  We study numerical (in)stability of the Method of characteristics (MoC)
applied to a system of non-dissipative hyperbolic partial differential
equations (PDEs) with periodic boundary conditions. We consider three different
solvers along the characteristics: simple Euler (SE), modified Euler (ME), and
Leap-frog (LF). The two former solvers are well known to exhibit a mild, but
unconditional, numerical instability for non-dissipative ordinary differential
equations (ODEs). They are found to have a similar (or stronger, for the
MoC-ME) instability when applied to non-dissipative PDEs. On the other hand,
the LF solver is known to be stable when applied to non-dissipative ODEs.
However, when applied to non-dissipative PDEs within the MoC framework, it was
found to have by far the strongest instability among all three solvers. We also
comment on the use of the fourth-order Runge--Kutta solver within the MoC
framework.
