Numerical integration is the family of algorithms that approximate definite integrals and advance differential equations in time when closed-form solutions are unavailable. In physics simulation it denotes the time-stepping schemes (such as explicit and implicit Euler, Verlet, and Runge-Kutta methods) that integrate equations of motion to update positions and velocities each frame. The choice of scheme balances accuracy, numerical stability, and computational cost.
- Numerical Integration approximates integrals and advances equations of motion when no analytic solution exists, forming the time-stepping core of Physics Simulation.
- It underpins Rigid Body Dynamics by integrating forces into velocities and velocities into positions each step.
- Scheme selection trades off accuracy against numerical stability, a concern shared with Finite Element Analysis and broader Simulation.
Definition wikilinks
- Related foundational concepts include Real-Time Rendering and Collision Detection.
Overview
- Numerical integration converts continuous differential equations into discrete update rules, allowing computers to evolve a system state forward in fixed or adaptive time steps.
- In interactive simulation, explicit schemes are cheap but can become unstable at large time steps, whereas implicit schemes are more stable at greater cost.
- The accuracy of a scheme is characterised by its order; higher-order methods reduce error per step but require more force evaluations.
Mechanisms
- Explicit Euler: simplest first-order scheme, updating state directly from current derivatives.
- Semi-implicit and Verlet integration: widely used in games for stable, energy-conserving particle and rigid-body updates.
- Runge-Kutta family: higher-order multi-stage schemes offering improved accuracy for stiff or sensitive systems.
- Adaptive step sizing: error estimation drives the step length to maintain a target tolerance.
- Implicit solvers: solve a system each step for unconditional stability in stiff problems such as cloth and soft bodies.
Applications
- Advancing equations of motion in real-time game and robotics physics engines.
- Time integration within finite-element and continuum-mechanics solvers.
- Trajectory and orbital propagation in scientific and engineering simulation.
- Particle systems, fluids, and deformable bodies in spatial-computing experiences.