Here is the PostScript file theory3.ps and the LaTeX file theory3.tex.

Math 228B Theoretical Assignment 3 Winter 1999

Due: Monday, March 8, 1999

Problem 1   If f(x) has at least q^th continuous derivative and f_s(x) is the q^th order polynomial which interpolates f(x) at the points x_j; i.e.,

$$f_s(x_j) = f(x_j), j = 0, \cdots, M-1 ,$$

then show that for all $x \in [0, D]$

$$\vert\vert f-f_s\vert\vert _{\infty} \:\leq C\cdot (\Delta {x})^{q+1}$$

Problem 2   What slope $\Delta u$ will ensure that the end points of the line segment which passes through the point ( x_j, u_j ) does not exceed the limits set by u_j-1 and u_j+1, provided that either

$$u_{j-1} \le u_j \le u_{j+1} \qquad {\rm or} u_{j+1} \le u_j \le u_{j-1} .$$

(The figure shows an example of the first case.)

Problem 3   Recall that the definition of the corrected antidiffusive flux A^c_j+1/2 is given by

$$A^c_{j+1/2} = S_{j+1/2} \, \max \{ 0 , K \}$$

where $S_{j+1/2} = \rm {sign} \{ A_{j+1/2} \}$ and

$$K = \, \min \big \{ \vert A_{j+1/2} \vert , \, S_{j+1/2} {... ...\over \Delta {t}} ( u^{TD}_{j+2} - u^{TD}_{j+1}\vert) \big \}$$

where

$$u^{TD}_{j} = u^n_{j} + {\Delta {t}\over \Delta {x}} [ F^L_{j-1/2} - F^L_{j+1/2} ]$$

is the low-order solution. Show that the second and third terms inside the curly brackets in the definitition of K ensure that

$$u^{n+1}_{j} = u^{TD}_{j} + {\Delta {t}\over \Delta {x}} [ A^c_{j-1/2} - A^c_{j+1/2} ]$$

does not exceed the limits set by u_j-1 and u_j+1, provided that either

$$u_{j-1} \le u_j \le u_{j+1} \qquad {\rm or} \qquad u_{j+1} \le u_j \le u_{j-1}.$$

MAT 228B Webpage | Theory HW #1 | Theory HW #2 |Theory HW #3