Comparison of Numerical Integrators for N-Body Simulation of the Solar System: Euler, Velocity Verlet, RK4 and Adaptive RK45

Simulating planetary motion requires numerical integration of the gravitational N-body problem, which has no closed-form solution for more than two bodies. We built an interactive, browser-based three-dimensional solar-system simulator that runs four integrators in parallel from identical initial conditions: explicit Euler, velocity Verlet, the classical fourth-order Runge-Kutta method (RK4), and the adaptive Dormand-Prince RK45 method. Initial conditions come from JPL’s approximate Keplerian elements at epoch J2000, and calculations use astronomical units, years and solar masses. We compare the methods by relative energy error, angular-momentum error and position error against a high-accuracy reference solution. Over 100 simulated years with a 1.83-day step, explicit Euler loses 18% of its energy and its orbits spiral outward, velocity Verlet keeps energy error near 3.5×10⁻⁷ and conserves angular momentum to machine precision, fixed-step RK4 reaches 1.7×10⁻⁵ with a slow upward drift, and adaptive RK45 reaches 6.8×10⁻¹². RK4 shows approximately fourth-order convergence in position. The results show that no single method wins on every measure: Verlet is the more robust choice for long, coarse-step runs, while RK4 and RK45 are more accurate at fine steps. The software, tests and data are open source. Code: https://github.com/wabix2/realistic-solar-system (folder rk4-labs). Live demo: https://rk4-labs.vercel.app