Explicit Time Integration
Explicit methods compute the next state directly from known current values, cheaply per step but limited by a stability time-step bound.
Marching forward directly
An explicit time integrator computes the solution at the next time level using only quantities already known at the current (and earlier) levels. Forward Euler, u_{n+1} = u_n + dt f(u_n), is the simplest example; explicit Runge-Kutta methods build higher accuracy from several such evaluations. Each step is cheap because no system of equations must be solved: the update is an explicit formula.
This simplicity and low per-step cost make explicit methods the default for wave-dominated and hyperbolic problems, and for large kinetic simulations where an implicit solve would be prohibitive.
The stability constraint
The price is a stability limit on the time step. For explicit schemes the step size is bounded by the fastest process in the system: the Courant-Friedrichs-Lewy (CFL) condition ties dt to the grid spacing divided by the fastest wave or particle speed. Diffusive terms impose an even tighter dt proportional to the square of the grid spacing. Exceed the bound and the solution blows up.
When explicit wins
Explicit methods are ideal when the time scales of interest are comparable to the stability limit anyway, so no accuracy is wasted. They parallelize trivially because each update is local. When the physics contains much faster, uninteresting time scales (stiffness), the stability limit forces tiny steps and implicit or semi-implicit methods become preferable.
- Next state computed by direct formula, no linear solve
- CFL bound: dt limited by fastest wave or particle speed
- Diffusion imposes dt proportional to grid spacing squared
- Excellent parallel scaling from purely local updates
The choice between explicit and implicit integration is one of the most consequential in a simulation, balancing per-step cost against the number of steps stability allows.