Stiff equations are differential equations where certain numerical methods, like
explicit Euler's method, require very small step sizes to maintain stability, even
though the solution itself may change slowly. These equations are characterized by
the presence of rapidly decaying components or vastly differing timescales.
Characteristics of Stiff Equations
1. Large variation in timescales: A stiff system has some components that change
very quickly and others that change slowly. To capture the behavior of the fast-
changing components, explicit methods need extremely small time steps, even
though the slow components don't need such precision.
2. Rapidly decaying solutions: Stiff equations often have solutions that decay rapidly
to equilibrium. A large step size in these cases could lead to numerical instability
because the fast components may cause the solution to diverge or oscillate.
3. Numerical instability with large step sizes: Explicit methods, like Euler's method,
require very small step sizes for stiff equations to avoid instability. Even though the
step size could theoretically be larger based on the slow components, the presence
of fast dynamics forces a much smaller step size for stability.
Example of a Stiff Equation
A classic example of a stiff equation is:
dy −x
=−1000 y +3000−2000 e
dx
- This equation has a rapidly decaying transient term and a slowly varying term.
- The exact solution has a fast component that decays quickly and a slow
component that evolves smoothly. To capture the fast decay, explicit methods would
need very small steps, even if you're mostly interested in the slower behavior.
Why Stiffness Affects Stability
In stiff equations, the fast-changing components force the numerical method to use
a small step size, even if you're more interested in capturing the slower behavior of
the system. For instance, the eigenvalues of the system's Jacobian may have large
negative real parts, causing the solution to exhibit rapid decays, which explicit
methods are not well-equipped to handle with larger steps.
For explicit Euler's method, applying it to a test equation of the form y ' =λ y (where
λ is large and negative) yields:
y n +1=( 1+h λ ) y n
For stability, (|1+ h λ|≤1 ) , so h must be extremely small when λ is large, leading to
impractically small step sizes in stiff problems.
### Methods for Solving Stiff Equations
For stiff problems, implicit methods are generally preferred because they have
better stability properties, even for large step sizes. Some common methods
include:
1. Implicit Euler Method:
- This method is **A-stable**, meaning it can handle stiff problems without step
size restrictions. It involves solving a system of algebraic equations at each step but
offers superior stability for stiff equations.
y n +1= y n +h f ( y n+1 , t n +1)
2. Backward Differentiation Formulas (BDF):
- BDF methods are multistep implicit methods that are particularly effective for
stiff problems. They allow larger step sizes while maintaining stability.
3. Semi-implicit Methods:
- These methods handle the stiff components implicitly and the non-stiff ones
explicitly, making them a good balance between efficiency and stability.
MATLAB for Stiff Equations
MATLAB provides solvers specifically designed for stiff problems, such as:
-ode15s: This solver is used for stiff problems and differential-algebraic equations.
-ode23s: Another stiff solver, suitable for moderate accuracy problems.
These solvers use implicit methods to handle the stiffness and allow you to use
larger step sizes without compromising stability.
Conclusion
Stiff equations require special attention because explicit methods like Euler's
method can become unstable unless very small step sizes are used. Implicit
methods and specialized solvers are typically used to solve stiff equations
efficiently, allowing for larger time steps while maintaining stability.
