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%\chapter{Nonlinear Advection-Diffusion}

In this chapter we solve the 1-D nonlinear advection-diffusion problem
by numerical method.
In contrast to the linear equation described in the previous chapter, here we discuss the nonlinear advection.
Especially, we solve the {\it Burgers equation}. 

%======================================================================
\section{Theoretical Background}

%======================================================================
\subsection{Nonlinear Advection Equation}

Assuming the finite-amplitude wave in the fluid which propagate only in direction (1-D $x$-$t$ plane), the equation of motion is
\begin{equation}
    \frac{Du}{Dt} = 0, \label{eq2-1}
\end{equation}
where viscosity, heat conductance and pressure gradient are ignored.
In the Eulerian frame for x-t plane, the Lagrange differential of (\ref{eq2-1}) can be rewritten as
\begin{equation}
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} =0. \label{eq2-2}
\end{equation}
For the solution of (\ref{eq2-2}), $u$ is constant on the curve $Dx/Dt=u$. 
Each point of the wave form propagates at different speed, i.e., the speed is fast (slow) where $u$ is high (low).
Where the gradient $\partial u/\partial x$ is positive, it decreases with time to approach zero.
Meanwhile, where the gradient is negative, it once diverges to -$\infty$.
%The solution has no physical meanings at those points.
After the appearance of -$\infty$, (\ref{eq2-2}) can no longer describe the phenomenon correctly, which is caused by the omission of the viscosity.

%======================================================================
\subsection{Burgers Equation}

Adding the diffusive term in the right side of (\ref{eq2-2}) to incorporate for represents the effect of viscosity, the equation is written as,
\begin{equation}
   \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^{2}u}{\partial x^{2}}. \label{eq2-3}
\end{equation}
This is called {\it Burgers Equation}, which we can solve analytically  (Tatsumi, 1982).
Using the Cole-Hopf transformation (Hope,1950; Cole,1951),
\begin{equation}
   u = -2 \nu \frac{\partial}{\partial x}\log\theta, \label{eq2-4}
\end{equation}
\begin{equation}
   \theta (x,t) = \exp \left[ - \frac{1}{2 \nu}
                            \int^x u(x',t) dx' \right],
						\label{eq2-5}
\end{equation}
the diffusion equation for $\theta (x,t)$ is
\begin{equation}
   \frac{\partial \theta}{\partial t} = \nu \frac{\partial^{2}\theta}{\partial x^{2}}.	\label{eq2-6}
\end{equation}
The general solution of (\ref{eq2-6}) is
\begin{equation}
   \theta (x,t) = \frac{1}{\sqrt{4 \pi\nu t}}
                  \int_{-\infty}^{\infty} \theta(x',0)
                  \exp \left[ - \frac{(x-x')^2}{4 \nu t} \right] dx',
						\label{eq2-7}
\end{equation}
where $\theta(x,0)$ is the initial condition (see also \ref{C01-diffusion}).
Consequently the general solution of the Burgers equation is as follow,
\begin{equation}
u = - 2 \nu \frac{\partial}{\partial x} \log \left( \frac{1}{\sqrt{4 \pi\nu t}}
       \int_{-\infty}^{\infty} \exp 
          \left[ -\frac{1}{2\nu} \left\{ \frac{(x-x')^2}{2t} + U(x') \right\}
          \right]   dx'   \right),		\label{eq2-8}
\end{equation}
\begin{equation}
\mathrm{where} U(x) = \int^x u(x',0) dx'.		\label{eq2-9}
\end{equation}

%======================================================================
\section{Exercises}
%======================================================================
\subsection{Nonlinear Advection Equation}

We solve the equation (\ref{eq2-2}) with no viscosity ($\nu=0$) by the numerical method, and study the time development for some initial conditions. 
Observe the wave form be gradually distorted and finally break. 

\begin{description}
 \item[Example 1] \mbox{}
\end{description}
\begin{center}
\begin{tabular}{ccc}
  sine  &  4 &  0  \\
 Initial function & Amplitude & Diffusion
\end{tabular}
\end{center}

Since the wave breaking is caused by the limitation of the numerical method, there is no meaning of the solution. 
Change the amplitude, and observe the result that the starting time of the wave breaking depends on the advection speed.
We can also examine the distorting processes by
changing the initial wave forms (sine, cone, and step functions).

To study nonlinear advection and diffusion numerically, carry out the program "C3" in GFD Menu.

\begin{figure}
  \begin{center}
    \includegraphics[width=10cm,keepaspectratio,clip]{fig/C02_01.epsf}
    \caption{Result of Example 1.}
    \label{fig:C02_01}
  \end{center}
\end{figure}

%======================================================================
\subsection{Burgers Equation}

We solve the Burgers equation (\ref{eq2-3}) with viscosity ($\nu\ne 0$).
\begin{description}
 \item[Example 2] \mbox{}
\end{description}
\begin{center}
\begin{tabular}{ccc}
  sine  &  4 &  2  \\
 Initial function & Amplitude & Diffusion
\end{tabular}
\end{center}

When the local diffusion is large, there is an appropriate range of parameters in which the wave keeps its standing form. 
Find out the range by changing the ratio {\tt amplitude}/{\tt diffusion}.

%======================================================================
\section{Appendix}

We use the 4th order {\it Runge Kutta Gill Method} for integration with respect to time, and also use the {\it Central Difference} for space differences in the numerical experiments. 
These numerical schemes are unstable for $\nu > 7$ (Nishizawa, 1999).
Here the stability of the difference method is investigated according to Horinouchi (1996).

%======================================================================
\section*{Reference}

\begin{description}
\itemsep   -1.55mm
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\item Cole, J., 1951: On a quasi-linear parabolic equation occuring
      inaerodynamics.
      {\it Q.Appl.Math.}, {\bf 9}, 225-236. 
\item Hopf, E., 1950: The partial differential equation
      $u_t+uu_x=u_{xx}$.
      {\it Common.Pure Appl.Math.}, {\bf3}, 201-230.
\item Horinouchi T., 1996: Practice D5  \& Programs (in Japanese).

\item Nishizawa., S., 1999: Note for numerical calculation of Burgers equation (in Japanese).

\item Tatsumi, T., 1982: Fluid Dynamics, {\it Baifukan}, 453pp (in Japanese).
\end{description}

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