Degenerate parabolic models for sand slides

Abstract The morphodynamic evolution of the shape of dunes and piles of granular material is largely dictated by avalanching phenomena, acting when the local slope gets steeper than a critical repose angle. A class of degenerate parabolic models are proposed closing a mass balance equation with several viscoplastic constitutive laws to describe the motion of the sliding layer. Comparison among them is carried out by means of computational simulations putting in evidence the features that depend on the closure constitutive assumption and the robust aspects of the models. The versatility of the model is shown applying it to the movement of sand in presence of walls, open ends, columns, doors, and in complicated geometries

Wind-blown particulate transport: A review of computational fluid dynamics models

The transport of particulate by wind constitutes a relevant phenomenon in environmental sciences and civil engineering, because erosion, transport and deposition of particulate can cause serious problems to human infrastructures. From a mathematical point of view, modeling procedure for this phenomenon requires handling the interaction between different constituents, the transfer of a constituent from the air to the ground and viceversa, and consequently the ground-surface interaction and evolution. Several approaches have been proposed in the literature, according to the specific particulate or application. We here review these contributions focusing in particular on the behavior of sand and snow, which almost share the same mathematical modeling issues, and point out existing links and analogies with wind driven rain. The final aim is then to classify and analyze the different mathematical and computational models in order to facilitate a comparison among them. A first classification of the proposed models can be done distinguishing whether the dispersed phase is treated using a continuous or a particle-based approach, a second one on the basis of the type of equations solved to obtain particulate density and velocity, a third one on the basis of the interaction model between the suspended particles and the transporting fluid

Splitting schemes for coupled differential equations: Block Schur-based approaches & Partial Jacobi approximation

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. In a broad sense, one can categorise the numerical solution strategies for coupled problems into two classes: monolithic approaches and sequential (also known as split, decoupled, partitioned or segregated) approaches. The monolithic approaches treat the entire problem as one, whereas the sequential approaches are iterative decoupling techniques where the different sub-problems are treated separately. Although the monolithic approaches often offer the most robust solution strategies, they tend to require ad-hoc preconditioners and numerical implementations. Sequential methods, on the other hand, offer the possibility to add and remove equations from the model flexibly and rely on existing black-box solvers for each specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, making the sequential approaches an even more viable alternative. The cost of running inner iterations to recover the multi-physics coupling could, however, easily become prohibitive. Moreover, the sequential approaches might not converge at all. In this work, we present a general formulation of splitting schemes for continuous operators with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator, we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought of as a preconditioner and constructed based on an approximate Schur complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. The schemes presented are not explicitly dependent on any particular discretisation methodologies. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to investigate the practical implications of the theoretical results