FTCS scheme

In numerical analysis, the FTCS (Forward-Time Central-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 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 first-order convergence in time. 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: one-dimensional 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}-2u_{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}-2u_{i}^{n}+u_{i-1}^{n}\right)

Stability

The FTCS method, for one-dimensional 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.

References

^ John C. Tannehill; Dale A. Anderson; Richard H. Pletcher (1997). Computational Fluid Mechanics and Heat Transfer (2nd ed.). Taylor & Francis. ISBN 1-56032-046-X.
^ Patrick J. Roache (1972). Computational Fluid Dynamics (1st ed.). Hermosa. ISBN 0-913478-05-9.
^ Patrick J. Roache (1998). Computational Fluid Dynamics (2nd ed.). Hermosa. ISBN 0-913478-09-1.

[1] John C. Tannehill; Dale A. Anderson; Richard H. Pletcher (1997). Computational Fluid Mechanics and Heat Transfer (2nd ed.). Taylor & Francis. ISBN 1-56032-046-X.

[2] Patrick J. Roache (1972). Computational Fluid Dynamics (1st ed.). Hermosa. ISBN 0-913478-05-9.

[3] Patrick J. Roache (1998). Computational Fluid Dynamics (2nd ed.). Hermosa. ISBN 0-913478-09-1.

[1]

[2]

[3]

The method

Illustration: one-dimensional heat equation

Stability

See also

References