
Software components for heterogeneous many-core architectures 83
derivatives are replaced with finite difference approximations, leaving only the
temporal derivative as unknown. The spatial derivatives are approximated
from u
n
, where u
n
represents the approximate solution to u(t
n
) at a given
time t
n
with time step size δt such that t
n
= nδt for n = 0, 1, . . . The finite
difference approximation can be interpreted as a matrix-vector product as
sketched in (6.2), and so the semi-discrete heat conduction problem becomes
∂u
∂t
= Au, A ∈ R
N×N
, u ∈ R
N
, (6.6)
where A is the sparse finite difference matrix and N is the number of un-
knowns in the discrete system. The ...