We adopt the linear shallow water equations in Riemann invariant form and use the method of characteristics, in a procedure applicable to various formations of submerged rectangular bars. The distance from the parallel bar (or bars) to the beach determines the phase differences between right running waves in the beach basin and whether they superpose destructively or constructively before hitting the beach, to define the safest and the most dangerous cases. Our numerical calculations for one bar, two bars and for periodic rectangular bars confirm the analytical formulae obtained.

This paper discusses the development of an invariant finite difference scheme to simulate the microphase separation of copolymers in one-dimensional thin liquid films. The film phenomena are modelled using two-phase shallow water equations and the Ohta-Kawasaki potential, which governs the phase separation of the copolymer. Non-positive volume fractions and spurious oscillations are eventually eliminated, in simulating the one-dimensional phase separation lamellar pattern.

In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory (WENO) schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms. Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values. Extensive simulations are performed, which indicate that, the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy, and are more cost effective than WENO scheme with Runge-Kutta time discretization, while still maintaining nonoscillatory properties.

This paper adopts the finite-volume multi-stage (FMUSTA) scheme to the two-dimensional coupled system combining the shallow water equations and the advection-diffusion equation. For the convection part, the numerical flux is estimated by adopting the FMUSTA scheme, where high order accuracy is achieved by the data reconstruction using the monotonic upstream schemes for conservation laws method. For the diffusion part, the evaluations of first-order derivatives are solved via the method of Jacobian transformation. The hydrostatic reconstruction method is employed for treatment of source terms. The overall accuracy of resulting scheme is second-order both in time and space. In addition, the scheme is non-oscillatory and conserves the pollutant mass during the transport process. For scheme validation, six advection and diffusion transport tests are simulated. The influences of the grid spacing and limiters on the numerical performance are also discussed. Furthermore, the scheme is employed in the simulation of suspended sediment transport in natural-irregular river topography. From the satisfactory agreements between the simulated results and the field measured data, it is demonstrated that the proposed FMUSTA scheme is practically suitable for hydraulic engineering applications.

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension. The governing equations consist of two components, namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation. Based on different formulations of the morphodynamic equations, we propose a family of three finite volume methods. The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport. The obtained results for several morphodynamic problems are considered to be representative, and might be helpful for a fair rating of finite volume solution schemes, particularly in long time computations.

A modified Verlet method which involves a kind of mid-point rule is constructed and applied to the one-dimensional motion of elastic balls of finite size, falling under constant gravity in space and then under the chemical potential in the interface region of phase separation within a two-liquid film. When applied to the simulation of two balls falling under constant gravity in space, the new method is found to be computationally superior to the usual Verlet method and to Runge–Kutta methods, as it allows a larger time step for comparable accuracy. The main purpose of this paper is to develop an efficient numerical method to simulate balls in the interface region of phase separation within the two-liquid film, where the ball motion is coupled with two-phase flow. The two-phase flow in the film is described via shallow water equations, using an invariant finite difference scheme that accurately resolves the interface region. A larger time step in computing the ball motion, more comparable with the time step in computing the two-phase flow, is a significant advantage. The computational efficiency of the new method in the coupled problem is demonstrated for the case of four elastic balls in the two-liquid film.

2D shallow water equations with depth-averaged k−εmodel is considered. A meshless method based on multiquadric radial basis functions isdescribed. This methods is based on the collocation formulation and does not require thegeneration of a grid and any integral evaluation. The application of this method to a flowin horizontal channel, taken as an experimental device, is presented. The results ofcomputations are compared with experimental data and are found to be satisfactory

We present a hybrid finite-volume-particle numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the one- and two-dimensional Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation.This paper is an extension of our previous work [Chertock, Kurganov and Petrova, J. Sci. Comput. (to appear)], where the one-dimensional finite-volume-particle method has been proposed.The core idea behind the finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilizedat the right place. A special attention is given to the recovery of the point values of the numerical solution from its particle distribution. The reconstruction is obtained using a dual equation for the pollutant concentration.This results in a significantly enhanced resolution of the computed solution and also makes it much easier to extendthe finite-volume-particle method to the two-dimensional case.