Throughout history, coastal zones have been vulnerable to the dual risks of erosion and flooding. With climate change likely to exacerbate these risks in the coming decades, coasts are becoming an ever more critical location on which to apply hydro-morphodynamic models blended with advanced numerical and statistical techniques, to assess risk.

We implement a novel depth-averaged hydro-morphodynamic model using a discontinuous Galerkin based finite element discretisation within the coastal ocean model {\em Thetis}. Our model is the first with this discretisation to simulate both bedload and suspended sediment transport, and is validated for test cases in fully wet and wet-dry domains. These test cases show our model is more accurate, efficient and robust than industry-standard models. Additionally, we use our model to implement the first fully flexible and freely available adjoint hydro-morphodynamic model framework which we then successfully use for sensitivity analysis, inversion and calibration of uncertain parameters. Furthermore, we implement the first moving mesh framework with a depth-averaged hydro-morphodynamic model, and show that mesh movement can help resolve the multi-scale issues often present in hydro-morphodynamic problems, improving their accuracy and efficiency.

We present the first application of the multilevel Monte Carlo method (MLMC) and multilevel multifidelity Monte Carlo method (MLMF) to industry-standard hydro-morphodynamic models as a tool to quantify uncertainty in erosion and flood risk. We use these methods to estimate expected values and cumulative distributions of variables which are of interest to decision makers. MLMC, and more notably MLMF, significantly reduce computational cost compared to the standard Monte Carlo method whilst retaining the same level of accuracy, enabling in-depth statistical analysis of complex test cases that was previously unfeasible.

The comprehensive toolkit of techniques we develop provides a crucial foundation for researchers and stakeholders seeking to assess and mitigate coastal risks in an accurate and efficient manner.