Piecewise Bézier curve-based numerical schemes for improved accuracy in convection-dominated transport problems

In convection-dominated transport phenomena, accurately capturing sharp gradients while minimizing numerical diffusion and avoiding spurious oscillations remains a fundamental challenge in computational fluid dynamics (CFD). This study introduces a novel class of bounded finite-volume schemes based on piecewise Bézier curve interpolation, named PUBICK (Piecewise Upwind Bézier Interpolation for Convective Kinematics). The proposed family of schemes is constructed within the Normalized Variable Formulation and is designed to satisfy key stability and boundedness criteria, including Total Variation Diminishing (TVD), the Convective Boundedness Criterion (CBC), and the Boundedness and Accuracy-Increasing Requirement (BAIR). The performance of the PUBICK scheme is evaluated through comprehensive numerical experiments in one- and two-dimensional problems, including linear advection, viscous Burgers’ equation, and multiphase flow in porous media. Comparisons with established high-resolution schemes demonstrate that PUBICK achieves competitive accuracy while ensuring stability and boundedness. To further characterize the numerical behavior, a modified wavenumber (Fourier) analysis is conducted in a linearized sense to assess the dispersive and dissipative properties of the schemes in the frequency domain. The results show that PUBICK provides a favorable balance between phase accuracy and numerical dissipation among bounded schemes, while maintaining robustness in convection-dominated regimes. These findings highlight the potential of the proposed Bézier-based formulation as an efficient and reliable approach for simulating complex transport phenomena.