Abstract: We propose a novel semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes, which are designed by approximating the advective form of equation via direct characteristics tracing, the scheme we proposed approximates the conservative form of equation. This essential difference makes the proposed scheme conservative by nature, and extendable to equations with variable coefficients. The proposed semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and capture sharp interfaces without introducing oscillations. The scheme is extended to high dimensional problem by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, several challenging examples of rigid body rotation and swirling deformation in multi-dimensions, as well as the Vlasov Poisson system for plasma applications. As the information is propagating along characteristics, the semi-Lagrangian scheme does not have CFL time step restriction, allowing for a cheaper and more flexible numerical realization than the regular finite difference scheme.
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