Numerical schemes for a class of nonlocal conservation laws: a general approach

In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. Thereby, we approximate in a first step the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical findings and demonstrate that the approach can be applied to further nonlocal problems

Modelling and numerical analysis of energy-dissipating systems with nonlocal free energy

The broad objective of this thesis is to design finite-volume schemes for a family of energy-dissipating systems. All the systems studied in this thesis share a common property: they are driven by an energy that decreases as the system evolves. Such decrease is produced by a dissipation mechanism, which ensures that the system eventually reaches a steady state where the energy is minimised. The numerical schemes presented here are designed to discretely preserve the dissipation of the energy, leading to more accurate and cost-effective simulations. Most of the material in this thesis is based on the publications [16, 54, 65, 66, 243]. The research content is structured in three parts. First, Part II presents well-balanced first-, second- and high-order finite-volume schemes for a general class of hydrodynamic systems with linear and nonlinear damping. These well-balanced schemes preserve stationary states at machine precision, while discretely preserving the dissipation of the discrete free energy for first- and second-order accuracy. Second, Part III focuses on finite-volume schemes for the Cahn-Hilliard equation that unconditionally and discretely satisfy the boundedness of the phase eld and the free-energy dissipation. In addition, our Cahn-Hilliard scheme is employed as an image inpainting filter before passing damaged images into a classification neural network, leading to a significant improvement of damaged-image prediction. Third, Part IV introduces nite-volume schemes to solve stochastic gradient-flow equations. Such equations are of crucial importance within the framework of fluctuating hydrodynamics and dynamic density functional theory. The main advantages of these schemes are the preservation of non-negative densities in the presence of noise and the accurate reproduction of the statistical properties of the physical systems. All these fi nite-volume schemes are complemented with prototypical examples from relevant applications, which highlight the bene fit of our algorithms to elucidate some of the unknown analytical results.Open Acces

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

In this paper we develop a family of very high-order central (up to 6th-order) non-oscillatory schemes for mixed-element unstructured meshes. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size therefore simplifying their implementation. Their key ingredient is the non-linear combination in a CWENO style similar to Dumbser et al [1] of a high-order polynomial arising from a central stencil with lower-order polynomials from directional stencils. Therefore, in smooth regions of the computational domain the optimum order of accuracy is recovered, while in regions of sharp-gradients the larger influence of the reconstructions from the directional stencils suppress the oscillations. It is the compactness of the directional stencils that increases the chances of at least one of them lying in a region with smooth data, that greatly enhances their robustness compared to classical WENO schemes. The two variants developed are CWENO and CWENOZ schemes, and it is the first time that such very-high-order schemes are designed for mixed-element unstructured meshes. We explore the influence of the linear weights in each of the schemes, and assess their performance in terms of accuracy, robustness and computational cost through a series of stringent 2D and 3D test problems. The results obtained demonstrate the improved robustness that the schemes offer, a parameter of paramount importance for and their potential use for industrial-scale engineering applications

Lyapunov stabilization for nonlocal traffic flow models

Using a nonlocal second-order traffic flow model we present an approach to control the dynamics towards a steady state. The system is controlled by the leading vehicle driving at a prescribed velocity and also determines the steady state. Thereby, we consider both, the microscopic and macroscopic scales. We show that the fixed point of the microscopic traffic flow model is asymptotically stable for any kernel function. Then, we present Lyapunov functions for both, the microscopic and macroscopic scale, and compute the explicit rates at which the vehicles influenced by the nonlocal term tend towards the stationary solution. We obtain the stabilization effect for a constant kernel function and arbitrary initial data or concave kernels and monotone initial data. Numerical examples demonstrate the theoretical results

An overview of non-local traffic flow models

We give an overview of mathematical traffic flow models with non-local velocity. More precisely, we consider conservation laws with flux functions depending on an integral evaluation of the density of vehicles through a convolution product. We summarize the analytical results recently obtained for this kind of models and we provide some numerical simulations illustrating the behavior of different groups of drivers or vehicles

A Space-time Nonlocal Traffic Flow Model: Relaxation Representation and Local Limit

We propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2\times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.Comment: 32 page

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions