1,721,025 research outputs found
Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction
No full textHigh-order finite volume schemes for conservation laws are very useful in applications, due to their ability to compute accurate solutions on quite coarse meshes and with very few restrictions on the kind of cells employed in the discretization. For balance laws, the ability to approximate up to machine precision relevant steady states allows the scheme to compute accurately, also on coarse meshes, small perturbations of such states, which are very relevant for many applications. In this paper, we propose third- and fourth-order accurate finite volume schemes for the shallow water equations. The schemes have the well-balanced property thanks to a path-conservative approach applied to an appropriate non-conservative reformulation of the equations. High-order accuracy is achieved by designing truly two-dimensional (2D) reconstruction procedures of the central WENO (CWENO) type. The novel schemes are tested for accuracy and well-balancing and shown to maintain positivity of the water height on wet/dry transitions. Finally, they are applied to simulate the Tohoku 2011 tsunami event
Central WENO subcell finite volume limiters for Ader discontinuous Galerkin schemes on fixed and moving unstructured meshes
No full textWe present a novel a posteriori subcell finite volume limiter for high order discontinuous Galerkin (DG) finite element schemes for the solution of nonlinear hyperbolic PDE systems in multiple space dimensions on fixed and moving unstructured simplex meshes. The numerical method belongs to the family of high order fully discrete one-step ADER-DG schemes [12, 45] and makes use of an element-local space-time Galerkin finite element predictor. Our limiter is based on the MOOD paradigm, in which the discrete solution of the high order DG scheme is checked a posteriori against a set of physical and numerical admissibility criteria, in order to detect spurious oscillations or unphysical solutions and in order to identify the so-called troubled cells. Within the detected troubled cells the discrete solution is then discarded and recomputed locally with a more robust finite volume method on a fine subgrid. In this work, we propose for the first time to use a high order ADER-CWENO finite volume scheme as subcell finite volume limiter on unstructured simplex meshes, instead of a classical second order TVD scheme. Our new numerical scheme has been developed both for fixed Eulerian meshes as well as for moving Lagrangian grids. It has been carefully validated against a set of typical benchmark problems for the compressible Euler equations of gas dynamics and for the equations of ideal magnetohydrodynamics (MHD)
Adaptivity for systems of conservation laws driven by an error indicator based on entropy production
No full textin: La matematica e le sue applicazioni, 13-2009, Dip. di Matematica, Politecnico di Torino. ISSN 1974-305
Evaluating the temporal variability of concentrations of POPs in a glacier-fed stream food chain using a combined modeling approach
Full text linkFalling snow acts as an efficient scavenger of contaminants from the atmosphere and, accumulating on the ground surface, behaves as a temporary storage reservoir; during snowaging and metamorphosis, contaminants may concentrate and be subject to pulsed release during intense snow melt events. In high-mountain areas, firn and ice play a similar role. The consequent concentration peaks in surfacewaters can pose a risk to high-altitude ecosystems, since snow and ice melt often coincide with periods of intense biological activity. In such situations, the role of dynamic models can be crucialwhen assessing environmental behavior of contaminants and their accumulation patterns in aquatic organisms. In the present work, a dynamic fatemodeling approach was combined to a hydrological module capable of estimating water discharge and snow/ice melt contributions on an hourly basis, starting from hourly air temperatures. The model was applied to the case study of the Frodolfo glacierfed stream (Italian Alps), for which concentrations of a number of persistent organic pollutants (POPs), such as polychlorinated biphenyl (PCBs) and p,p′-dichlorodiphenyldichloroethylene (p,p′-DDE) in stream water and four macroinvertebrate groups were available. Considering the uncertainties in input data, results showed a satisfying agreement for both water and organism concentrations. This study showed the model adequacy for the estimation of pollutant concentrations in surface waters and bioaccumulation in aquatic organisms, as well as its possible role in assessing the consequences of climate change on the cycle of POPs
Analysis of a multi-population kinetic model for traffic flow
Full text linkIn this work we extend a recent kinetic traffic model [G. Puppo, M. Semplice, A. Tosin, G. Visconti, Kinet. Relat. Models, in press, 2016] to the case of more than one class of vehicles, each of which is characterized by few different microscopic features. We consider a Boltzmann-like framework with only binary interactions, which take place among vehicles belonging to the various classes. Our approach differs from the multi-population kinetic model proposed in [G. Puppo, M. Semplice, A. Tosin, G. Visconti, Commun. Math. Sci., 14:643-669, 2016] because here we assume continuous velocity spaces and we introduce a parameter describing the physical velocity jump performed by a vehicle that increases its speed after an interaction. The model is discretized in order to investigate numerically the structure of the resulting fundamental diagrams and the system of equations is analyzed by studying well posedness. Moreover, we compute the equilibria of the discretized model and we show that the exact asymptotic kinetic distributions can be obtained with a small number of velocities in the grid. Finally, we introduce a new probability law in order to attenuate the sharp capacity drop occurring in the diagrams of traffic
