Topic-based study materials for Mathematical and Numerical Methods in Engineering in the Biomedical Engineering degree programme at Politecnico di Milano. The document covers: 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
Topic-based study materials for Mathematical and Numerical Methods in Engineering in the Biomedical Engineering degree programme at Politecnico di Milano. The document covers: 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
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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
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