A second-order accurate semi-Lagrangian method for convection-diffusion equations with interfacial jumps

TL;DR

This paper introduces a second-order accurate semi-Lagrangian finite-difference method for convection-diffusion equations with interfacial jumps, featuring a novel interpolation scheme and ghost fluid approach for sharp interface capturing.

Contribution

The paper develops a new semi-Lagrangian scheme with a specialized interpolation and ghost fluid method for accurate convection-diffusion solutions with interfaces.

Findings

Achieves second-order convergence for convection-diffusion equations with interfaces.

Provides a sharp capturing method with minimal local truncation error near interfaces.

Demonstrates the effectiveness of the level-set advection algorithm in jump conditions.

Abstract

In this paper, we present a second-order accurate finite-difference method for solving convectiondiffusion equations with interfacial jumps on a moving interface. The proposed method is constructed under a semi-Lagrangian framework for convection-diffusion equations; a novel interpolation scheme is developed in the presence of jump conditions. Combined with a second-order ghost fluid method [3], a sharp capturing method with a first-order local truncation error near the interface and second-order truncation error away from the interface is developed for the convectiondiffusion equation. In addition, a level-set advection algorithm is presented when the velocity gradient jumps across the interface. Numerical experiments support the conclusion that the proposed methods for convection-diffusion equations and level-set advection are necessary for the second-order convergence solution and the interface position.