Semi-Lagrangian difference approximations with different stability requirements

Abstract

The paper demonstrates different ways of using the semi-Lagrangian approximation depending on the fulfillment of conservation laws. A one-dimensional continuity equation and a parabolic one are taken as simple methodological examples. For these equations, the principles of constructing discrete analogues are demonstrated for three different conservation laws (or the requirements of stability in the related discrete norms similar to the L1, L2, L∞-norms). It is significant that different conservation laws yield difference problems of different types as well as different ways to justify their stability.

“Mean Field Games” as Mathematical Models for Control and Optimization of Business Activity

“…The standard widespread approximations provide monotone schemes for both problems and give the total task for the minimization of each i α over all domain Ω . Last time, the implementation of special approximations (Shaidurov, Efremov and Gileva, 2018;Shaidurov, Viatkin, Kuchunova, 2018) for these problems led to the disintegration of the total minimization to local pointwise ones (Lachapelle, Salomon and Turinici, 2010;Shaydurov, Zhang and Kornienko, 2019). Therefore, the discretization of the above differential problem looks as follows.…”