The Convergence Analysis of Higher Order Approximation for the Transport Equation With Nonhomogeneous and Homogeneous Boundary Conditions

The study examines the convergence analysis of a higher order approximation for a transport equation with both nonhomogeneous and homogeneous boundary conditions by using the Crank–Nicolson and their modified schemes. Using the Taylor series expressions, the suggested numerical schemes are created. Von Neumann stability analysis combined with the error-boundedness criterion provides a complete examination of stability and convergence. The analysis’s findings show that both Crank–Nicolson systems exhibit unconditional stability and are convergent with second-order accuracy in both directions. Using the current two approaches, extensive numerical experiments are carried out. The outcomes of carried-out trials are compared to the accuracy of the previous schemes in the literature. The comparative analysis demonstrates that the current approaches yield higher accuracy than the earlier techniques. In addition, we contrast the results of the two approaches and compare them. Based on these findings, the accuracy of the modified Crank–Nicolson scheme is better than that of the original Crank–Nicolson scheme. Overall, this research demonstrates that current methods effectively solve partial differential equations.