High order Finite Volume Schemes for Balance Laws with Stiff Relaxation
Full text linkThe paper deals with the construction and analysis of efficient high order finite volume shock capturing schemes for the numerical solution of hyperbolic systems with stiff relaxation. In standard high order finite volume schemes it is difficult to treat the average of the source implicitly, since the computation of such average couples neighboring cells, making implicit schemes extremely expensive. The main novelty of the paper is that the average of the source is split into the sum of the source evaluated at the cell average plus a correction term. The first term is treated implicitly, while the small correction is treated explicitly, using IMEX-Runge-Kutta methods, thus resulting in a very effective semi-implicit scheme. This approach allows the construction of effective high order schemes in space and time. An asymptotic analysis is performed for small values of the relaxation parameter, giving an indication on the structure of the IMEX schemes that have to be adopted for time discretization. Several numerical tests confirm the accuracy and efficiency of the approach
Kinetic models for traffic flow resulting in a reduced space of microscopic velocities
Full text linkThe purpose of this paper is to study the properties of kinetic models for traffic flow described by a Boltzmann-type approach and based on a continuous space of microscopic velocities. In our models, the particular structure of the collision kernel allows one to find the analytical expression of a class of steady-state distributions, which are characterized by being supported on a quantized space of microscopic speeds. The number of these velocities is determined by a physical parameter describing the typical acceleration of a vehicle and the uniqueness of this class of solutions is supported by numerical investigations. This shows that it is possible to have the full richness of a kinetic approach with the simplicity of a space of microscopic velocities characterized by a small number of modes. Moreover, the explicit expression of the asymptotic distribution paves the way to deriving new macroscopic equations using the closure provided by the kinetic model
On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws
No full textHigh order numerical methods for networks of hyperbolic conservation laws have recently gained increasing popularity. Here, the crucial part is to treat the boundaries of the single (one-dimensional) computational domains in such a way that the desired convergence rate is achieved in the smooth case but also stability criterions are fulfilled, in particular in the presence of discontinuities. Most of the recently proposed methods rely on a WENO extrapolation technique introduced by Tan and Shu (2010). Within this work, we refine and in a sense generalize these results for the case of a third order scheme. Numerical evidence for the analytically found parameter bounds is given as well as results for a complete third order scheme based on the proposed boundary treatment
Central Weighted ENO Schemes for Hyperbolic Conservation Laws on Fixed and Moving Unstructured Meshes
Full text linkWe present a novel family of arbitrary high order accurate central Weighted ENO (CWENO) finite volume schemes for the solution of nonlinear systems of hyperbolic conservation laws on fixed and moving unstructured simplex meshes in two and three space dimensions. Starting from the given cell averages of a function on a triangular or tetrahedral control volume and its neighbors, the nonlinear CWENO reconstruction yields a high order accurate and essentially nonoscillatory polynomial that is defined everywhere in the cell. Compared to other WENO schemes on unstructured meshes, the total stencil size is the minimum possible one, as in classical pointwise WENO schemes of Jiang and Shu. However, the linear weights can be chosen arbitrarily, which makes the practical implementation on general unstructured meshes particularly simple. We make use of the piecewise polynomials generated by the CWENO reconstruction operator inside the framework of fully discrete and high order accurate one-step ADER finite volume schemes on fixed Eulerian grids as well as on moving arbitrary-Lagrangian-Eulerian meshes. The computational efficiency of the high order finite volume schemes based on the new CWENO reconstruction is tested on several two- and three-dimensional benchmark problems for the compressible Euler equations and is found to be more efficient in terms of memory consumption and computational efficiency with respect to classical WENO reconstruction schemes on unstructured meshes. We also provide evidence that the new algorithm is suitable for implementation on massively parallel distributed memory supercomputers, showing a numerical example in three dimensions that was run with more than one billion high order elements in space and using more than 10,000 CPU cores
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