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FTCS scheme

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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 and second-order 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: 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} - 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 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.

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

  1. ^ John C. Tannehill;  
  2. ^ Patrick J. Roache (1972). Computational Fluid Dynamics (1st ed.).  
  3. ^ Patrick J. Roache (1998). Computational Fluid Dynamics (2nd ed.).  
  4. ^ LeVeque, Randy (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.  
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