  The bond-propagation (BP) algorithm for the specific heat of the two
dimensional Ising model is developed and that for the internal energy is
completed. Using these algorithms, we study the critical internal energy and
specific heat of the model on the square lattice and triangular lattice with
free boundaries. Comparing with previous works [X.-T. Wu {\it et al} Phys. Rev.
E {\bf 86}, 041149 (2012) and Phys. Rev. E {\bf 87}, 022124 (2013)], we reach
much higher accuracy ($10^{-26}$) of the internal energy and specific heat,
compared to the accuracy $10^{-11}$ of the internal energy and $10^{-9}$ of the
specific heat reached in the previous works. This leads to much more accurate
estimations of the edge and corner terms. The exact values of some edge and
corner terms are therefore conjectured. The accurate forms of finite-size
scaling for the internal energy and specific heat are determined for the
rectangle-shaped square lattice with various aspect ratios and various shaped
triangular lattice. For the rectangle-shaped square and triangular lattices and
the triangle-shaped triangular lattice, there is no logarithmic correction
terms of order higher than 1/S, with S the area of the system. For the
triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist
logarithmic correction terms of order higher than 1/S for the internal energy,
and logarithmic correction terms of all orders for the specific heat.
