Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
A well-balanced scheme for the numerical processing of source terms in hyperbolic equations
SIAM Journal on Numerical Analysis
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Computational Physics
Flux difference splitting and the balancing of source terms and flux gradients
Journal of Computational Physics
The surface gradient method for the treatment of source terms in the shallow-water equations
Journal of Computational Physics
A well-balanced gas-kinetic scheme for the shallow-water equations with source terms
Journal of Computational Physics
Central Schemes for Balance Laws of Relaxation Type
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Upwind Schemes with Exact Conservation Property for One-Dimensional Open Channel Flow Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
WENO schemes for balance laws with spatially varying flux
Journal of Computational Physics
Journal of Computational Physics
Fourth-order balanced source term treatment in central WENO schemes for shallow water equations
Journal of Computational Physics
ADER schemes for the shallow water equations in channel with irregular bottom elevation
Journal of Computational Physics
Computers & Mathematics with Applications
Shallow Water Flows in Channels
Journal of Scientific Computing
Binary weighted essentially non-oscillatory (BWENO) approximation
Journal of Computational and Applied Mathematics
Advances in Engineering Software
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The goal of this work is to extend finite volume WENO and central WENO schemes to the hyperbolic balance laws with geometrical source term and spatially variable flux function. In particular, we apply proposed schemes to the shallow water and the open-channel flow equations where the source term depends on the channel geometry. For obtaining stable numerical schemes that are free of spurious oscillations, it becomes crucial to use the decomposed source term evaluation, which maintains the balancing between the flux gradient and the source term. In addition, the open-channel flow equations contain spatially variable flux function. The appropriate definitions of the terms that arise in the source term decomposition, in combination with the Roe approximate Riemann solver that includes the spatial derivative of the flux function, lead to the finite volume WENO scheme that satisfies the exact conservation property - the property of preserving the quiescent flow exactly. When the central WENO schemes are applied, additional reformulations are introduced for the transition from the staggered values to the nonstaggered ones and vice versa by using the WENO reconstruction procedure. The proposed central WENO schemes also preserve the quiescent flow, but only in prismatic channels. In various test problems the obtained balanced schemes show improvements in comparison with the standard versions of the proposed type schemes, as well as with some other first- and second-order numerical schemes.