Second order discontinuous Galerkin scheme for compound natural channels with movable bed. Applications for the computation of rating curves

A new higher order 1D numerical scheme for the propagation of flood waves in compound channels with a movable bed is presented. The model equations are solved by means of an ADER Discontinuous Galerkin explicit scheme which can, in principle, reach any order of space–time accuracy. The higher order nature of the scheme allows the numerical coupling between flux and source terms appearing in the governing equa- tions and, importantly, to handle moderately stiff and stiff source terms. Stiff source terms arise in the case of abrupt changes of river geometry such as in the case of hydraulic structures like bridges and weirs. Hydraulic interpretation of these conditions with 1D numerical modelling requires particular attention; for instance, a 1st order scheme might either lead to inaccurate solutions or impossibility to simulate these complex con- ditions. Validation is carried out with several test cases with the aim to check the scheme capability to deal with abrupt geometric changes and to capture the direction and celerity of propagation of bed and water surface disturbances. Validation is done also in a real case by using stage-discharge field measurements in the Ombrone river (Tuscany). The proposed scheme is further employed for the computation of flow rating curves in cross-sections just upstream of an abrupt narrowing, considering both fixed and movable bed con- ditions and different ratios of contraction for cross-section width. This problem is of particular relevance as, in common engineering practice, rating curves are derived from stage-measuring gauges installed on bridges with flow conditions that are likely to be influenced by local width narrowing. Results show that a higher order scheme is needed in order to deal with stiff source terms and reproduce realistic flow rating curves, unless a strong refinement of the computational grid is performed. This capability appears to be crucial for the computation of rating curves on coarse grids as it allows the modeling of abrupt contractions and jumps in bed bottom elevations, which often occur near cross-sections where stage measuring gauges are installed.