This study examines the use of stochastic methods to model input uncertainties and their propagation through the nonlinear shallow-water equations for flood hazard mapping. Finite volume models with a Godunov-type scheme mimic breaking waves as bores and conserve flow volume across discontinuities for accurate description of the runup process. A Galerkin projection and a spectral sampling approach based on the polynomial chaos method provide a framework to describe the uncertainties in the model results associated with the input conditions.;The polynomial chaos method expands the conserved variables into series of orthogonal polynomial chaos with deterministic coefficients that in turn define the statistical properties of the variables. The Galerkin projection makes use of the orthogonal property to provide a series of coupled, nonlinear shallow-water equations for time integration of the coefficients in stochastic space. The spectral sampling technique, on the other hand, provides a series of uncoupled equations to determine the time-evolution of the polynomial coefficients from ensemble averages of modeled events in random space. Numerical examples of long-wave transformation over a hump and runup on a plane beach and a conical island illustrate the uncertainty propagation and the stochastic properties associated with the moving waterline.;Both stochastic solutions agree well with the results from the Monte Carlo method, but at small fractions of its computing cost. The spectral sampling approach, which is highly efficient and does not require modification of the deterministic code, has demonstrated its advantages over the Galerkin projection approach. The results demonstrate its efficacy in capturing nonlinear and localized processes and producing flood hazard maps for given exceedance probabilities.
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