Definition
In an explicit scheme, the solution at the next time step t_{n+1} is computed directly from quantities that are already known at time step t_n.
General form
| u_{n+1}=u_n + Δt \cdot F(u_n, t_n) |
Everything on the right side is known → you just evaluate the formula.
Characteristics
✅ No system of equations to solve
✅ Very fast
✅ Easy to implement
❌ Conditionally stable → time step must be small
(e.g., Courant condition or diffusion stability condition)
Typical use
Wave propagation
Advection / transport
Lagrangian particle methods
Many CFD solvers
Explicit hydraulic fracturing simulators (early-time PKN/Perkins-Kern)