Hybrid Well-balanced WENO Schemes with Different Indicators for Shallow Water Equations

Abstract

In (J. Comput. Phys. 229: 8105-8129, 2010), Li and Qiu investigated the hybrid weighted essentially non-oscillatory (WENO) schemes with different indicators for Euler equations of gas dynamics. In this continuation paper, we extend the method to solve the one- and two-dimensional shallow water equations with source term due to the non-flat bottom topography, with a goal of obtaining the same advantages of the schemes for the Euler equations, such as the saving computational cost, essentially non-oscillatory property for general solution with discontinuities, and the sharp shock transition. Extensive simulations in one- and two-dimensions are provided to illustrate the behavior of this procedure.NSFC [10931004, 40906048]; Nanjing University of Information Science Technology [20090203

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

“…Using the fifth-order SWENO described in Section 2.2, we reconstruct point values at interface , +1/2 ; meantime, we calculate point values of ( ) , +1/2 and ( ) , +1/2 . We have to point out that only one order lower accuracy approximation for is needed compared with that of , as the extra Δ factor multiplied by the second term in flux F in (31). In the same way, we need the third accuracy order of approximation for , which is in the third term in (31).…”

“…We have to point out that only one order lower accuracy approximation for is needed compared with that of , as the extra Δ factor multiplied by the second term in flux F in (31). In the same way, we need the third accuracy order of approximation for , which is in the third term in (31). Indeed, for linear polynomial 2 ( ), 3 ( ), the first derivative is constant, the second derivative is zero, the nonlinear weights do not need to calculate for reconstruction of , , and only the same linear weight is used, which makes the process easier.…”

“…In recent years, WENO scheme has been generalized for hyperbolic conservation laws [22,26,27] after the first WENO scheme was originally derived in [28] for the thirdorder finite volume frame based on ENO type schemes [29,30], such as CWENO and hybrid WENO [31] arising from various applications. In 2016, a successful type of WENO [32] was proposed to approximate hyperbolic conservation laws.…”

High Order Well-balanced WENO Scheme for the Gas Dynamics Equations under Gravitational Fields

“…The main idea there is to decompose the source term into a sum of two terms first, and discretize each of them independently using a finite difference formula consistent with that of approximating the flux derivative terms in the conservation law. The same technique has been generalized to other hyperbolic balance laws in [21] and to hybrid WENO schemes in [10]. Similar idea has also been employed in designing well-balanced method for the Euler equation in [5], where the same discrete gradient operator has been used to approximate the pressure gradient and gravitational potential gradient.…”

“…Our motivation of using ±α f (U), hence the steady state solution is preserved as before, if we simply split the derivatives in the source term as: 10) and apply the same flux splitting WENO procedure to approximate them with the nonlinear coefficients a k coming from the WENO approximations to f ± (U) respectively. We have thus proved that At the end, we present how the high order well-balanced WENO method can be designed for the more general steady state (2.5).…”