An optimized cubic B-spline algorithm for high-precision approximation of nonlinear transport phenomena

This study aims to investigate a numerical scheme based on extended cubic B-spline functions for solving the nonlinear gas dynamics equation, which plays a crucial role in the study of physical phenomena such as explosions, combustion, detonation and condensation within moving flows. The standard finite difference formulation has been employed to approximate the time derivative, while the solution curve in spatial direction is interpolated using extended cubic B-spline functions. A comprehensive stability analysis of the scheme is provided to ensure that errors do not propagate over time. Additionally, a convergence analysis for cubic B-spline interpolation is conducted to assess the accuracy of the solution. The effectiveness and efficiency of the proposed method are tested through numerical simulations. The findings indicate that the proposed technique provides better error estimates compared to other methods discussed in the literature. The straightforward algorithm, high accuracy and minimal computational efforts are the major advantages of this approach. Therefore, the proposed method may serve as a promising and efficient alternative for the numerical solution of nonlinear partial differential equations.