Well balanced HWENO scheme for Shallow Water Equations

HWENO (Hermite Weighted Essentially Non-Oscillatory) reconstructions are introduced in literature, in the context of Euler equations for gas dynamics, to obtain high-order accuracy schemes characterized by high compactness property. Such a property allows an easier treatment of the boundary conditions and of the internal interfaces. To obtain this result, in HWENO schemes both the conservative variables and their first derivative values are evolved in time, while only the conservative variables values are evolved in the original WENO schemes. In this work, HWENO reconstructions are applied to the SWE (Shallow Water Equations) space integration to obtain a fourth-order accurate compact scheme satisfying well-balancing requirements. Time integration is performed by a Strong Stability Preserving Runge-Kutta method. Beside the classical SWE, the non-homogeneous equations (including the derivative of the source term due to bottom variations), describing the time and space evolution of the derivative of the conservative variables, are considered here. In particular, an original well-balanced treatment of the source term involved in this adjoint equations is developed and tested. Several standard one-dimensional test cases are used to verify the high-order accuracy, the well-balancing, and the good resolution properties of the model.