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Now we focus on different explicit methods to solve advection equation (2.1) nu-merically on the periodic domain [0,L] with a given initial condition u0 =u(x,0). 2.1 FTCS Method We start the discussion of Eq. (2.1)with a so-called FTCS (forwardin time, centered in space) method. As discussed in Sec. 1.2 we introduce the discretization in time
1. General properties of the one-dimensional advection equation. The advection equation in one dimension states that the velocity, u(x,t), of a fluid particle is conserved following the particle motion (x is distance and t is time). Without external forces, this equation is. du. = 0, dt. or. € ∂u ∂u. u = 0 . ∂t ∂x. €
Advection refers to the transport mechanism of a substance (or conserved property i.e. mass) by a fluid due to the fluid's bulk motion (i.e. to the flow). Diffusion refers to another transport mechanism which occurs without any motion of the fluid ́s bulk.
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During advection, a fluid transports some conserved quantity or material via bulk motion. The fluid's motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space.
This chapter incorporates advection into our diffusion equation (deriving the advective diffusion equation) and presents various methods to solve the resulting partial differential equation for different geometries and contaminant conditions. 2.1 Derivation of the advective diffusion equation.
Advection Equations and Hyperbolic Systems. Hyperbolic partial differential equations (PDEs) arise in many physical problems, typi-cally whenever wave motion is observed. Acoustic waves, electromagnetic waves, seis-mic waves, shock waves, and many other types of waves can be modeled by hyperbolic equations.
