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, it is unstable. The abbreviation FTCS was first used by Patrick Roache.[2][3]
Contents
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:
\[\frac{u_{i}^{n + 1} - u_{i}^{n}}{\Delta t} = \frac{\alpha}{\Delta x^2} \left(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n} \right)\]
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.
See also
References
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