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Hyperbolic Equation in one space variable:

The simplest hyperbolic problem is that of the vibrating string

$\displaystyle \frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial^{2}u}{\partial x^{2}}$ (24)

in the domain $ R=[0\leq x\leq 1]\times[t>0],$ satisfying the following initial conditions

(25)

and boundary conditions

$\displaystyle \left. \begin{aligned}u(0,t)=g_{1}(t)\\ u(1,t)=g_{2}(t) \end{aligned} \right\}\qquad t>0$ (26)

We place a mesh of points $ (x_{m},t_{n})$ on R, where $\displaystyle x_{m}=m h\qquad m=0,1,2,......,M,\quad Mh=1$

$\displaystyle t_{n} = nk\qquad n=0,1,2,......$

The exact difference replacement of (24) at the nodal points $ (x_{m},t_{n})$ is given by

$\displaystyle \left(sinh ^{-1}\frac{\delta t}{2}\right)^{2} u(x_{m},t_{n})= p^{2} \left(sinh ^{-1} \frac{\delta x}{2}\right)^{2}u(x_{m},t_{ n})$ (27)

where $ p=k/h$ is mesh ratio and

$\displaystyle 4\left(sinh^{-1} \frac{\delta}{2}\right)^{2}=\delta^{ 2}-\frac{1}{12}\delta^{4}+\frac{1}{90}\delta^{ 6}-.......$

The explicit and implicit difference scheme for (24) will be obtained by approximating equation (27).

Explicit Difference Schemes:
An explicit difference scheme for (24) is given by

$\displaystyle \delta^{2}_{t}U^{n}_{m}=p^{2}\delta^{ 2}_{x}U^{n}_{m}$

which may be written in the form

$\displaystyle U^{n+1}_{m}=2(1-p^{2})U^{n}_{m}+ p^{2}(U^{n}_{m-1}+U^{n}_{m+1})-U^{n-1}_{m}$ (28)

where $ U^{n}_{m}$ is the approximation to $ u^{n}_{m} \equiv u(x_{m},t_{n})$.

If each term in (28) is expanded in Taylor's series about the nodal point $ (x_{m},t_{n})$ and the function $ u(x_{m},t_{ n})$ satisfies (24), then we get the truncation error

For p=1, the truncation error vanishes and so the difference representation of (24) is obtained as

$\displaystyle U^{n+1}_{m}=U^{n}_{m-1}+U^{n}_{m+1}-U^{n-1}_{m}$

In order to start computation, we require data on the two lines $ t=0$ and $ t=k$. The first condition in (25) gives $ U^{0}_{m}$ on the initial line as

$\displaystyle U^{0}_{m}=f_{1}(mh)\qquad 0\leq m\leq M$

We can use the second condition in (25) to find values on the line $ t=k$. Using the central difference approximation for the derivative, i.e

$\displaystyle \frac{\partial u^{0}_{m}}{\partial t} \simeq \frac{U^{1}_{m}-U^{-1}_{m}}{2k}$

In the second condition in (25) and eliminating $ U^{-1}_{m}$ from (28) for n=0, we get the formula to give the values on the first time level,

The boundary conditions (26) become $ U^{n}_{0}=g^{n}_{1} \quad$ and $ U^{n}_{M}=g^{n}_{2};\quad n=1,2,.......$

Formula (27) may now be used to advance computation for $ n\geq1$.

To examine the finite difference formula (27) for stability, we replace $ U^{n}_{m}$ by $ \xi^{ n } e^{i \beta m h}$ and get

$\displaystyle \xi+\frac{1}{\xi}=2-4p^{2}sin^{2}\left(\frac{\beta h}{2}\right)$

or

$\displaystyle \xi^{2}-2A\xi+1=0$ (29)

where $ A=1-2p^{2}sin^{2}\left(\frac{\beta h}{2}\right)$
The solution of (29) is given by

$\displaystyle \xi_{1}=A+\sqrt{A^{2}-1}$   and$\displaystyle \quad \xi_{2}=A-\sqrt{A^{2}-1}$

This gives that $ \displaystyle \xi_{1}=\frac{1}{\xi_{2}}$.

\begin{displaymath}\left . \begin{array}{ccc} If & A>1 & \vert\xi_{1}\vert>1 \\... ...1 & \vert\xi_{1}\vert=\vert\xi_{2}\vert=1 \end{array}% \right .\end{displaymath}

Thus for stability $ -1\leq A\leq 1$ or $ -1\leq 1-2 p^{2}sin^{2}(\frac{\beta h}{2})\leq1$, which gives $ p\leq 1$.
This is the condition of stability.

Implicit Difference Scheme:
In order to improve stability, we now consider an implicit difference replacement of equation (25). This takes the form

$\displaystyle p^{2}\left\{ a \left[U^{n+1}_{m+1}-2U^{n+1}_{m}+U^{n+1}_{m-1}\rig... ...{n}_{m-1}\right]+a\left[U^{n-1}_{m+1}-2U^{n-1}_{m}+U^{n-1}_{m-1}\right]\right\}$

where . One can examine this scheme for stability and find that the scheme is stable for all $ p>0$ if $ a\geq\frac{1}{4}$. Thus, if $ a=\frac{1}{4}$, the implicit difference formula

$\displaystyle \left[U^{n+1}_{m+1}-2U^{n+1}_{m}+U^{n+1}_{m-1}\right]+2\left[U^{n... ...}_{m}+U^{n}_{m-1}\right]+ \left[U^{n-1}_{m+1}-2U^{n-1}_{m}+U^{n-1}_{m-1}\right]$

$\displaystyle =\frac{4}{p^{2}}\left[U^{n+1}_{m}-2U^{n}_{m}+U^{n-1}_{m}\right]$

The consistency of a finite difference approximation to a hyperbolic equation can be defined briefly. A difference scheme to a hyperbolic equation is consistent; if $ \displaystyle \frac{Truncation}{k^{2}}\rightarrow 0$ as $ h, k\rightarrow 0$.

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