A Novel Numerical Treatment to One Dimensional Heat and Wave Equations with Second Order Accuracy

Numerical solutions to partial differential equations (PDEs) play a vital role in modeling complex physical phenomena across scientific computing and engineering disciplines. Achieving stable, accurate solutions requires careful selection of numerical methods and parameters. However, guidelines for appropriate technique selection across PDE types and boundary scenarios remain insufficiently codified. This study aims to demonstrate solution behaviors and stability constraints for key one-dimensional parabolic and hyperbolic PDEs solved using second-order accurate methods under differing boundary conditions. The analysis employs finite difference techniques alongside multi-step Adams-Bashforth, Lax-Wendroff, CrankNicolson and Runge-Kutta time integration schemes for addressing heat (diffusion), wave (advection), and convection-diffusion equations over periodic and bounded domains. Stability and consistency are assessed in relation to grid resolution, time step constraints, boundary condition variations and source term introduction. The second-order accurate solutions reliably capture propagating and diffusing phenomena for periodic and bounded settings. However, the solutions exhibit high sensitivity to time step size and grid spacing, indicating the vital need for careful parameter tuning. Stability strictly demands sufficiently small time steps, while precision requires adequate grid resolution. Appropriate boundary condition choices are also essential in shaping accurate solution behavior.The results validate the capacity of second-order techniques to elucidate complex transient heat and wave dynamics under diverse scenarios in one dimension. The insights gained on stability requirements and parameter constraints will guide selection of suitable methods and discretization for related multiphysics simulations. Extensions to higher dimensions should leverage advanced paradigms like adaptive mesh refinement and high-order schemes.