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Math 228C Computational Assignment 4 Spring 1999

Due: Monday, Jun 21, 1999

Compute problem to time T=2 on 3 grids $\Delta x=h, \Delta x=\frac h 2,\Delta x=\frac h 4$Let

$$\vec {\textbf u}_{ij}^{AVE} \frac 14(\vec u_{ij}^{\frac h2}+\vec u_{ij+1}^{\frac h2}+\vec u_{i+1j}^{\frac h2}+\vec u_{i+1j+1}^{\frac h2})$$ (1) =

NOw $2\times N^2$ velocities $\vec{\textbf u}^h_{ij}$ and $\vec {\textbf u}_{ij}^{AVE}$Compute $\vert\vert\vec {\textbf u}^h-\vec {\textbf u}^{AVE}\vert\vert _2$, $\vert\vert\vec {\textbf u}^h-\vec {\textbf u}^{AVE}\vert\vert _1$, $\vert\vert\vec {\textbf u}^h-\vec {\textbf u}^{AVE}\vert\vert _{\infty}$,where

$$\vert\vert\vec {\textbf u}^h-\vec {\textbf u}^{AVE}\vert\vert _{\infty} \max_{ij}(\vert\vec {\textbf u}^h_{ij}-\vec {\textbf u}^{AVE}_{ij})$$ (2) =

This is in the col with $h-\frac h2$;e.g. when $h=\frac 1 {64}$ this is in col 64-128

Also compute $\vert\vert\vec{\textbf u}^{\frac h 2}-\vec{\textbf u}^{AVE \frac h 4}\vert\vert$

Now let

$$\vert\vert\vec{\textbf u}^h-\vec{\textbf u}^{AVE \frac 12}\vert\vert$$ (3) err^h =

$$err^{\frac h2} \vert\vert\vec{\textbf u}^{\frac h 2}-\vec{\textbf u}^{AVE \frac h 4}\vert\vert$$ (4) =

The discretization of $\vec{\textbf u}_t+(\vec{\textbf u}\times\nabla)\vec{\textbf u}=-\nabla p$ is more complicated than $\rho_t+(\vec{\textbf u}\times\nabla)\rho=0$

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