  Exact diagonalization (ED) of small model systems gives the thermodynamics of
spin chains or quantum cell models at high temperature $T$. Density matrix
renormalization group (DMRG) calculations of progressively larger systems are
used to obtain excitations up to a cutoff $W_C$ and the low-$T$ thermodynamics.
The hybrid approach is applied to the magnetic susceptibility $\chi(T)$ and
specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the
linear Heisenberg antiferromagnet (HAF) and the frustrated $J_1-J_2$ model with
ferromagnetic (F) $J_1 < 0$ and antiferromagnetic (AF) $J_2 > 0$. The hybrid
approach is fully validated by comparison with HAF results. It extends
$J_1-J_2$ thermodynamics down to $T \sim 0.01|J_1|$ for $J_2/|J_1| \geq
\alpha_c = 1/4$ and is consistent with other methods. The criterion for the
cutoff $W_C(N)$ in systems of $N$ spins is discussed. The cutoff leads to
bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$
at system size $N$.
