High-Order Numerical Method for 1D Non-local Diffusive Equation

In this paper we present a non-local numerical scheme based on the Local Discontinuous Galerkin method for a non-local diffusive partial differential equation with application to traffic flow. In this model, the velocity is determined by both the average of the traffic density as well as the changes in the traffic density at a neighborhood of each point. We discuss nonphysical behaviors that can arise when including diffusion, and our measures to prevent them in our model. The numerical results suggest that this is an accurate method for solving this type of equation and that the model can capture desired traffic flow behavior. We show that computation of the non-local convolution results in $\mathcal{O}(n^2)$ complexity, but the increased computation time can be mitigated with high-order schemes like the one proposed.Comment: 17 pages and 8 figure

A finite volume method for scalar conservation laws with stochastic time-space dependent flux function

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions

A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework

Godunov-like numerical fluxes for conservation laws on networks

summary:We describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads. On individual roads, we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters. In order to solve traffic flows on networks, we construct suitable numerical fluxes at junctions based on preferences of the drivers. Numerical experiment comparing different approaches is presented

Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)

This workshop brought together leading experts, as well as the most promising young researchers, working on nonlinear hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in modeling, analysis, and numerics. Particular topics included ill-/well-posedness, randomness and multiscale modeling, flows in a moving domain, free boundary problems, games and control

Construction of fluxes at junctions for the numerical solution of traffic flow models on networks

summary:We deal with the simulation of traffic flow on networks. On individual roads we use standard macroscopic traffic models. The discontinuous Galerkin method in space and explicit Euler method in time is used for the numerical solution. We apply limiters to keep the density in an admissible interval as well as prevent spurious oscillations in the numerical solution. To solve traffic networks, we construct suitable numerical fluxes at junctions. Numerical experiments are presented