The spatial discretization of these equations was developed via the spectral multidomain penalty method (SMPM). This work presents a high-order element-based numerical simulation of an experimental granular avalanche, in order to assess the potential of these spectral techniques to handle geophysical conservation laws. Simulation results clearly indicate the importance of the new phase-eigenvalues and strongly support for the implementation of the complete phase-eigenvalues for the enhanced and appropriate descriptions of real two-phase landslides, avalanches, debris flows, particle-laden flows and flash-floods. Results are also compared by applying the derived phase-eigenvalues that incorporate the strong phase-interactions in the two-phase debris movements against the simple and classical solid-only, and fluid-only eigenvalues without the phase interactions. This resulted in an appropriate determination of the enhanced flow dynamical quantities, including the evolution of the solid- and fluid-phase, fluid volume fraction, and the total debris height. Enhanced simulations for two-phase mass flows down an inclined channel have been carried out by applying these exact eigenvalues together with the high-resolution TVD-NOC simulation schemes and computational codes. Associated phase-Froudenumbers and phase-wave-speeds are also defined and determined. We call these phase-eigenvalues the solid- and fluid-phase-eigenvalues. Based on this model, here, we analytically construct several novel and general exact eigenvalues for both the solid- and fluid-phases. The model, which includes several important physical aspects of the real two-phase mass flows, reveals strong interactions between the phases, is written as a set of a well-structured, highly non-linear, hyperbolic-parabolic partial differential equations. As the physical mathematical model, we consider the general two-phase debris flow equations developed by Pudasaini (2012) as a mixture of viscous fluid and solid particles. For the correct and reliable descriptions of flow behaviour and the numerical stability, we need the exact descriptions of eigenvalues of the system representing the dynamics of the mass flows. Reliable methods are required to predict accurately the flow evolution, run-out distances, inundation areas, deposition behaviour, impact forces, and the overall flow dynamics from inception in the high elevations to standstill in the run-out zones. Landslides, debris flows and tsunamis are some examples of large-scale geophysical mass transports that are widely observed in mountainous regions, valleys and the lower plains, and extremely destructive natural hazards. Effects due to a pressure dependence of the bed friction angle and lateral variations of the basal topography are therefore also numerically examined. Parameter investigations show that avalanche flows are much more sensitive against variations of the bed friction angle than that of the internal angle of friction. In the SH theory the material response is expressed by only two phenomenological parameters – the internal and the bed friction angles. Results show that the high-resolution schemes, particularly the NOC scheme with the Minmod TVD limiter or the van Leer limiter, provide excellent performances. central and upstream difference schemes, as well as high-resolution NOC (Non-Oscillatory Central Differencing) schemes, in which several second-order TVD (Total Variation Diminishing) limiters and a third-order ENO (Essentially Non-Oscillatory) cell reconstruction scheme are used. In this paper several numerical methods are applied to solve the SH equations and compared, including traditional difference schemes, e.g.
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Numerical schemes solving these free surface flows must be able to cope with smooth as well as non-smooth solutions. for a moving front or possibly formed shock waves in avalanche flows if velocities change from supercritical to subcritical e.g. Because of the hyperbolicity of the equations, successful numerical modelling is challenging, particularly when large gradients of the physical variables occur, e.g. We review the equations and point out the geometrical complexities to which these equations have been generalized. The Savage-Hutter (SH) equations of granular avalanche flows are a hyperbolic system of equations determining the distribution of depth and depth-averaged velocity components tangential to the sliding bed.
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