Fixedstep

Fixedstep refers to numerical integration and simulation methods that advance time in uniform increments, t_{n+1} = t_n + h, where h is a constant step size. This contrasts with variable-step methods that adapt the step size to control error. Fixed-step methods are widely used for solving ordinary differential equations in engineering and scientific computing, including explicit Euler, improved Euler (Heun), midpoint, and higher-order Runge-Kutta schemes. In discrete-time simulations, fixed time steps align with the sampling rate of digital systems and hardware.

In control systems and real-time simulation, fixed-step solvers offer determinism and reproducibility, enabling predictable CPU timing and easier synchronization with hardware-in-the-loop testing. They are also common in digital signal processing and embedded software, where fixed sampling rates drive processing. However, fixed-step methods must choose h small enough to resolve dynamics and maintain numerical stability; stiff systems may require prohibitively small steps or specialized stiff solvers, where explicit fixed-step methods become unstable.

Choosing a fixed step involves trade-offs: smaller h increases accuracy but raises computational cost; larger h reduces cost but may degrade stability and accuracy. In software environments like MATLAB/Simulink, users can select fixed-step solvers and specify h; in real-time systems, fixed stepping is often a design constraint.

Limitations include reduced efficiency for smooth problems that variable-step methods handle adaptively, potential resonance with system dynamics, and error accumulation over long simulations. Overall, fixed-step techniques are foundational in deterministic, time-discretized simulations and real-time applications, where consistency and timing are paramount.