Euler Integration Stability

Calculus Level pending

Consider the formula for explicit Euler integration of a complex exponential:

y ( t ) = y ( t − Δ t ) + y ˙ ( t − Δ t ) Δ t y ( t ) = e λ t y ˙ ( t ) = λ y ( t ) λ = a + j b j = − 1 y(t) = y(t - \Delta t) + \dot{y}(t - \Delta t) \, \Delta t \\ y(t) = e^{\lambda \, t} \\ \dot{y}(t) = \lambda \, y(t) \\ \lambda = a + j \, b \\ j = \sqrt{-1}

The complex exponential is processed sequentially in time with a discrete time interval Δ t \Delta t . In order for the algorithm to be stable (non-divergent) for a given Δ t \Delta t , the following condition must hold true:

∣ y ( t ) y ( t − Δ t ) ∣ ≤ 1 \large{\Big| \frac{y(t)}{y(t - \Delta t)} \Big | \leq 1}

In other words, the modulus of the ratio of the present value of y y to the previous value of y y must be less than or equal to 1 1 . Note that this ratio will generally be a complex number.

Let's examine the case where Δ t = 1 \Delta t = 1 . Suppose we were to plot all non-divergent ( a , b ) (a,b) points on a two-dimensional plane, with the horizontal axis corresponding to the a a values and the vertical axis corresponding to the b b values. This is known as the region of stability.

What is the area of the resulting region?


The answer is 3.14159.

Steven Chase by Steven Chase , 37, USA

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1 solution

Steven Chase
May 26, 2019

Re-write the first equation:

y ( t ) = y ( t − Δ t ) + y ˙ ( t − Δ t ) Δ t y ( t ) = y ( t − Δ t ) + λ y ( t − Δ t ) Δ t y ( t ) y ( t − Δ t ) = 1 + λ Δ t = 1 + λ = ( 1 + a ) + j b \large{y(t) = y(t - \Delta t) + \dot{y}(t - \Delta t) \, \Delta t \\ y(t) = y(t - \Delta t) + \lambda \, y(t - \Delta t) \, \Delta t \\ \frac{y(t)}{y(t - \Delta t)} = 1 + \lambda \, \Delta t = 1 + \lambda = (1 + a) + j b}

Examine the boundary of the stability region:

∣ 1 + λ ∣ 2 = 1 ∣ ( 1 + a ) + j b ∣ 2 = 1 ( 1 + a ) 2 + b 2 = 1 \large{|1 + \lambda|^2 = 1 \\ |(1 + a) + j b|^2 = 1 \\ (1 + a)^2 + b^2 = 1}

Clearly, the boundary of the stability region is a circle centered at ( a , b ) = ( − 1 , 0 ) (a,b) = (-1,0) with a radius of 1 1 . The enclosed area is therefore π \pi . Note that the algorithm only converges for a < 0 a < 0 , which corresponds to exponential decay. However, the stability region does not incorporate the entire left-hand plane, which means that stability is conditional on the size of the time step in relation to λ \lambda . The implicit Euler method is more stable alternative to the explicit Euler method.

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