Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

In this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples.

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problem between its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, and tends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction with one's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension, oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions.

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 utilized at 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 extend the finite-volume-particle method to the two-dimensional case.

This paper concerns numerical methods for two-phase flows. The governing equations are the compressible 2-velocity, 2-pressure flow model. Pressure and velocity relaxation are included as source terms. Results obtained by a Godunov-type central scheme and a Roe-type upwind scheme are presented. Issues of preservation of pressure equilibrium, and positivity of the partial densities are addressed.

The Darcy–Boussinesq equations at infinite Darcy–Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number $Ra$. High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu, from onset at $Ra \,{=}\, 4\pi^2$ up to $Ra\,{=}\,10^4$. Over an intermediate range of increasing Rayleigh numbers, the simulations display the ‘classical’ heat transport $\hbox{\it Nu} \,{\sim}\, Ra$ scaling. As the Rayleigh number is increased beyond $Ra \,{=}\, 1255$, we observe a sharp crossover to a form fitted by $\hbox{\it Nu} \,{\approx}\, 0.0174 \times Ra^{0.9}$ over nearly a decade up to the highest $Ra$. New rigorous upper bounds on the high-Rayleigh-number heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: $\hbox{\it Nu} \,{\le}\, 0.0297 \times Ra$. The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of $Ra$, including hysteretic effects observed in the simulations as the Rayleigh number is decreased from $1255$ back down to onset.

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes, proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently applied in the context of the new schemes. The method is tested on the one-dimensional Euler equations, subject to different initial data, and the results are compared to the numerical solutions, computed by other second-order central schemes. The numerical experiments clearly illustrate the advantages of the proposed technique.