Model Equation: Analytical and Numerical Solutions

We are discussing the model equation: du/dt = λu with u(0) = 1.
λ can be any complex number. The exact solution is u(t) = exp(λt).
The stability of the numerical solution depends on hλ, and the numerical method used.
Click on the hλ-plane canvas to select a value of λ.
The plot canvas compares the real parts of the numerical solution and the exact solution.

Numerical Solver: Step size (h): Number of steps to display (N):

Growth functions of the given numerical ODE solving methods
Method	Growth function: G(hλ)
Explicit: Euler Forward Step (RK₁)	1 + hλ
Explicit: RK₂	1 + hλ + (hλ)²/2
Explicit: RK₃	1 + hλ + (hλ)²/2 + (hλ)³/6
Explicit: RK₄	1 + hλ + (hλ)²/2 + (hλ)³/6 + (hλ)⁴/24
Implicit: Euler Backward Step	1/(1 - hλ)
Implicit: Euler Symmetrical	(1 + hλ/2)/(1 - hλ/2)
Implicit: Hammer-Hollingsworth	[1 + hλ/2 + (hλ)²/12] /[1 - hλ/2 + (hλ)²/12]