  We begin with the theoretical study of spectral energy cascade due to the
propagation of high amplitude sound in the absence of thermal sources. To this
end, a first-principles-based system of governing equations, correct up to
second order in perturbation variables is derived. The exact energy corollary
of such second-order system of equations is then formulated and used to
elucidate the spectral energy dynamics of nonlinear acoustic waves. We then
extend this analysis to thermoacoustically unstable waves -- i.e. amplified as
a result of thermoacoustic instability. We drive such instability up until the
generation of shock waves. We further study the nonlinear wave propagation in
geometrically complex case of waves induced by the spark plasma between the
electrodes. This case adds the geometrical complexity of a curved,
three-dimensional shock, yielding vorticity production due to baroclinic
torque. Finally, detonation waves are simulated by using a low-order approach,
in a periodic setup subjected to high pressure inlet and exhaust of combustible
gaseous mixture. An order adaptive fully compressible and unstructured Navier
Stokes solver is currently under development to enable higher fidelity studies
of both the spark plasma and detonation wave problem in the future.
