Lecture on Stiff Systems (Section 1-6 of Chua and Lin) ECE 546 Jan. 15, 2008

The Circuit

The State Equations Definition of Eigenvalues/Eigenvectors Single-Input Single-Output State Model Selecting capacitor voltages as state variables, ICBS

Unit Step Response To solve for c 1 and c 2

Unit Step Response (cont) Using Matlab, “Exact” or analytical solution

Definition of Stiff System System is called stiff if spread of time constants is large For given example

Short-Term Unit Step Response (Fig of Chua and Lin)

Reasonable Time Step/Grid Define reasonable time step/grid to be one in which numerical solution approximates analytical solution (with acceptable accuracy) at grid points, i.e. without requiring an excessive number of grid points

Short-Term Response Trapezoidal algorithm most accurate for predicting short term response with smallest number of grid points Forward Euler least accurate Backward Euler produces well-damped numerical solution that “lags” analytical solution

Long-Term Unit Step Response

Long-Term Response Previous grid is unreasonable –Many more points than needed to predict long- term response Larger time step needed after fast transients subside –Cannot use Forward Euler (unstable)

Long-Term Response (Larger Time Step) Forward Euler unstable

Long-Term Response Trapezoidal algorithm produces artificial oscillations when time step increased Backward Euler appears best suited for predicting long term response if we are restricted to fixed time step Other strategies possible –Use trapezoidal with h = 0.2e-7 and after fast transients subside, switch to Backward Euler with larger time step

Conclusions No single best algorithm for all systems/cases Forward Euler unstable – typically the worst choice If restricted to fixed time step, Backward Euler best (of three considered) for predicting long-term response Many, many, many other algorithms exist – continuing area of research in CS/Math