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 ${\displaystyle u(x,t)}$, ${\displaystyle z(x,t)}$ of one spatial variable ${\displaystyle x}$ representing the Lagrangian specification of the fluid flow field and the time variable ${\displaystyle t}$, functions ${\displaystyle f(w)}$, ${\displaystyle \phi (w)}$ of one variable ${\displaystyle w}$, and positive constants ${\displaystyle k,q,B}$, the Majda model is a pair of coupled partial differential equations:^[2]

${\displaystyle {\frac {\partial u(x,t)}{\partial t}}+q\cdot {\frac {\partial z(x,t)}{\partial t}}+{\frac {\partial f(u(x,t))}{\partial x}}=B\cdot {\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}}$

${\displaystyle {\frac {\partial z(x,t)}{\partial t}}=-k\cdot \phi (u(x,t))\cdot z(x,t)}$^[2]

the unknown function ${\displaystyle u=u(x,t)}$ 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 ${\displaystyle z=z(x,t)\in [0,1]}$ is the mass fraction in a simple one-step chemical reaction scheme;

the given flux function ${\displaystyle f=f(w)}$ is a nonlinear convex function;

the given ignition function ${\displaystyle \phi =\phi (w)}$ is the starter for the chemical reaction scheme;

${\displaystyle k}$ is the constant reaction rate;

${\displaystyle q}$ is the constant heat release;

${\displaystyle B}$ 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 ${\displaystyle 2\times 2}$ 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]