Main idea

In rocket trajectory simulation, you're basically doing a "particle" simulation of a discrete number of particles to know where they may end up after a certain time. You could do this in a continuum by propagating a probability distribution of where the particle could be over time.

Motivation

Technical Details

Derivation of PDE

Symbols:

• $\mathbf{X}(t)$: a state vector in the state space of the system.
• $P(\mathbf{X}, t)$: the probability density of observing a state $\mathbf{X}$ at time $t$.
• $\mathbf{f}(\mathbf{X}, t)$: deterministic system response.
• $\mathbf{g}(\mathbf{X}, t)$: stochastic system response.
• $\Delta x_{i}, \Delta t$: Discrete steps in state and time the computation domain.
• $P_{i,j,\dots}^k$: probability of observing a discrete state $X_{i,j,\dots}$ at timestep $t_{k}$.

Deterministic differential equation of the state vector: $$\dot{\mathbf{X}}=\mathbf{f}(\mathbf{X}, t) \qquad (1)$$ By introducing a bit of noise (wind gusts, etc) in the form of a multivariate Wiener process $\mathbf{W}$, we get a stochastic differential equation of the state vector: $$d\mathbf{X}=\mathbf{f}(\mathbf{X}, t)dt + \mathbf{g}(\mathbf{X}, t)\odot d\mathbf{W} \qquad (2)$$ Applying the Fokker-Planck equation to $(2)$:

$$\frac{\partial P(\mathbf{X}, t)}{\partial t} =-\frac{\partial}{\partial \mathbf{X}}\left[\mathbf{f}(\mathbf{X}, t)P(\mathbf{X}, t)\right] + \frac{\partial^2}{\partial \mathbf{X}^2}\left[\frac{1}{2}\mathbf{g}^2(\mathbf{X}, t)P(\mathbf{X}, t)\right]$$

Finite Volume Discretisation

Conservation of probability means that each discrete state-space volume $\Omega$ has a time-change in probability equal to the probability flux along the boundary $\partial \Omega$: $$\frac{\partial}{\partial t} \int_{\Omega} P(\mathbf{X}, t),dV=-\oint_{\partial\Omega} \mathbf{J}(\mathbf{X}, t)\cdot d\mathbf{S}$$ The Fokker-Planck PDE can be written in conservative form as: $$\frac{\partial P(\mathbf{X}, t)}{\partial t} = \nabla_{\mathbf{x}}^2\left[\tfrac{1}{2}\mathbf{g}^2(\mathbf{X, t})P(\mathbf{X}, t) \right] - \nabla_{\mathbf{x}}\cdot[\mathbf{f}(\mathbf{X}, t)P(\mathbf{X}, t)]$$ We can derive the finite-volume form of the conservative equation: $$ \int_{\Omega}\frac{\partial P(\mathbf{X}, t)}{\partial t},dV = \int_{\Omega}\nabla_{\mathbf{x}}^2\left[\tfrac{1}{2}\mathbf{g}^2(\mathbf{X}, t)P(\mathbf{X}, t) \right],dV - \int_{\Omega}\nabla_{\mathbf{x}}\cdot[\mathbf{f}(\mathbf{X}, t)P(\mathbf{X}, t)],dV$$ $$\frac{\partial}{\partial t}\int_{\Omega}P(\mathbf{X}, t),dV = \oint_{\partial\Omega}\nabla_{\mathbf{x}}\left[\tfrac{1}{2}\mathbf{g}^2(\mathbf{X}, t)P(\mathbf{X}, t) \right]\cdot d\mathbf{S} - \oint_{\partial\Omega}(\mathbf{f}(\mathbf{X}, t)P(\mathbf{X}, t))\cdot d\mathbf{S}$$ $$\frac{\partial}{\partial t}\int_{\Omega}P(\mathbf{X}, t),dV = \oint_{\partial\Omega}\left(\tfrac{1}{2}\nabla_{\mathbf{x}}\left[\mathbf{g}^2(\mathbf{X}, t)P(\mathbf{X}, t) \right] - \mathbf{f}(\mathbf{X}, t)P(\mathbf{X}, t)\right)\cdot d\mathbf{S}$$

In a discrete hypercube in $N$-dimensional state-space ($V$ is the cell volume, $S$ is the area of every wall of the cell):

Discretizing the spatial derivative as the difference between neighboring volumes: