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We study numerical (in)stability of the Method of characteristics (MoC)applied to a system of non-dissipative hyperbolic partial differentialequations (PDEs) with periodic boundary conditions. We consider three differentsolvers along the characteristics: simple Euler (SE), modified Euler (ME), andLeap-frog (LF). The two former solvers are well known to exhibit a mild, butunconditional, numerical instability for non-dissipative ordinary differentialequations (ODEs). They are found to have a similar (or stronger, for theMoC-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 wasfound to have by far the strongest instability among all three solvers. We alsocomment on the use of the fourth-order Runge--Kutta solver within the MoCframework.

Stability analysis of the numerical Method of characteristics applied to a class of energy-preserving systems. Part I: Periodic boundary conditions