Investigating the Stability Behavior of Euler's Method in First-Order Linear Initial Value Problems

Abstract

We analyze stability and convergence of the explicit Euler method when applied to first order linear IVPs of the form y'= λy + g(t). We analytically determine the region of stability of Euler's method on the complex hλ-plane to be the unit disk |1 + hλ| ≤ 1 from which we can obtain the limiting step-size h* = 2/|λ| when λ is real and negative. We choose five example test problems which include pure exponential, pure sinusoidal forcing, stiff oscillation, variable-coefficient and exponential forcing and demonstrate O(h) convergence numerically for many step sizes. We examine the behavior of the amplification factor |R(z)| = |1 + z| on the real axis and compare the method to Heun's method (RK2) and the fourth-order Runge-Kutta method (RK4). We verify numerically an analytical global error bound of the form (h|λ|/2)(e^{|λ|t} − 1) and show it to be quite conservative by 3–20×. Work-precision diagrams show RK4 to be 10–100× more accurate per function evaluation than Euler. We show the effects of stiffness by noting that large |λ| drives h* → 0 which makes Euler impossible to apply to stiff problems.