Finite difference method

Abstract: After introducing the concept of central differences of a function to approximate its derivatives by simple numerical expressions, the writer shows how a problem involving a linear differential equ...

Abstract: The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker–Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

Abstract: The development and application of a one-dimensional heat transport model in highly transient streams governed by unsteady flow is described here. The resultant framework consists of two modules called the hydrodynamic and heat transport modules. In the hydrodynamic module, the hydraulic variables such as flow depth and velocity are simulated. Based on this information, the heat transport module is executed to calculate temperatures. A new approach—coupling heat transport in the surface water and diffusion in the sediment zone—is developed and applied for this module. In this approach, an interaction term accounts for the flux of heat energy between the water and sediments, which affects the distribution of water temperature in clear and shallow streams. Implicit finite difference methods called the Preissmann four-point and the Crank-Nicolson schemes are used to solve each module. Application of this framework demonstrates that the model effectively simulates the hydraulic variables and temperatures.