substepping

Substepping is a technique in numerical simulation in which a single time step is subdivided into multiple smaller steps. The simulation state is advanced through these substeps within one frame or overall time increment. Substepping is used to improve accuracy and stability when integrating dynamics that are stiff or highly constrained, or when the main time step is large relative to the fastest dynamics of the system. It is common in real-time physics engines and in numerical solvers for ordinary differential equations.

Implementation typically involves selecting a substep size or a target number of substeps. If dt is the frame time, the algorithm computes N = ceil(dt/dt_sub) or uses a fixed N, and then applies N successive integration steps of size dt_sub (the final substep may be shorter to fit exactly). For each substep, forces are evaluated, motion is integrated, and constraints or contacts are resolved. Some designs perform constraint solving at every substep to maintain stability of contacts.

Benefits include increased numerical stability, reduced energy drift, and more accurate collision handling in dynamics simulations with stiff interactions or large time steps. Substepping can allow a simulation to run with a larger frame time without sacrificing accuracy. The main trade-offs are higher computational cost and potential complexity in ensuring consistent state across substeps, especially for event-based logic and asynchronous inputs.

Substepping is widely used in real-time physics, animation, and some numerical ODE solvers. It is one tool among several time integration strategies, including fixed-step, adaptive, and implicit methods. While substepping improves stability in many cases, it does not replace the need for appropriate time integration choices based on the system's characteristics.