In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations.
[1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.
When used as a method for advection equations, or more generally hyperbolic partial differential equations, it is unstable unless artificial viscosity is included.
The abbreviation FTCS was first used by Patrick Roache.
[2][3] The FTCS method is based on the forward Euler method in time (hence "forward time") and central difference in space (hence "centered space"), giving first-order convergence in time and second-order convergence in space.
For example, in one dimension, if the partial differential equation is then, letting
, the forward Euler method is given by: The function
must be discretized spatially with a central difference scheme.
can be explicitly computed (no need of solving a system of algebraic equations) if values of
The FTCS method is often applied to diffusion problems.
As an example, for 1D heat equation, the FTCS scheme is given by: or, letting
: As derived using von Neumann stability analysis, the FTCS method for the one-dimensional heat equation is numerically stable if and only if the following condition is satisfied: Which is to say that the choice of
must satisfy the above condition for the FTCS scheme to be stable.
for one-, two-, and three-dimensional applications, respectively.
[4] A major drawback of the FTCS method is that for problems with large diffusivity
, satisfactory step sizes can be too small to be practical.