Luka Sopta

Extension of ENO and WENO schemes to one-dimensional sediment transport equations

Essentially nonoscillatory and weighted essentially nonoscillatory schemes are high order resolution schemes constructed for the hyperbolic conservation laws. In this paper we extend these schemes to the one-dimensional bed-load sediment transport equations. The difficulties that arise in the numerical modelling come from the fact that a nonconservative product is present in the system. Our specific numerical approximations for the nonconservative product are based on two ideas. First is to include the influence of that term in the system upwinding and the second is to define the numerical approximation in such a way that the obtained scheme solves the system for the quiescent flow case exactly. As a consequence, the resulting schemes give excellent results, as it can be seen from the numerical tests we present. On the opposite, the numerical results obtained by applying the pointwise evaluation of nonconservative product on the same tests present unacceptably large numerical errors.

Upwind Schemes with Exact Conservation Property for One-Dimensional Open Channel Flow Equations

In this paper we present an application and extension of the upwind schemes with source terms decomposed, developed by Bermudez, Vazquez, Hubbard, and Garcia-Navarro, to the one-dimensional open channel flow equations with general, i.e., nonprismatic and nonrectangular, geometries. Our specific numerical approximations for terms that appear in these equations and are related to the channels geometrical properties are quite straightforward and natural, and at the same time respect the balancing of the flux gradient and the source term. As a consequence, the resulting upwind schemes have the exact conservation property. In several test problems we illustrate the achieved improvement, particularly significant for applications to natural watercourses due to their irregular riverbed geometries.

Balanced Central NT Schemes for the Shallow Water Equations

The numerical method we consider is based on the nonstaggered central scheme proposed by Jiang, Levy, Lin, Osher, and Tadmor (SIAM J. Numer. Anal. 35, 2147(1998)) that was obtained by conversion of the standard central NT scheme to the nonstaggered mesh. The generalization we propose is connected with the numerical evaluation of the geometrical source term. The presented scheme is applied to the nonhomogeneous shallow water system. Including an appropriate numerical treatment for the source term evaluation we obtain the scheme that preserves quiescent steady-state for the shallow water equations exactly. We consider two different approaches that depend on the discretization of the riverbed bottom. The obtained schemes are well balanced and present accurate and robust results in both steady and unsteady flow simulations.

High-Order ENO and WENO Schemes with Flux Gradient and Source Term Balancing

We developed a new set of numerical schemes particularly designed for hyperbolic conservation laws with significant source term. These schemes are based on one hand on the essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) schemes (Harten, Osher, Engquist, Chakravarthy, Shu, Balsara) and on the other hand on the concept of the flux gradient and source term balancing(Bermudez, Vazquez, Hubbard, LeVeque). In this paper we present results of extended numerical testing of the original ENO and WENO schemes and new schemes on one-dimensional shallow water equations. We perform computations using 2-step and 3-step Runge-Kutta time operator approximation and from lower to higher formal order reconstruction via primitive function for the space operator. On the basis of the obtained numerical results we examine effects of the scheme order increasing in interaction with the introduced improvement by the source term decomposition. We also discuss the time evolution of the numerical error due to variable bed depth in quiescent flow, steady state flow and unsteady flow for the original and for the newly developed schemes.

Numerical Simulations of Water Wave Propagation and Flooding

In this paper we present main points in the process of application of numerical schemes for hyperbolic balance laws to water wave propagation and flooding. The appropriate mathematical models are the one-dimensional open channel flow equations and the two-dimensional shallow water equations. Therefore good simulation results can only be obtained with well-balanced numerical schemes such as the ones developed by Bermudez and Vazquez, Hubbard and Garcia-Navarro, LeVeque, etc. as well as the ones developed by the authors of this paper. We also propose a modification of the well-balanced Q-scheme for the two-dimensional shallow water equations that solves the wetting and drying problem. Finally, we present numerical results for three simulation tasks: the CADAM dam break experiment, the water wave propagation in the Toce river, and the catastrophic dam break on the Malpasset river.

Order of Accuracy of Extended Weno Schemes

Extended finite difference WENO schemes for hyperbolic balance laws with spatially varying flux and geometrical source term were developed by the authors. In these schemes high order ENO and WENO reconstruction for flux and source term characteristicwise components are used, therefore schemes give high resolution results.

Numerical Approximations of the Sediment Transport Equations

The mathematical model describing the physical phenomenon of one-dimensional bed-load sediment transport in channels and rivers consists of three equations. Two of them represent conservation laws for one-dimensional shallow water equations, and third is the conservation law governing bed-load sediment transport. Here we considerone possibletype of the sediment flux proposed by Hudson and Sweby [7]. We compare numerical results for test problems using different numerical schemes: Q-scheme, Hubbard’s scheme, ENO Roe and ENO locally Lax-Friedrichs scheme. The obtained results illustrate good properties of ENO schemes with the source term decomposition, developed by authors. We also prove that these schemes have the exact C-property when applied to the sediment transport equations.