  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.
