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Unformatted text preview: Chapter 2 Method of characteristics for solving conservation equation As indicated in section 1.2 , the conservation equation for the one lane traffic is given by ∂ρ ∂t + ∂ ∂x ( u · ρ ) = 0 , where the density, ρ ( x, t ) is taken as the prime variable. Suppose that the velocity-density relation is modelled by u = u ( ρ ) . Then the conservation equation becomes that ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x = 0 . In this chapter we always adopt the Greenshields model , i.e. u ( ρ ) = U max parenleftbigg 1- ρ ρ max parenrightbigg . This leads to ∂ ∂x ( u · ρ ) = ∂ ∂x parenleftbigg U max parenleftbigg ρ- ρ 2 ρ max parenrightbiggparenrightbigg = U max parenleftbigg 1- ρ ρ opt parenrightbigg . ∂ρ ∂x i.e. c ( ρ ) = U max parenleftbigg 1- ρ ρ opt parenrightbigg . 2.1 General solution We are going to solve the following initial value problem ( IVP ): ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x = 0 x ∈ (-∞ , ∞ ) , t > ρ ( x, 0) = ρ ( s ) Theorem The general solution of the above IVP is given by ρ ( x, t ) = ρ ( s ) , where the s is defined by s = x- c ( ρ ) · t, if the discriminant Δ negationslash = 0 and the Δ is defined by Δ ≡ 1 + c prime ( ρ ) ρ prime ( s ) · t . 2 Proof Notice that ∂ρ ∂t = ρ prime ( s ) ∂s ∂t = ρ prime ( s ) parenleftbigg- c prime ( ρ ) ∂ρ ∂t · t- c ( ρ ) parenrightbigg and ∂ρ ∂x = ρ prime ( s ) ∂s ∂x = ρ prime ( s ) parenleftbigg 1- c prime ( ρ ) ∂ρ ∂x · t parenrightbigg Substituting in the conservation equation, we have ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x = ρ prime ( s ) bracketleftbiggparenleftbigg- c prime ( ρ ) ∂ρ ∂t · t- c ( ρ ) parenrightbigg + c ( ρ ) parenleftbigg 1- c prime ( ρ ) ∂ρ ∂x · t parenrightbiggbracketrightbigg = ρ prime ( s ) c prime ( ρ ) · t bracketleftbigg ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x bracketrightbigg . This leads to that bracketleftbigg ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x bracketrightbigg (1 + c prime ( ρ ) ρ prime ( s ) · t ) = 0 Then the conservation equation is satisfied, i.e. bracketleftbigg ∂ρ ∂t + c ( ρ ) · ∂ρ ∂x bracketrightbigg = 0 , if the discriminant Δ negationslash = 0. 2.2 Characteristic lines In this section we will study the new variable s defined by s = x- c ( ρ ( s )) · t This is a non-linear relation. The methodology of linearization is to study the arbitrary, but fixed s . It is clear that s = x , at t = 0. Thus the physical meaning of s is the initial position. For each given s on x-axis ( s is a constant now ), the equation x- c ( ρ ( s )) · t = s ( constant ) is, in fact, a straight line on the t- x plane, which is called the phase plane, as illustrated in Figure 2.2.1r la ....
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- Spring '11
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