  The study of generalising the central difference for integer order Laplacian
to fractional order is discussed in this paper. Analysis shows that, in
contrary to the conclusion of a previous study, difference stencils evaluated
through fast Fourier transform prevents the convergence of the solution of
fractional Laplacian. We propose a composite quadrature rule in order to
efficiently evaluate the stencil coefficients with the required convergence
rate in order to guarantee convergence of the solution. Furthermore, we propose
the use of generalised higher order lattice Boltzmann method to generate
stencils which can approximate fractional Laplacian with higher order
convergence speed and error isotropy. We also review the formulation of the
lattice Boltzmann method and discuss the explicit sparse solution formulated
using Smolyak's algorithm, as well as the method for the evaluation of the
Hermite polynomials for efficient generation of the higher order stencils.
Numerical experiments are carried out to verify the error analysis and
formulations.
