利用者:Lordbookren/sandbox2

FTCS法（英: Forward time-centered space method）とは、偏微分方程式の数値解法の一つである。有限差分法の一種であり、熱伝導方程式やそれに似た放物型偏微分方程式に対して用いられる^[1]。時間方向に1次の精度をもつ陽解法であり、熱伝導方程式で使用した場合、計算条件によっては不安定になりうる。また移流方程式といった双曲型偏微分方程式に適用した場合は、人工粘性を導入しない限り不安定である。「FTCS」という略語はPatrick Roacheによって初めて使用された^[2]^[3]。

計算手法

FTCS法は時間方向の離散化に前進オイラー法（forward time）、空間方向に中心差分（centered space）を用いるスキームであり、時間に関して1次精度、空間に関しては2次精度である。 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.

1次元熱伝導方程式における例

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}}=\alpha {\frac {u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}}{\Delta x^{2}}}

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)

安定性

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

\Delta t\leq {\frac {\Delta x^{2}}{2\alpha }}.

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

\Delta t\leq {\frac {1}{4\alpha \left({\frac {1}{\Delta x^{2}}}+{\frac {1}{\Delta y^{2}}}\right)}}.

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