Numerical Solutions for One-Dimensional Steady-State Convection-Diffusion Equation without Source Term Using Four Different Discretization Schemes

Four numerical discretization schemes are available for solving the one-dimensional steady-state convection-diffusion equation without source terms: Central Differencing, First-Order Upwind, Hybrid Scheme, and QUICK Scheme. Each scheme can provide accurate solutions, but careful selection is required to avoid numerical instability and achieve optimal results. The Central Differencing scheme represents a straightforward implementation approach using symmetric difference approximations for both convective and diffusive terms. While easily codable with simple matrix operations, it demonstrates lower accuracy and may require additional stabilization techniques for high Peclet numbers. Upwind and Hybrid schemes offer improved accuracy through directional differencing that accounts for flow direction. Implementation typically involves conditional statements or sign-based coefficient adjustments in the discretization matrix. These schemes require more computational time due to their enhanced physical consistency but provide better stability characteristics. The QUICK (Quadratic Upstream Interpolation for Convective Kinematics) scheme employs quadratic interpolation for higher-order accuracy. Its implementation involves complex stencil calculations with weighted coefficients for neighboring nodes, making it computationally intensive but delivering superior accuracy and stability. When selecting a scheme, developers must balance computational efficiency against numerical accuracy, considering factors like grid resolution, flow conditions, and stability requirements in their code architecture.