Advection Equation

Abstract

Let us consider a continuity equation for the one-dimensional drift of incompress-ible fluid. In the case that a particle density u(x,t) changes only due to convection processes one can write u(x, t + △t) = u(x − c △t, t). If △t is sufficient small, the Taylor-expansion of both sides gives u(x,t) + △t ∂ u(x,t) ∂t ≃ u(x,t) − c△t ∂ u(x,t) ∂ x or, equivalently ∂ u ∂t + c ∂ u ∂ x = 0. (2.1) Here u = u(x,t), x ∈ R, and c is a nonzero constant velocity. Equation (2.1) is called to be an advection equation and describes the motion of a scalar u as it is advected by a known velocity field. According to the classification given in Sec. 1.1, Eq. (2.1) is a hyperbolic PDE. The unique solution of (2.1) is determined by an initial condition u 0 := u(x, 0) u(x,t) = u 0 (x − ct) , (2.2) where u 0 = u 0 (x) is an arbitrary function defined on R. One way to find this exact solution is the method of characteristics (see App. B). In the case of Eq. (2.1) the coefficients A = c, B = 1, C = 0 and Eqn. (B.2) read dt ds = 1 ⇔ |t(0) = 0| ⇔ t = s, dx ds = c ⇔ |x(0) = x 0 | ⇔ x = x 0 + ct. Hence, Eq. (B.3) becomes 13