  In this paper, we propose a mass conservative semi-Lagrangian finite
difference scheme for multi-dimensional problems without dimensional splitting.
The semi-Lagrangian scheme, based on tracing characteristics backward in time
from grid points, does not necessarily conserve the total mass. To ensure mass
conservation, we propose a conservative correction procedure based on a flux
difference form. Such procedure guarantees local mass conservation, while
introducing time step constraints for stability. We theoretically investigate
such stability constraints from an ODE point of view by assuming exact
evaluation of spatial differential operators and from the Fourier analysis for
linear PDEs.
  The scheme is tested by classical two dimensional linear passive-transport
problems, such as linear advection, rotation and swirling deformation. The
scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high
order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order
characteristics tracing scheme is generalized to the nonlinear guiding center
Vlasov model and incompressible Euler system. The effectiveness of the proposed
conservative semi-Lagrangian scheme is demonstrated numerically by our
extensive numerical tests.
