A Novel Richardson Extrapolation Technique for Numerical Approximation of Singularly Perturbed Convection-Diffusion Problems With Integral Boundary Conditions

This study presents the novel Richardson extrapolation techniques for solving numerical approx- imation of singularly perturbed convection-diffusion problems (SPCDP) with integral boundary conditions (IBC). A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) and exponential (eXp) mesh. To handle the integral boundary con- ditions, the trapezoidal rule is applied. The parameter-uniform error bound for the numerical derivative is established which leading to a first-order convergence rate. The study establishes an error bound for numerical solutions and determines the numerical approximation as well as analyze a upwind finite difference scheme on a piecewise uniform mesh ( Shishkin mesh) and exponential (eXp) for singularly perturbed convection diffusion equations with integral boundary conditions. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from O N −1 ln N to O N −2 ln2 N using this technique, where N is the number of mesh intervals. Numerical results are presented to validate the theoretical findings, demonstrating the effectiveness and accuracy of the proposed technique.