Why does the Euler method pendulum gain energy and swing higher, which seems to violate physics?

The energy gain is a numerical artifact, not a physical phenomenon. The Forward Euler method has a property called numerical instability for this type of ODE. Its simplistic linear extrapolation consistently overestimates the change in angular velocity, adding a small amount of energy each time step. In a real conservative system, total mechanical energy is constant. This error accumulation highlights a key limitation of low-order methods for long-term simulations.

Is RK4 always better than Euler? Why would anyone use Euler?

While RK4 is far more accurate for a given step size, it requires four function evaluations per step compared to Euler's one, making it computationally more expensive. The Euler method is still used in scenarios where simplicity and speed are paramount, and error is acceptable, such as in some real-time video game physics or for very simple systems with extremely small time steps. It also serves as a foundational concept for understanding more advanced methods.

What happens if I make the time step (h) very small for both methods?

As the time step decreases, the solution from both methods will converge toward the true solution, and the energy drift in Euler's method will become less severe over a fixed time interval. However, for Euler to match RK4's accuracy at a moderate step size, it would require a vastly smaller (and computationally prohibitive) step. This illustrates the concept of 'order': RK4's error scales with h^4, while Euler's scales only with h.

Does this simulation model a real pendulum perfectly?

No, it models an idealized, nonlinear pendulum. Key simplifications include the absence of friction or air drag (damping), a perfectly rigid and massless rod, and a point mass bob. These assumptions allow us to isolate and study the core numerical integration errors without confounding physical effects. In a real pendulum, energy would gradually decrease due to damping, not increase as seen with the Euler method.

Euler vs RK4 (Pendulum)

Numerical integration methods are essential tools for solving differential equations that lack analytical solutions, such as the motion of a nonlinear pendulum. This simulator visualizes a direct comparison between two fundamental algorithms: the simple (or Forward) Euler method and the fourth-order Runge-Kutta (RK4) method. Both methods are tasked with solving the same underlying ordinary differential equation (ODE) derived from Newton's second law for a pendulum: d²θ/dt² = -(g/L) sin(θ), where θ is the angular displacement, g is gravitational acceleration, and L is the pendulum length. This equation is simplified by neglecting air resistance and assuming a massless rod, focusing purely on the nonlinear restoring force. The simulator converts this second-order ODE into a system of two first-order ODEs: dθ/dt = ω and dω/dt = -(g/L) sin(θ). Starting from identical initial conditions and using the same fixed time step (h), the two algorithms propagate the state (θ, ω) forward in time. The Euler method uses a simple linear projection: θ_{n+1} = θ_n + h * ω_n and ω_{n+1} = ω_n + h * [-(g/L) sin(θ_n)]. In contrast, RK4 calculates four weighted intermediate slopes, providing a much more accurate estimate of the average slope over the interval. By interacting with the simulation, students can observe how the Euler method's simplicity leads to significant energy growth (amplitude increase) over time—a non-physical artifact—while RK4 maintains a nearly constant energy and period for much longer, closely approximating the true conservative system. This demonstrates critical concepts in computational physics: local truncation error, global error accumulation, stability, and the trade-off between computational cost and accuracy.

Who it's for: Undergraduate students in physics, engineering, or computational science courses studying numerical methods for differential equations or nonlinear dynamics.

Key terms

Numerical Integration
Runge-Kutta Methods
Euler Method
Ordinary Differential Equation (ODE)
Nonlinear Pendulum
Truncation Error
Energy Conservation
Phase Space

How it works

Same nonlinear pendulum θ'' = −(g/L) sin θ, same initial angle, same fixed step h: explicit Euler (θ then ω update) gains energy and phase drifts badly for large h; RK4 stays much closer to a high-resolution reference for the same work per step. Toggle h to see Euler diverge first.

Key equations

Euler: θ += ωh, ω += a(θ)h · RK4: four-stage average on (θ, ω)

Frequently asked questions

Why does the Euler method pendulum gain energy and swing higher, which seems to violate physics?: The energy gain is a numerical artifact, not a physical phenomenon. The Forward Euler method has a property called numerical instability for this type of ODE. Its simplistic linear extrapolation consistently overestimates the change in angular velocity, adding a small amount of energy each time step. In a real conservative system, total mechanical energy is constant. This error accumulation highlights a key limitation of low-order methods for long-term simulations.
Is RK4 always better than Euler? Why would anyone use Euler?: While RK4 is far more accurate for a given step size, it requires four function evaluations per step compared to Euler's one, making it computationally more expensive. The Euler method is still used in scenarios where simplicity and speed are paramount, and error is acceptable, such as in some real-time video game physics or for very simple systems with extremely small time steps. It also serves as a foundational concept for understanding more advanced methods.
What happens if I make the time step (h) very small for both methods?: As the time step decreases, the solution from both methods will converge toward the true solution, and the energy drift in Euler's method will become less severe over a fixed time interval. However, for Euler to match RK4's accuracy at a moderate step size, it would require a vastly smaller (and computationally prohibitive) step. This illustrates the concept of 'order': RK4's error scales with h^4, while Euler's scales only with h.
Does this simulation model a real pendulum perfectly?: No, it models an idealized, nonlinear pendulum. Key simplifications include the absence of friction or air drag (damping), a perfectly rigid and massless rod, and a point mass bob. These assumptions allow us to isolate and study the core numerical integration errors without confounding physical effects. In a real pendulum, energy would gradually decrease due to damping, not increase as seen with the Euler method.