Majda's model

Majda's model is a qualitative model (in mathematical physics) introduced by Andrew Majda in 1981 for the study of interactions in the combustion theory of shock waves and explosive chemical reactions.[1]

The following definitions are with respect to a Cartesian coordinate system with 2 variables. For functions , of one spatial variable representing the Lagrangian specification of the fluid flow field and the time variable , functions , of one variable , and positive constants , the Majda model is a pair of coupled partial differential equations:[2]

[2]
the unknown function is a lumped variable, a scalar variable formed from a complicated nonlinear average of various aspects of density, velocity, and temperature in the exploding gas;
the unknown function is the mass fraction in a simple one-step chemical reaction scheme;
the given flux function is a nonlinear convex function;
the given ignition function is the starter for the chemical reaction scheme;
is the constant reaction rate;
is the constant heat release;
is the constant diffusivity.[2]

Since its introduction in the early 1980s, Majda's simplified "qualitative" model for detonation ... has played an important role in the mathematical literature as test-bed for both the development of mathematical theory and computational techniques. Roughly, the model is a system consisting of a Burgers equation coupled to a chemical kinetics equation. For example, Majda (with Colella & Roytburd) used the model as a key diagnostic tool in the development of fractional-step computational schemes for the Navier-Stokes equations of compressible reacting fluids ...[3]

  1. ^ Majda, Andrew (1981). "A qualitative model for dynamic combustion". SIAM J. Appl. Math. 41 (1): 70–93. doi:10.1137/0141006.
  2. ^ a b c Humphreys, Jeffrey; Lyng, Gregory; Zumbrun, Kevin (2013). "Stability of viscous detonations for Majda's model". Physica D: Nonlinear Phenomena. 259: 63–80. arXiv:1301.1260. Bibcode:2013PhyD..259...63H. doi:10.1016/j.physd.2013.06.001. S2CID 119301730.
  3. ^ Lyng, Gregory D. (2015). "Spectral and nonlinear stability of viscous strong and weak detonation waves in Majda's qualitative model" (PDF).