Manoj Solanki

In this work, we present a high-order finite difference method for solving the one-dimensional heat and advection–diffusion equations. The proposed approach combines a fourth-order compact finite difference scheme (CFDM)for spatial discretization and the Crank–Nicolson method for time integration. The compact f inite difference scheme provides higher accuracy by incorporating information from neighboring grid points, while the Crank–Nicolson method ensures stability and accuracy in time evolution. The method achieves fourth-order accuracy in both space and time, meaning the error decreases at a rate of O(h4,k4) as the grid size h and time step k are refined. One of the key advantages of the proposed method is its unconditional stability, which allows for accurate solutions even at larger time steps without imposing strict stability constraints. We also investigated the behavior of the advection–diffusion equation under different Péclet numbers to assess the method’s performance in convection-dominated scenarios. The numerical results and graphical analysis demonstrate that the method produces highly accurate solutions that closely match the exact solutions, confirming the robustness and efficiency of the approach. The combination of high accuracy, unconditional stability, and efficiency makes this method well-suited for solving complex advection–diffusion problems in various scientific and engineering applications.

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