Trapezoidal Rule Properties

Calculus Level pending

Consider a decaying exponential function of time.

y ( t ) = e − t y(t) = e^{-t}

Suppose we use the Trapezoidal integration method to discretely model this function.

y k = y k − 1 + 1 2 y ˙ k − 1 Δ t + 1 2 y ˙ k Δ t y ˙ k = − y k y_k = y_{k-1} + \frac{1}{2} \dot{y}_{k-1} \Delta t + \frac{1}{2} \dot{y}_k \Delta t \\ \dot{y}_k = - y_k

In the above equation, y k y_k is the present value of the function and y k − 1 y_{k-1} is the previous value of the function. The simulated function is "monotonic" if y k y k − 1 > 0 \frac{y_k}{y_{k-1}} > 0 , and "oscillatory" if y k y k − 1 < 0 \frac{y_k}{y_{k-1}} < 0 . The function "converges" if ∣ y k y k − 1 ∣ < 1 \Big| \frac{y_k}{y_{k-1}} \Big | < 1 , and "diverges" if ∣ y k y k − 1 ∣ > 1 \Big| \frac{y_k}{y_{k-1}} \Big | > 1 . Different behaviors are exhibited for different values of the time step Δ t \Delta t .

Subject to the constraints y k y k − 1 ≠ 0 \frac{y_k}{y_{k-1}} \neq 0 and Δ t > 0 \Delta t > 0 , which types of behavior are possible?

Monotonic Convergence and Monotonic Divergence Monotonic Convergence and Oscillatory Divergence Monotonic Convergence and Oscillatory Convergence Only Monotonic Convergence

Steven Chase by Steven Chase , 37, USA

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1 solution

Karan Chatrath
Nov 3, 2020

y = e − t ⟹ y ˙ = − y y = \mathrm{e}^{-t} \ \implies \dot{y} = -y y k = y k − 1 + y ˙ k − 1 Δ t 2 + y ˙ k Δ t 2 y_k = y_{k-1} + \frac{\dot{y}_{k-1}\Delta t}{2} + \frac{\dot{y}_{k}\Delta t}{2} y k = y k − 1 − y k − 1 Δ t 2 − y k Δ t 2 y_k = y_{k-1} - \frac{y_{k-1}\Delta t}{2} - \frac{y_{k}\Delta t}{2} y k ( 1 + 0.5 Δ t ) = y k − 1 ( 1 − 0.5 Δ t ) y_k\left(1 + 0.5\Delta t\right) = y_{k-1}\left(1 - 0.5\Delta t\right) R = y k y k − 1 = 1 − 0.5 Δ t 1 + 0.5 Δ t R = \frac{y_k}{y_{k-1}} = \frac{ 1 - 0.5\Delta t}{1 + 0.5\Delta t}

R R becomes negative when Δ t > 2 \Delta t >2 . So when Δ t < 2 \Delta t<2 the behaviour is monotonic and it is oscillatory otherwise. By inspection, it can also be seen that the absolute value of R R always remains below 1, which indicates convergence.

Therefore, the types of possible behaviour are 'monotonic convergence and oscillatory convergence'.

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