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6.3 Non-Linear equations with CIP


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The same approach is applicable more generally for non-linear and vector equations

$\displaystyle \frac{\partial \vec{f}}{\partial t} + \frac{\partial}{\partial x}(u\vec{f})=\vec{g}$ (1)

where $ u=u(\vec{f})$ and $ \vec{g}=\vec{g}(\vec{f})$ . The problem is again decomposed in alternating phases with / without advection describing the evolution of the function

$\displaystyle \left\{ \begin{array}{ll} \displaystyle \frac{\partial \vec{f}}{\...
...c{G} \qquad &\textrm{(non-advection with compression term)} \end{array} \right.$ (2)

and by differentiation of (eq.6.3#eq.1), the evolution of the derivative

$\displaystyle \left\{ \begin{array}{ll} \displaystyle \frac{\partial \vec{f^\pr...
...ac{\partial u}{\partial x} \qquad &\textrm{(non-advection)} \end{array} \right.$ (3)

Starting with the non-advection phase, the discretized function is first evolved according to

$\displaystyle \vec{f_j^*} = \vec{f_j^t} +\vec{G_j}\Delta t$ (4)

where the super-script ($ ^*$ ) refers to an intermediate step. To avoid an explicit evaluation of $ \vec{G_j^\prime}$ , the equation for the derivative is computed with
$\displaystyle \frac{\vec{f_j^{\prime\;*}}-\vec{f_j^{\prime\;t} }}{\Delta t}$ $\displaystyle =$ $\displaystyle \left[
\frac{\vec{G_{j+1}}-\vec{G_{j-1}}}{2\Delta x}
-\vec{f_j^{\prime\;t}}\frac{u_{j+1}-u_{j-1}}{2\Delta x} = \right]$  
  $\displaystyle =$ $\displaystyle \frac{ \vec{f_{j+1}^{\prime\;*}}-\vec{f_{j-1}^{\prime\;*}}
-\vec{...
...}}
{2\Delta x \Delta t}
-\vec{f_j^{\prime\;t}}\frac{u_{j+1}-u_{j-1}}{2\Delta x}$ (5)

The advection phase can then be evolved in the same manner as before (eq.6.2#eq.4), by shifting the cubic-Hermite polynomials along the characteristics (exercise 6.04).

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