A high-accuracy differential quadrature method with optimized spline basis for Burgers’ equation

In this study, a new numerical approach is presented by combining the differential quadrature method with particle swarm optimization for an efficient solution of nonlinear partial differential equations. The proposed methodology is illustrated using the Burgers’ equation, which is used to model complicated physical phenomena including convection, diffusion, and reaction processes. Optimization methods are used in the presented technique to find the best parameter selection in the spline-based basis functions that considerably increase the accuracy and stability of the solution. The effectiveness of the framework is guaranteed through a wide range of numerical experiments, which are in excellent agreement with analytical/exact solutions. Stability analysis according to the Courant-Friedrichs-Lewy criterion ensures the stability of the proposed method. The ability of method to capture detailed nonlinear behaviors renders it a potential agent for treating a broad array of convection-diffusion-reaction problems in applied science and engineering.