Numerical integration of ODE
General formulation: \(u'(t)=\mathcal{F}(u,t)\), \(u(t)=(p(t),v(t))\).
- Explicit Euler
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- \(u^{k+1}=u^{k}+\Delta t\,\mathcal{F}(u^k,t^k)\)
- (+) Easy to implement
- (-) Worst scheme in all cases (divergence, low accuracy)
- Explicit Runge-Kutta
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- \(u^{k+1}=u^{k}+\Delta t\,\sum_j \alpha_j k_j\)
- (+) Good accuracy
- (+) Efficient to apply
- (+/-) Stability OK for non-stiff problem, diverge on stiff problem
- (-) Artificial damping for constant energy system
- Implicit methods
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- \(u^{k+1}=u^{k}+\Delta t\,\mathcal{F}(u^{k+1},t^{k+1})\)
- (+) Good to deal with stiff problem - very stable
- (-) Add numerical damping (converge even if solution oscillates)
- (-) Hard/computationally costly to apply on non linear problem
- Symplectic integrator
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- \(v^{k+1}=v^{k}+\Delta t\,F^k/m\)
- \(p^{k+1}=p^{k}+\Delta t\,v^{k+1}\)
- (+) Handle well constant energy system, preserves energy (Hamiltonian systems)
- (+) Simple and efficient to implement
- (-) Less accurate than RK
- (-) Diverge on stiff problem