Definition
In an implicit scheme, the solution at time step n+1n+1 depends on unknown values at time step n+1n+1.
General form
| u_{n+1}=u_n + Δt \cdot F(u_{n+1}, t_{n+1}) |
The unknown appears on both sides → you must solve an equation or solve a matrix system.
Characteristics
❗ Requires solving linear or nonlinear systems (Newton, LU, iterative solvers)
✅ Can use large time steps
✅ Often unconditionally stable
✅ Allows solving “stiff” problems
❌ Much more computationally expensive
Typical use
Heat conduction (diffusion equation)
Reservoir simulation pressure equation (IMPES / fully-implicit)
Geomechanics and poroelasticity
Structural mechanics (FEM)
Slow, diffusive or stiff processes