Fractional-step High-order and Bound-preserving Method for Convection Diffusion Equations

TL;DR

This paper introduces a high-order, bound-preserving fractional-step method for nonlinear convection-diffusion equations, extending to 2D with ADI, ensuring stability, mass conservation, and computational efficiency.

Contribution

The paper develops a novel fractional-step, high-order compact scheme that preserves bounds and mass, and extends it to two dimensions using ADI for improved efficiency.

Findings

Scheme is weakly monotonic and bound-preserving under mild step constraints.

Numerical experiments confirm theoretical stability and accuracy.

Extension to 2D reduces computational cost while maintaining properties.

Abstract

In this paper, we derive two bound-preserving and mass-conserving schemes based on the fractional-step method and high-order compact (HOC) finite difference method for nonlinear convection-dominated diffusion equations. We split the one-dimensional equation into three stages, and employ appropriate temporal and spatial discrete schemes respectively. We show that our scheme is weakly monotonic and that the bound-preserving property can be achieved using the bound-preserving limiter under some mild step constraints. By employing the alternating direction implicit (ADI) method, we extend the scheme to two-dimensional problems, further reducing computational cost. We also provide various numerical experiments to verify our theoretical results.