Notes on the Euler Equations

These notes describe how to do a piecewise linear or piecewise parabolic method for the Euler equations. 1 Euler equation properties The Euler equations in one dimension appear as: ∂ρ ∂t + ∂(ρu) ∂x = 0 (1) ∂(ρu) ∂t + ∂(ρuu + p) ∂x = 0 (2) ∂(ρE) ∂t + ∂(ρuE + up) ∂x = 0 (3) These represent conservation of mass, momentum, and energy. Here ρ is the density, u is the one-dimensional velocity, p is the pressure, and E is the total energy / mass, and can be expressed in terms of the specific internal energy and kinetic energies as: E = e + 1 2 u 2 (4) The equations are closed with the addition of an equation of state: p = ρe(γ − 1) (5) where γ is the ratio of specific heats for the gas/fluid (for an ideal, monatomic gas, γ = 5/3). In this form, the equations are said to be in conservative form. They can be written as: U t + [F(U)] x = 0 (6) with U =   ρ ρu ρE   F(U) =   ρu ρuu + p ρuE + up   (7) An alternate way to express these equations is using the primitive variables: ρ, u, p. Exercise 1: Show that the Euler equations in primitive form can be written as q t + A(q)q x = 0 (8) where q =   ρ u p   A(q) =   u ρ 0 0 u 1/ρ 0 γp u   (9) The eigenvalues of A can be found via |A − λI| = 0, where |. .. | indicates the determinant and λ are the eigenvalues. Exercise 2: Show that the eigenvalues of A are λ (−) = u − c, λ (•) = u, λ (+) = u + c where the speed of sound is c = γp/ρ.