General Considerations

Numerical Approximations

Differential Equations

Given an ordinary differential equation (ODE) for simulating a quantity $f$ over time $t$:

$$\frac{df(t)}{dt}=F(f(t),t)$$

where $F$ is some expression that can depend on $f$ and/or $t$. Point $f(t)$ can be propagated to the next point $f(t+\mathrm{\Delta }t)$ via explicit Euler method:

$$f(t+\mathrm{\Delta }t)\approx f(t)+\mathrm{\Delta }t\cdot F(f(t),t)$$

where $\mathrm{\Delta }t$ is the time step. The approximation can be quite crude, especially for large $\mathrm{\Delta }t$. In case of an ODE that converges towards a fixed value, this can lead to overshooting the target, which is why clamping $f$ is often advised. For ODEs of the form below:

$$\frac{df(t)}{dt}=k\cdot (f_{\infty }-f(t))k>0$$

with rate constant $k$ and target $f_{\infty }$. The target $f_{\infty }$ gets approached with exponential convergence and the clamped version of explicit Euler can be simplified with linear interpolation ($\mathrm{lerp}$):

$${\xi }_{\mathrm{progress}}=\min (k\cdot \mathrm{\Delta }t,1)$$ $$f(t+\mathrm{\Delta }t)\approx \mathrm{lerp}(f(t),f_{\infty },{\xi }_{\mathrm{progress}})$$