Wavelet-based numerical methods adaptive modelling of shallow water flows

Mesh adaptation techniques are commonly coupled with the numerical schemes in an attempt to improve the modelling efficiency and capturing of the different physical scales which are involved in the shallow water flow problems. This work designs an adaptive technique that avails from the wavelets theory for transforming the local single resolution information into multiresolution information in which these data information became accessible. The adaptivity of wavelets was first comprehensively tested via using an arbitrary function in which the spatial resolution adaptivity was achieved from the local solution itself and it was based on a single user-prescribed parameter. Secondly, the adaptive technique was combined with two standard numerical modelling schemes (i.e. finite volume and discontinuous Galerkin schemes) to produce two wavelet-based adaptive schemes. These schemes are designed for modelling one-dimensional shallow water flows and are referred to the Haar wavelets finite volume (HWFV) and multiwavelet discontinuous Galerkin (MWDG) schemes. Both adaptive schemes were systematically tested using hydraulic test cases. The results demonstrated that the proposed adaptive technique could serve as lucid foundation on which to construct holistic and smart adaptive schemes for simulating real shallow water flow.