  We present a new, updated version of the EuclidEmulator (called
EuclidEmulator2), a fast and accurate predictor for the nonlinear correction of
the matter power spectrum. Percent-level accurate emulation is now supported in
the eight-dimensional parameter space of $w_0w_a$CDM$+\sum m_\nu$models between
redshift $z=0$ and $z=3$ for spatial scales within the range 0.01 $h$/Mpc $\leq
k \leq$ 10 $h$/Mpc. In order to achieve this level of accuracy, we have had to
improve the quality of the underlying N-body simulations used as training data:
(1) we use self-consistent linear evolution of non-dark matter species such as
massive neutrinos, photons, dark energy and the metric field, (2) we perform
the simulations in the so-called N-body gauge, which allows one to interpret
the results in the framework of general relativity, (3) we run over 250
high-resolution simulations with $3000^3$ particles in boxes of 1
(Gpc/$h$)${}^3$ volumes based on paired-and-fixed initial conditions and (4) we
provide a resolution correction that can be applied to emulated results as a
post-processing step in order to drastically reduce systematic biases on small
scales due to residual resolution effects in the simulations. We find that the
inclusion of the dynamical dark energy parameter $w_a$ significantly increases
the complexity and expense of creating the emulator. The high fidelity of
EuclidEmulator2 is tested in various comparisons against N-body simulations as
well as alternative fast predictors like Halofit, HMCode and CosmicEmu. A blind
test is successfully performed against the Euclid Flagship v2.0 simulation.
Nonlinear correction factors emulated with EuclidEmulator2 are accurate at the
level of 1% or better for 0.01 $h$/Mpc $\leq k \leq$ 10 $h$/Mpc and $z\leq3$
compared to high-resolution dark matter only simulations. EuclidEmulator2 is
publicly available at https://github.com/miknab/EuclidEmulator2 .
