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Energy Analysis

The energetic analysis of scheme (6.33) is very similar to that of the wave equation itself. It is easiest to begin with scheme (6.33) in its condensed operator form, and consider the unbounded spatial domain $ {\mathcal D}={\mathbb{Z}}$. Taking the inner product of scheme (6.33) with the grid function defined by

$\displaystyle \delta_{t\cdot} u$ (6.46)

which is an approximation to the velocity, gives

$\displaystyle \langle \delta_{t\cdot} u, \delta_{tt}u \rangle_{{\mathcal D}}= \gamma^2 \langle \delta_{t\cdot} u, \delta_{xx}u\rangle_{{\mathcal D}}$ (6.47)

After employing summation by parts, one has

$\displaystyle \langle \delta_{t\cdot} u, \delta_{tt}u \rangle_{{\mathcal D}}+ \gamma^2 \langle \delta_{x+}\delta_{t\cdot} u, \delta_{x+}u\rangle_{{\mathcal D}}=0$ (6.48)

This may be written as the total difference

$\displaystyle \delta_{t+}\left(\frac{1}{2}\Vert\delta_{t-}u\Vert _{{\mathcal D}...
...amma^2}{2}\langle \delta_{x+}u, e_{t-}\delta_{x+}u\rangle_{{\mathcal D}}\right)$ (6.49)

or, after identifying the terms inside the brackets above with kinetic and potential energy, as

$\displaystyle \delta_{t+}{\mathfrak{h}} = 0$ (6.50)

with

$\displaystyle {\mathfrak{t}} = \frac{1}{2}\Vert\delta_{t-}u\Vert _{{\mathcal D}...
...x+}u\rangle_{{\mathcal D}}\qquad {\mathfrak{h}} = {\mathfrak{t}}+{\mathfrak{v}}$ (6.51)

The total difference (6.50) above is a statement of conservation of numerical energy for the scheme (6.33), and it thus follows that

$\displaystyle {\mathfrak{h}}^{n} = {\mathfrak{h}}^{0}$ (6.52)

Scheme (6.33) is thus exactly conservative, in this special case of a spatial domain of infinite extent. This is true regardless of the values chosen for the time step $ k$ and the grid spacing $ h$. One might thus wonder how the concept of numerical stability, which follows from frequency domain analysis, intervenes in the energetic framework. The key point, just as for the case of the simple harmonic oscillator, is that in contrast with the energy defined for the continuous system, $ {\mathfrak{h}}$ is not necessarily positive definite, due to the indefinite nature of the numerical potential energy term $ {\mathfrak{v}}$.

To determine conditions under which the numerical energy $ {\mathfrak{h}}$ is positive definite, one may proceed as in the case of the simple harmonic oscillator, and write, for the potential energy,

$\displaystyle {\mathfrak{v}}$ $\displaystyle =$ $\displaystyle \frac{\gamma^2}{2}\langle \delta_{x+}u, e_{t-}\delta_{x+}u\rangle...
...c{2\gamma^2}{h^2}\Vert u\Vert _{{\mathcal D}}\Vert e_{t-}u\Vert _{{\mathcal D}}$ (6.53)

Figure: Numerical energy conservation for scheme (6.33). The parameters, sample rate and initial conditions have been chosen identical to those given in the caption to Figure 6.2. At left, the total energy $ {\mathfrak h}$ (solid black line), $ {\mathfrak v}$ (solid grey line) and $ {\mathfrak t}$ (dotted grey line) are plotted against time; notice that it is possible, at certain instants, for the numerical potential energy $ {\mathfrak v}$ to take on negative values. At right, the variation of the energy $ ({\mathfrak h}^n-{\mathfrak h}^0)/{\mathfrak h}^0$ is also plotted against time; multiples of the quantization error are plotted as solid grey lines.


next up previous contents index
Next: Numerical Boundary Conditions Up: A Simple Finite Difference Previous: Numerical Dispersion   Contents   Index
Stefan Bilbao 2006-11-15