In numerical analysis, the FTCS (ForwardTime CentralSpace) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations.^{[1]} It is a firstorder 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 equation, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache.^{[2]}^{[3]}
The method
The FTCS method is based on central difference in space and the forward Euler method in time, giving firstorder convergence in time and secondorder convergence in space. For example, in one dimension, if the partial differential equation is

\frac{\partial u}{\partial t} = F\left(u, x, t, \frac{\partial^2 u}{\partial x^2}\right)
then, letting u(i \,\Delta x, n\, \Delta t) = u_{i}^{n}\,, the forward Euler method is given by:

\frac{u_{i}^{n + 1}  u_{i}^{n}}{\Delta t} = F_{i}^{n}\left(u, x, t, \frac{\partial^2 u}{\partial x^2}\right)
The function F must be discretized spatially with a central difference scheme. This is an explicit method which means that, u_{i}^{n+1} can be explicitly computed (no need of solving a system of algebraic equations) if values of u at previous time level (n) are known. FTCS method is computationally inexpensive since the method is explicit.
Illustration: onedimensional heat equation
The FTCS method is often applied to diffusion problems. As an example, for 1D heat equation,

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}
the FTCS scheme is given by:

\frac{u_{i}^{n + 1}  u_{i}^{n}}{\Delta t} = \frac{\alpha}{\Delta x^2} \left(u_{i + 1}^{n}  2 u_{i}^{n} + u_{i  1}^{n} \right)
or, letting r = \frac{\alpha\, \Delta t}{\Delta x^2}:

u_{i}^{n + 1} = u_{i}^{n} + r \left(u_{i + 1}^{n}  2 u_{i}^{n} + u_{i  1}^{n} \right)
Stability
The FTCS method, for onedimensional equations, is numerically stable if and only if the following condition is satisfied:

r = \frac{\alpha\, \Delta t}{\Delta x^2} \leq \frac{1}{2}.
The time step \Delta t is subjected to the restriction given by the above stability condition. A major drawback of the method is for problems with large diffusivity the time step restriction can be too severe.
For hyperbolic partial differential equations, the linear test problem is the constant coefficient advection equation, as opposed to the heat equation (or diffusion equation), which is the correct choice for a parabolic differential equation. It is well known that for these hyperbolic problems, any choice of \Delta t results in an unstable scheme.^{[4]}
See also
References

^ John C. Tannehill;

^ Patrick J. Roache (1972). Computational Fluid Dynamics (1st ed.).

^ Patrick J. Roache (1998). Computational Fluid Dynamics (2nd ed.).

^ LeVeque, Randy (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.