On The Gas Dynamics Of Majda Combustion Model, Some Numerical Results

Majda introduced a qualititive model to study shock-wave chemistry interactions in combustion theory:where f(u) is a convex strongly nonlinear function satisfyingIn the above model, u is a lumped variable representing some features of density, velocity ,and temperature while z represents the mass fraction of unburnt gas in a simplified kinetics scheme.The parameter β > 0 is a lumped parameter representing the effects of diffusion and heat conduction ,K is the rate of reaction and q₀ represents the amount of unit energy liberated by chemical reaction.In paticular, q₀ > 0 for exothermic reactions. Under the assumptions of " ignition temperature" kinetics ( we assume that (T_i) is the "ignition temperature " ,then we can conclude : Φ(u) = 0 , when u ≤(T_i); Φ(u)≠0, when u > T_i), Φ(u) is typically a smooth increasing function with 0 ≤ Φ≤ 1 and(see [2]).The structure and formula of the weak entropy solution of the model (*) is unknown.Levy proved that the weak entropy solution of (*) tends to the traveling wave solution obtained by Majda( the difference is only a trivial constant).Levy also did a numerical experiment to check the above theoretical results, (see [1]).