Innumerical analysis, theFTCS (forward time-centered space) method is afinite difference method used for numerically solving theheat equation and similarparabolic partial differential equations.^[1] It is a first-order method in time,explicit in time, and isconditionally stable when applied to the heat equation. When used as a method foradvection equations, or more generallyhyperbolic 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 theforward Euler method in time (hence "forward time") andcentral difference in space (hence "centered space"), giving first-order convergence in time and second-order convergence in space. For example, in one dimension, if thepartial differential equation is

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}=F\left(u,x,t,\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}\right)}$

then, letting${\displaystyle u(i\mathrm{\Delta }x,n\mathrm{\Delta }t)=u_i^n}$, the forward Euler method is given by:

${\displaystyle \frac{u_i^{n+1}-u_i^n}{\mathrm{\Delta }t}=F_i^n\left(u,x,t,\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}\right)}$

The function${\displaystyle F}$ must be discretized spatially with acentral difference scheme. This is anexplicit method which means that,${\displaystyle u_i^{n+1}}$ can be explicitly computed (no need of solving a system of algebraic equations) if values of${\displaystyle u}$ at previous time level${\displaystyle (n)}$ are known. FTCS method is computationally inexpensive since the method is explicit.

The FTCS method is often applied todiffusion problems. As an example, for 1Dheat equation,

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}=\alpha \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}}$

the FTCS scheme is given by:

${\displaystyle \frac{u_i^{n+1}-u_i^n}{\mathrm{\Delta }t}=\alpha \frac{u_{i+1}^n-2u_i^n+u_{i-1}^n}{\mathrm{\Delta }{x}^2}}$

or, letting${\displaystyle r=\frac{\alpha \mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}$:

${\displaystyle u_i^{n+1}=u_i^n+r\left(u_{i+1}^n-2u_i^n+u_{i-1}^n\right)}$

As derived usingvon Neumann stability analysis, the FTCS method for the one-dimensional heat equation isnumerically stable if and only if the following condition is satisfied:

${\displaystyle \mathrm{\Delta }t\le \frac{\mathrm{\Delta }{x}^2}{2\alpha }.}$

Which is to say that the choice of${\displaystyle \mathrm{\Delta }x}$ and${\displaystyle \mathrm{\Delta }t}$ must satisfy the above condition for the FTCS scheme to be stable. In two-dimensions, the condition becomes

${\displaystyle \mathrm{\Delta }t\le \frac{1}{2\alpha \left(\frac{1}{\mathrm{\Delta }{x}^2}+\frac{1}{\mathrm{\Delta }{y}^2}\right)}.}$

If we choose$h=\mathrm{\Delta }x=\mathrm{\Delta }y=\mathrm{\Delta }z$, then the stability conditions become$\mathrm{\Delta }t\le h^2/(2\alpha )$,$\mathrm{\Delta }t\le h^2/(4\alpha )$, and$\mathrm{\Delta }t\le h^2/(6\alpha )$ for one-, two-, and three-dimensional applications, respectively.^[4]

A major drawback of the FTCS method is that for problems with large diffusivity${\displaystyle \alpha }$, satisfactory step sizes can be too small to be practical.

Forhyperbolic partial differential equations, thelinear test problem is the constant coefficientadvection equation, as opposed to theheat equation (ordiffusion equation), which is the correct choice for aparabolic differential equation.It is well known that for thesehyperbolic problems,any choice of${\displaystyle \mathrm{\Delta }t}$ results in an unstable scheme.^[5]

• Partial differential equations
• Crank–Nicolson method
• Finite-difference time-domain method

• Numerical differential equations
• Computational fluid dynamics

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