SAMS Subject Classification Number: 23

Higher order unconditionally positive finite difference (HUPFD) methods are developed to solve linear and non-linear advection-diffusion-reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for convergence to the exact solution. The Von Neuman condition is used to analyze the stability since we are dealing with the Cauchy problem. The proposed method’s efficiency and effectiveness is investigated by calculating the error, convergence rate, and computational time. A comparison of the solutions obtained by the higher order unconditionally positive finite difference and analytical methods is conducted for validation purposes. The numerical results show that the developed method preserve the solution accuracy. The results also show that increasing the order of the unconditionally positive finite difference leads to the implicit scheme that is unconditionally stable with an increased order of accuracy with respect to time and space.