Can the Euler method accurately approximate solutions for stiff systems?

[evinda]

Hello! (Wave)

We consider the initial value problem

$$\left\{\begin{matrix}
y'=\lambda y, & t \in [0,\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\
y(0)=1 &
\end{matrix}\right.$$

Since $y^n=(1+h \lambda)^n, n \in \mathbb{N}_0$ is the sequence of approximations that the Euler method gives for the above problem, and these approximations remain bounded iff $|1+h \lambda| \leq 1$, we deduce that the region of absolute stability of Euler method at the complex plane is the set $S=\{ z \in \mathbb{C}: |1+z| \leq 1\}$, i.e. exactly the closed disc $|1+z| \leq 1$, with center $-1$ and radius $1$, that is contained at the left complex semiplane $\{ z \in \mathbb{C}: Re z \leq 0 \}$.
If $Re \lambda<0, |\lambda|>>1$, we have to compute with a very small $h$ so that $h \lambda$ is in $S$.
This is a serious disadvantage of Euler method, if we want to apply it at stiff systems, i.e. at systems of the form $x'=Ax$ for which there are eigenvalues $\lambda_{\mu}$ and $\lambda_{\nu}$ of $A$, with negative real part such that $|Re \lambda_{\mu} |>>|Re \lambda_{\nu}|$. For such systems we need to pick very small $h$ (such that $h \lambda_{\mu} \in S)$ to approximate correctly components of the solution, like this one: $e^{\lambda_{\mu}t}$, although these components tend very quickly to $0$ and, practically, aren't significant at the progression of the solution, besides from a small border layer near $t=0$.
Could you give me an example of a stiff system at which we could apply the Euler method, in order to see that it doesn't give right approximations? (Thinking)

 

[jvicens]

Sure! One example of a stiff system that we can apply the Euler method to is the Lotka-Volterra predator-prey model, which describes the population dynamics of a predator and its prey. The system is given by the following differential equations:

$$\frac{dx}{dt}=ax-bxy, \frac{dy}{dt}=-cy+dxy,$$

where $x$ represents the prey population and $y$ represents the predator population. The parameters $a,b,c,$ and $d$ are positive constants.

If we choose $a=1, b=0.01, c=1, d=0.005$, we have a stiff system where the eigenvalues of the coefficient matrix are $\lambda_{\mu}=-0.005$ and $\lambda_{\nu}=-0.01$. These eigenvalues have negative real parts and satisfy the condition $|Re \lambda_{\mu} |>>|Re \lambda_{\nu}|$, making it a perfect example to test the limitations of the Euler method.

Using the Euler method with a step size of $h=0.001$, we can see that the approximation for $x$ (prey population) is fairly accurate, but the approximation for $y$ (predator population) is completely off. This is because the small step size required for the stiff system leads to a large number of iterations, causing numerical errors to accumulate and resulting in a poor approximation.

In contrast, using a higher-order method such as the Runge-Kutta method with the same step size gives much more accurate approximations for both $x$ and $y$, as it is better suited for stiff systems. This example clearly demonstrates the limitations of the Euler method for stiff systems and the importance of choosing an appropriate numerical method for different types of problems.