Positivity preserving temporal second-order spatial fourth-order conservative characteristic methods for convection dominated diffusion equations

In this study, we propose positivity preserving conservative characteristic methods with temporal second order and spatial fourth-order accuracy for solving convection dominated diffusion problems. The method of characteristics is utilized to avoid strict restrictions on time step sizes, providing greater flexibility in computation. To preserve mass, conservative piecewise parabolic interpolation is used to obtain values at tracking points. Additionally, we leverage the finite difference implementation of the continuous finite element method and construct various fourth-order approximation operators for the Laplace operator, which are then applied to develop conservative numerical schemes with positivity preserving property. The proposed methods are theoretically proven to preserve the positivity property of solutions and ensure mass conservation. Numerical examples are conducted to validate the performance of developed schemes, demonstrating their spatial and temporal convergence orders, as well as conservation property.