A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Abstract

Hermite weighted essentially non-oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high-order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193:115). For example, classical fifth-order weighted essentially non-oscillatory (WENO) reconstructions are based on a five-cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three-cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth-order accurate well-balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five-step and fourth-order accurate method. Besides the classical SWE, the non-homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well-balanced treatment of the source term involved in such equations is developed and tested. Several standard one-dimensional test cases are used to verify the high-order accuracy, the C-property and the good resolution properties of the model

A New Well-Balanced Finite Volume CWENO Scheme for Shallow Water Equations over Bottom Topography

“…Following the original works [4,5], many researchers have made extensive attempts in the development of well-balanced schemes. Representative researches mainly include: kinetic scheme [7], gas-kinetic scheme [8], central-upwind scheme [9], weighted essentially non-oscillatory (WENO) schemes [10][11][12][13][14][15][16][17][18][19], Hermite WENO scheme [20], central schemes [21,22], Runge-Kutta discontinuous Galerkin (RKDG) methods [23][24][25], ADER (Arbitrary DERivatives in time and space) schemes [26,27], spectral element method [28], Godunov-type method [29], element-free Galerkin method [30], ADER discontinuous Galerkin (ADER-DG) method [31], and so on. The research on the well-balanced schemes has become a very popular subject.…”

Well-balanced fifth-order finite difference Hermite WENO scheme for the shallow water equations

“…Xing and Shu designed the fifth-order well-balanced finite difference [27] based on a special decomposition of the source term, then, they also designed the well-balanced finite volume WENO scheme and DG method [29] for a class of hyperbolic balance laws including SWEs based on the hydrostatic reconstruction [1]. Caleffi [3] developed a well-balanced fourth-order finite volume Hermite WENO scheme for the one-dimensional SWEs on the basis of [19]. For more related well-balanced high-order methods, e.g., finite difference schemes [6,7,8,14,16,25], finite volume schemes [5,11,18,28], and DG methods [12,30,31,33,34,35].…”

“…Motivated by these good properties, we devote to designing a well-balanced HWENO scheme to solve the SWEs. One work [3] has been done in using HWENO scheme to solve the SWEs with non-flat bottom topography, in which Caleffi [3] developed a well-balanced finite volume HWENO method in one-dimensional case, but it only has the fourth-order accuracy and loses the fifth-order accuracy of the original HWENO scheme [19]. Drawback of the finite volume HWENO scheme [19] is it cannot be extended to two dimensions straightforwardly by dimension-by-dimension manner.…”

Contradiction between the C‐property and mass conservation in adaptive grid based shallow flow models: cause and solution

“…conservation property) [10] of an SWE model. Cproperty essentially requires a model to numerically preserve a quiescent steady flow and is usually employed to reflect the well-balancedness of a numerical scheme, for example, in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. The well-balancedness refers to the capability of a numerical scheme solving the SWE to balance the flux gradients and the source terms [28][29][30][31].…”