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{{Redirect|FTCS|the scientific conference|International Conference on Dependable Systems and Networks}} | |||
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 equation]]s.<ref>{{cite book | title = Computational Fluid Mechanics and Heat Transfer | author1 = John C. Tannehill | author2 = Dale A. Anderson |authorlink2=Dale A. Anderson| author3 = Richard H. Pletcher | edition = 2nd | publisher = [[Taylor and Francis|Taylor & Francis]] | year = 1997 | isbn = 1-56032-046-X}}</ref> It is a first-order method in time, [[Explicit and implicit methods|explicit]] in time, and is [[Numerical stability|conditionally stable]] when applied to the heat equation. When used as a method for [[advection|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.<ref>{{cite book | title = Computational Fluid Dynamics | author = Patrick J. Roache | edition = 1st | publisher = [[Hermosa (publisher)|Hermosa]] | year = 1972 | isbn = 0-913478-05-9}}</ref><ref>{{cite book | title = Computational Fluid Dynamics | author = Patrick J. Roache | edition = 2nd | publisher = [[Hermosa (publisher)|Hermosa]] | year = 1998 | isbn = 0-913478-09-1}}</ref> | |||
==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 | |||
:<math>\frac{\partial u}{\partial t} = F\left(u, x, t, \frac{\partial^2 u}{\partial x^2}\right)</math> | |||
then, letting <math>u(i \,\Delta x, n\, \Delta t) = u_{i}^{n}\,</math>, the forward Euler method is given by: | |||
:<math>\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) </math> | |||
The function <math>F</math> must be discretized spatially with a [[central difference]] scheme. This is an [[explicit and implicit methods|explicit method]] which means that, <math>u_{i}^{n+1}</math> can be explicitly computed (no need of solving a system of algebraic equations) if values of <math>u</math> at previous time level <math>(n)</math> 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]], | |||
:<math>\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}</math> | |||
the FTCS scheme is given by: | |||
:<math>\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)</math> | |||
or, letting <math>r = \frac{\alpha\, \Delta t}{\Delta x^2}</math>: | |||
:<math>u_{i}^{n + 1} = u_{i}^{n} + r \left(u_{i + 1}^{n} - 2 u_{i}^{n} + u_{i - 1}^{n} | |||
\right)</math> | |||
==Stability== | |||
The FTCS method, for one-dimensional equations, is [[numerical stability|numerically stable]] if and only if the following condition is satisfied: | |||
:<math> r = \frac{\alpha\, \Delta t}{\Delta x^2} \leq \frac{1}{2}. </math> | |||
The time step <math>\Delta t </math> 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 differential equation|linear test problem]] is the constant coefficient | |||
[[Advection|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 partial differential equations|hyperbolic problems]], ''any'' choice of | |||
<math> \Delta t</math> results in an unstable scheme.<ref>{{cite book|last=LeVeque|first=Randy|title=Finite Volume Methods for Hyperbolic Problems|year=2002|publisher=Cambridge University Press|isbn=0-521-00924-3}}</ref> | |||
==See also== | |||
*[[Partial differential equations]] | |||
*[[Crank–Nicolson method]] | |||
==References== | |||
<references/> | |||
{{Numerical PDE}} | |||
{{DEFAULTSORT:Ftcs Scheme}} | |||
[[Category:Numerical differential equations]] | |||
[[Category:Computational fluid dynamics]] |
Latest revision as of 22:24, 19 January 2014
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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
then, letting , the forward Euler method is given by:
The function must be discretized spatially with a central difference scheme. This is an explicit method which means that, can be explicitly computed (no need of solving a system of algebraic equations) if values of at previous time level 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,
the FTCS scheme is given by:
Stability
The FTCS method, for one-dimensional equations, is numerically stable if and only if the following condition is satisfied:
The time step 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 results in an unstable scheme.[4]
See also
References
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534