  In this work we study a multi-step scheme on time-space grids proposed by W.
Zhao et al. [28] for solving backward stochastic differential equations, where
Lagrange interpolating polynomials are used to approximate the time-integrands
with given values of these integrands at chosen multiple time levels. For a
better stability and the admission of more time levels we investigate the
application of spline instead of Lagrange interpolating polynomials to
approximate the time-integrands. The resulting scheme is a semi-discretization
in the time direction involving conditional expectations, which can be
numerically solved by using the Gaussian quadrature rules and polynomial
interpolations on the spatial grids. Several numerical examples including
applications in finance are presented to demonstrate the high accuracy and
stability of our new multi-step scheme.
