Many methods for numerically solving ordinary differential equations are either Runge-Kutta methods or linear multistep methods.

These methods can either be explicit or implicit.

The table below shows the four combinations of these categories and gives some examples of each.

Runge-Kutta methods advance the solution of a differential equation one step at a time.

That is, these methods approximate the solution at the next time step using only the solution at the current time step and the differential equation itself.

Linear multistep methods approximate the solution at the next time step using the computed solutions at the latest several time steps.

Explicit methods express the solution at the next time step as an explicit function of other information, not including the solution itself.

The solution at the next time step appears on only one side of the equations.

Implicit methods express the solution at the next time step as a function of other information including the solution itself.

The solution at the next time step appears on both sides of the equations.

Additional work needs to be done to solve for the solution.

More on explicit vs implicit methods here.

In the table above, ERK stands for, not surprisingly, explicit Runge-Kutta methods.

DIRK stands for diagonally implicit Runge-Kutta.

SDIRK stands for singly diagonally implicit Runge-Kutta.

BDF stands for backward difference formulas.

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