  Self-gravitational force calculation for infinitesimally thin disks is
important for studies on the evolution of galactic and protoplanetary disks.
Although high-order methods have been developed for hydrodynamic and
magneto-hydrodynamic equations, high-order improvement is desirable for solving
self-gravitational forces for thin disks. In this work, we present a new
numerical algorithm that is of linear complexity and of high-order accuracy.
This approach is fast since the force calculation is associated with a
convolution form, and the fast calculation can be achieved using Fast Fourier
Transform. The nice properties, such as the finite supports and smoothness, of
B-splines are exploited to stably interpolate a surface density and achieve a
high-order accuracy in forces. Moreover, if the mass distribution of interest
is exclusively confined within a calculation domain, the method does not
require artificial boundary values to be specified before the force
calculation. To validate the proposed algorithm, a series of numerical tests,
ranging from 1st- to 3rd-order implementations, are performed and the results
are compared with analytic expressions derived for 3rd- and 4th-order
generalized Maclaurin disks. We conclude that the improvement on the numerical
accuracy is significant with the order of the method, with only little increase
of the complexity of the method.
