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Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.

We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star operator. Our discretization framework conserves this geometrical structure and provides for all DFs proper FE spaces such that the differential operators (here gradient and divergence) hold in strong form. We introduce lowest possible order discretizations of the split 1D wave equations, in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant FE spaces, omitting partial integrations. Approximating the Hodge-star by nontrivial Galerkin projections (GP), the two discrete metric equations follow by projections onto either the piecewise constant (GP0) or piecewise linear (GP1) space. Out of the four possible realizations, our framework gives us three schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the dispersion relation with the P1–P1 FE scheme that approximates both variables by piecewise linear spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the dispersion relation with the stable P1–P0 FE scheme that applies piecewise linear and piecewise constant (P0) spaces. However, the split schemes exhibit second order convergence for both quantities of interest. For the split scheme applying twice GP0, we are not aware of a corresponding standard formulation to compare with. Though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast, the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find FE schemes by new combinations of FE spaces.

We present an adaptation of the MOOD method, initially introduced in [1,2], for the two-dimensional shallow-water system with varying bathymetry, where the major novelty of the study is the non- conservative term discretization in the framework of the MOOD strategy. We derive a robust sixth-order well-balanced scheme and propose a large panel of numerical tests to assess the accuracy of the method and show that numerical solutions are free of oscillations in the vicinity of discontinuities. We also demonstrate that the MOOD method guarantees the height positivity as long as the first-order scheme does., This research was financed by FEDER Funds through Programa Operacional Factores de Competitividade COMPETE and by Portuguese Funds through FCT Fundacdo para a Ciencia e a Tecnologia, within the Projects UID/MAT/00013/2013 and FCT-ANR/MAT-NAN/0122/2012.

In the present work, by employing the multiple time scaling method, we studied the non-linear waves in shallow-water problem and obtained a set of Korteweg-deVries equations governing the various order terms in the perturbation expansion. By seeking a travelling wave type of solutions for the evolution equations, we have obtained a set of wave speeds associated with each time parameter. It is shown that the speed corresponding to the lowest order time parameter given the wave speed of the conventional reductive perturbation method, whereas the wave speeds corresponding to the higher order time parameters give the speed correction terms. The result obtained here is exactly the same with that of Demiray [H. Demiray, Modified reductive perturbation method as applied to long water waves: Korteweg-deVries hierarchy, Int. J. Nonlinear Sci. 6 (2008) 11-20] who employed the modified reductive perturbation method. Publisher's Version

An integrated computational structure for non-cohesive sediment-transport and bed-level changes in near-shore regions has been developed. It is basically composed of: (1) three hydrodynamic sub-models; (2) a dynamic equation for the sediment transport (of the Bailard-type); and (3) an extended sediment balance equation. A shallow-water approximation, or Saint-Venant-type model, is utilized for the computation and up-to-date field currents, initially and after each characteristic computational period. A Berkhoff-type wave model allows us to determine the wave characteristics in deep water and intermediate water conditions. These computations make it possible to define a smaller modeling area for a non-linear wave-current model of the Boussinesq-type, including breaking waves, friction effects and improved dispersion wave characteristics. Bed topography is updated after each wave period, or a multiple of this, called computational sedimentary period. Applicability of the computational structure is confirmed through laboratory experiments. Practical results of a real-world application obtained around the S. Lourenço fortification, Tagus estuary (Portugal), with the intention of preventing the destruction of the Bugio lighthouse, are shown. http://www.sciencedirect.com/science/article/B6V1P-44NM35Y-4/1/5d4ee1fe921aba0096ad08f76d2017f1