Page 202 - Numerical Methods for Chemical Engineering

Accuracy and stability of single-step methods 191

that are continually introduced do not accumulate but remain manageable, as their effects
become less and less important at later times. By contrast, if any |µ j | > 1, round-off errors
grow exponentially and may drown out the true response with random noise.

Stiff systems from discretized PDEs

These issues of stability and error rejection become very important for stiff systems, as
the time step must be chosen to accommodate the largest eigenvalue (fastest mode). We
have seen above an example of a stiff system from chemical kinetics, but another important
source of stiff systems is the simulation of time-dependent PDEs such as the diffusion
equation
2
∂ϕ ∂ ϕ
= (4.173)
∂t ∂x 2
The finite difference method yields the linear ODE system, ϕ j ≡ ϕ(x j ),
1
dϕ j ϕ j−1 − 2ϕ j + ϕ j+1
= ˙ ϕ =− Aϕ
dt ( x) 2 ( x) 2
 
2 −1
 −1 2 −1 


−1 2 ... (4.174)
 
A =  
... ...
 
 −1 
−1 2
The eigenvalues and eigenvectors Aw [k] = λ k w [k] of this diffusion matrix are

kπ kπ j
2 [k]
λ k = 4 sin w j = sin k = 1, 2,..., N (4.175)
2(N + 1) N + 1
The largest (fastest mode) eigenvalue occurs for k = N,

Nπ [N] Nπ j
2
λ N = 4 sin ≈ 4 w j = sin ≈ sin(π j) (4.176)
2(N + 1) N + 1
The wavelength of this eigenvector is the spacing of the grid; i.e., it describes variations on
the smallest length scale that can be resolved by the grid (Figure 3.5). The smallest (slowest)
eigenvalue occurs for k = 1,
2
π π π [1] π j

2
λ 1 = 4 sin ≈ 4 = w j = sin (4.177)
2(N + 1) 2(N + 1) N 2 N + 1
The slowest mode eigenvector describes variations across the entire domain on the largest
length scale that can be resolved by the grid.
The condition number in the limit of large N is
4N 2
λ N
κ = ≈ (4.178)
λ 1 π
Unless the number of grid points is very small, the set of first order ODEs obtained from
discretizing a partial differential equation is very stiff.