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Burger's Equation

A simple nonlinear PDE which is often used as a model problem for fluid dynamical systems is given by the inviscid Burger's equation [82]:

$\displaystyle \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0$ (B.4)

It is similar in form to the advection equation mentioned in §3.6, and as we will see, its circuit representation is identical. The problem is assumed to be defined for $ x\in \mathbb{R}$, $ t\geq 0$. $ u$ can be considered to be a current, as before, through a single loop, and Kirchoff's Voltage Law around the loop will give (B.4). The question however, is of the type of circuit elements to be included in this loop; clearly they must be nonlinear, and certainly reactive as well. We note that the viscous form of Burger's equation was approached in this way in [202].

Using coordinate transformation (3.18), (B.4) can be rewritten as

$\displaystyle \left(\frac{v_{0}+u}{\sqrt{2}}\right)\frac{\partial u}{\partial t_{1}}+\left(\frac{v_{0}-u}{\sqrt{2}}\right)\frac{\partial u}{\partial t_{2}} = 0$    

Assuming that the solution is differentiable% latex2html id marker 89892
\setcounter{footnote}{2}\fnsymbol{footnote}, this can be rewritten as

$\displaystyle \underbrace{\frac{1}{2}\left(L_{1}\frac{\partial u}{\partial t_{1...
...l u}{\partial t_{2}}+\frac{\partial L_{2}u}{\partial t_{2}}\right)}_{v_{2}} = 0$ (B.5)


$\displaystyle L_{1} = \frac{1}{\sqrt{2}}\left(v_{0} + \frac{2}{3}u\right)\hspace{0.5in} L_{2} = \frac{1}{\sqrt{2}}\left(v_{0} - \frac{2}{3}u\right)$ (B.6)

Thus Burger's equation, in the form of (B.5), can be interpreted as a series combination of two nonlinear inductances, as shown in Figure B.1(a). The resultant MDWD network, with port resistances

$\displaystyle R_{1} = \frac{2L_{1}}{T_{1}}\hspace{1.0in}R_{2} = \frac{2L_{2}}{T_{2}}$ (B.7)

appears in Figure B.1(b). We emphasize that this network is passive only if power-normalized wave variables are employed.

Figure B.1: The (1+1)D inviscid Burger's equation-- (a) MDKC and (b) MDWD network.
% graphpaper(0,0)(380,80) ...
\end{picture} \end{center} \vspace{0.2in}

The positivity condition on these inductances now depends on the solution itself, $ u$, and we must have

$\displaystyle v_{0} \geq \frac{2}{3}\max_{x\in \mathbb{R},t\geq 0}\vert u\vert$    

An a priori estimate of $ \max_{x\in \mathbb{R},t\geq 0}\vert u\vert$ must be available; this is a consistent feature of all the circuit-based methods (and, it would seem, any explicit method) for the fluids systems that we will examine presently.

Let us now examine the scattering operation. First choose $ T_{1}=T_{2}=\sqrt{2}\Delta$, so that the current grid function for the current at location $ x=i\Delta$ and $ t = nT$ can be written as $ u_{i}(n)$. The two power-normalized input wave variables entering the adaptor at the same location and time step are $ \underline{a}_{1,i}(n)$ $ \underline{a}_{2,i}(n)$, and we have

$\displaystyle u_{i}(n) = \frac{2}{R_{1,i}(n)+R_{2,i}(n)}\left(\sqrt{R_{1,i}(n)}\underline{a}_{1,i}(n)+\sqrt{R_{2,i}(n)}\underline{a}_{2,i}(n)\right)$    

which, from (B.6) and (B.7), and using $ T_{1}=T_{2}=\sqrt{2}\Delta$ and $ v_{0} = \Delta/T$ can be rewritten as

$\displaystyle u_{i}(n) = \frac{\sqrt{\Delta}}{v_{0}}\left(\sqrt{v_{0}+\frac{2}{...
...rline{a}_{1,i}(n)+\sqrt{v_{0}-\frac{2}{3}u_{i}(n)}\underline{a}_{2,i}(n)\right)$ (B.8)

This is precisely the nonlinear algebraic equation which is to be solved (in $ u_{i}(n)$); once $ u_{i}(n)$ is determined, then so are the port resistances, and the output wave variables $ \underline{b}_{1,i}(n)$ and $ \underline{b}_{2,i}(n)$ can be obtained through scattering as per (2.33). As mentioned before, it is not at all clear from the form of (B.8) whether a solution exists and is unique. We note, however, that we (and others [16,70]) have successfully programmed simulations for the gas dynamics equations (see next section), using simple iterative methods to solve the nonlinear algebraic systems; the results would appear to be in accord with published simulation results using differencing methods [171].

next up previous
Next: The Gas Dynamics Equations Up: Applications in Fluid Dynamics Previous: Nonlinear Circuit Elements
Stefan Bilbao 2002-01-22