Physically-based particle system

3. Numerical Solution

• ODE (Ordinary Differential Equation) formulated as an Initial Value Problem

• • ex. \(\left\{\begin{array}{l}p'(t)=v(t) \\ m v'(t) = F(p,v,t)\end{array}\right.\;\;\;,\;\;\) with \(v(0)=v_0,\; p(0)=p_0\)

• - Discretize in time \(t^k=k\,h\),\(\;\;\) \(h=\Delta t=\) time step.
• • \(\Rightarrow\) Build a discrete numerical solution \(p^k=p(t^k), v^k=v(t^k)\).

• - We can consider initially the following iterative scheme

• • \(\left\{\begin{array}{l}v^{k+1} = v^{k} + h\,F(p^k, v^k, t^k) \\ p^{k+1}=p^k+h\,v^{k+1} \end{array}\right.\)

• • Simple to implement, reasonably OK for simple examples (more details later).