Materiale diviso per argomento di Mathematical and Numerical Methods in Engineering per il corso di Biomedical Engineering presso Politecnico di Milano. Il documento tratta: 1 – The Transport Equation Introduction and basic finite difference schemes Model problems · Computational grid · FD approximation of derivatives · FEC, BEC and upwind schemes (ref. Matematica Numerica, ch. 12, sec. 5) 1. Model problems 1.1 Linear transport problem Find u( t , x…
Materiale diviso per argomento di Mathematical and Numerical Methods in Engineering per il corso di Biomedical Engineering presso Politecnico di Milano. Il documento tratta: 1 – The Transport Equation Introduction and basic finite difference schemes Model problems · Computational grid · FD approximation of derivatives · FEC, BEC and upwind schemes (ref. Matematica Numerica, ch. 12, sec. 5) 1. Model problems 1.1 Linear transport problem Find u( t , x…
Qualità dell’importazione: il testo è stato estratto direttamente dal documento originale.
Passaggi rappresentativi riconosciuti nelle diverse parti del materiale. Il testo completo resta presente nella pagina per la ricerca, mentre l’anteprima compatta rende più semplice la lettura.
1 – The Transport Equation Introduction and basic finite difference schemes Model problems · Computational grid · FD approximation of derivatives · FEC, BEC and upwind schemes (ref. Matematica Numerica, ch. 12, sec. 5) 1. Model problems 1.1 Linear transport problem Find u( t , x ) such that ∂t u+a ∂ x u=f , x ∈ ℝ , t >0 u( 0 , x )=u0 ( x ), x ∈ ℝ where a is the (constant) transport velocity and u 0 the initial datum. For f =0, along each line x=x 0 +a t the solution is constant: these lines are the characteristic lines of the problem. Hence the solution is the initial profile shifted with speed a : u( t , x )=u0 ( x − at ) The profile is translated without deformation: to the right if a>0 , to the left if a<0 . Figure 1 — Linear transport of a Gaussian profile: the shape moves along the characteristics x = x₀ + at without changing. With a source term, integrating along the characteristic through ( t , x ) gives (Duhamel): t u( t , x )=u0 ( x − at )+∫ f ( s , x − a( t − s ) ) ds 0 1 – The transport equation — 1
1.2 Transport–diffusion problem ∂t u+a ∂ x u − ε ∂ xx u=f The term a ∂ x u is the transport (trasporto) and ε ∂ xx u , with ε >0 , the diffusion (diffusione). The profile still moves in the direction of a, but it also spreads out and its peak decreases. When ε is small compared to ∣ a ∣ the problem is transport-dominated. Figure 2 — Solution with a = 1, ε = 10⁻²: transport plus progressive smoothing of the profile. 1.3 Nonlinear transport and conservation laws If the velocity is the solution itself we get the nonlinear transport problem (Burgers equation): ∂u ∂u +u =f ( t , x ), 0<t ≤ T , x ∈( α , β ) ∂t ∂x The nonlinear term can be written in conservative form: 1 u ∂ x u= ∂ x ( u2 ) 2 Each value of u travels with its own speed, so higher values overtake lower ones: the front steepens and, in finite time, a discontinuity (shock) can form. Figure 3 — Inviscid Burgers equation: steepening of the front before the formation of a shock. A more general conservation law is written in terms of a flux function F ( u ): 1 – The transport equation — 2
Meaning of c. In one time step the characteristic line leaving ( t n , x i −1 ) travels the distance aτ =c h : c is the fraction of a cell crossed by the information in one time step. Figure 5 — Meaning of c: horizontal distance aτ covered by the characteristic in one time step, compared to h. 3. Approximation of the derivatives 3.1 Time discretization: forward and backward The reference time is the time level at which the derivative is evaluated: Forward (reference time t n): 1 ∂t u( t n , x i )= ( u( t n+1 , x i )− u( t n , x i ) ) +O( τ ) τ Backward (reference time t n): 1 ∂t u( t n , x i )= ( u( t n , x i )− u( t n −1 , x i ) ) +O( τ ) τ Backward, shifted by +1 (reference time t n+1): 1 ∂t u( t n+1 , x i )= ( u( t n+1 , x i )− u( t n , x i ) ) +O( τ ) τ The difference quotient is the same in the first and third formula: what changes is the time level at which the rest of the equation is written. This is what distinguishes forward (explicit) from backward (implicit) schemes. 3.2 Space discretization: centered and one-sided Centered 1 ∂ x u( t n , x i )=D ⁰ x u( t n , x i )+O( h2 ), D ⁰ x u( t n , x i )= ( u( t n , x i+1 )− u( t n , x i −1 ) ) 2h Forward 1 ∂ x u( t n , x i )=D ⁺ x u( t n , x i )+O( h ), D ⁺ x u( t n , x i )= ( u( t n , x i+1 )− u( t n , x i ) ) h Backward 1 ∂ x u( t n , x i )=D ⁻ x u( t n , x i )+O( h ), D ⁻ x u( t n , x i )= ( u( t n , x i )− u( t n , x i −1 ) ) h 1 – The transport equation — 4
Figure 7 — Foot of the characteristic at t_n: on the left of x_i if a > 0, on the right if a < 0. UPWIND scheme (one-sided “on the upwind side”) a>0: 1 n+1 a τ ( U i − U in ) + ( U in − U in−1 ) =f ( t n , x i ) h a<0: the upwind point of x i is x i+1 1 n+1 ( U − U in ) + ah ( U i+1 τ i n − U in ) =f ( t n , x i ) For a>0 and f =0, with c=aτ /h, the scheme reads aτ n U in+1=U in − h ( U i −U in−1 ) ⟺ U in+1=c U in−1 +(1− c )U in The new value is a weighted average of the two upwind values: it is exactly the linear interpolation n n between U i −1 and U i at the foot of the characteristic (Figure 7), which lies at distance ch from x i and (1 − c )h from x i −1. 4.4 Direction of propagation The one-sided difference must use the node on the side from which the flow comes (upwind); the node on the other side (downwind) must not be used. Figure 8 — Nodes used by the upwind scheme depending on the sign of a. Warning. Using the downwind node (e.g. x i+1 when a>0) is wrong: the scheme would take information from where the flow is going, not from where it comes. 1 – The transport equation — 6
forward in time backward in time one sided UPWIND (UP, UW) — centered EAC = FEC EIC = BEC Update formulas with the source term ( c=aτ /h ): Scheme Update formula Type Orders c n O(τ) in time, O(h²) FEC (EAC) U in+1=U in − 2 ( U i+1 −U in−1 ) +τ f ( t n , x i ) explicit in space c n+1 O(τ) in time, O(h²) BEC (EIC) U in+1 + 2 ( −1 ) =U i +τ f ( t n+1 , x i ) U i+1 − U in+1 n implicit in space O(τ) in time, O(h) Upwind, a > 0 U in+1=U in − c ( U in − U in−1 ) +τ f ( t n , x i ) explicit in space O(τ) in time, O(h) U in+1=U in − c ( U i+1 − U in ) +τ f ( t n , x i ) n Upwind, a < 0 explicit in space 1 – The transport equation — 8
Prima pagina del documento.
