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

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.