SOLVING DIFFERENTIAL GAME PROBLEMS USING THE WEIGHTED AVERAGE NON-STANDARD FINITE DIFFERENCE METHOD

Abstract

Differential game problems involving multiple players arise naturally in econom-ics, engineering, and optimal control, yet their numerical solution remains challenging dueto nonlinear dynamics, interaction among agents, and the need to preserve qualitative properties of the continuous model. In this study, we develop and apply a weighted average non-standard finite difference (WANSFD) technique for the numerical solution of multi-player differential games. The proposed method combines the advantages of weighted averagingwith a non-standard finite difference (NSFD) discretization, enabling the preservation ofkey qualitative features such as stability and convergence while maintaining numerical accuracy. Unlike classical finite difference schemes, the WANSFD approach remains stable for awide class of nonlinear and non-smooth payoff structures and provides reliable approximations even for coarse discretizations. Numerical experiments demonstrate that the proposedmethod outperforms standard schemes in terms of stability, accuracy, and robustness acrossvarious differential game scenarios. These results confirm that the WANSFD technique offers an efficient and dependable numerical framework for analyzing complex differential gamemodels and optimal control systems.