Hamilton-Jacobi WENO - 1 Hamilton ‐ Jacobi ENO WENO,…...

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Unformatted text preview: 11/17/2010 1 Hamilton ‐ Jacobi ENO, WENO,… Higher accurate spatial differencing HJ ‐ ENO • Depending u>0 or <0, we use D ‐ φ or D + φ to approximate . This is of first order. • ENO will still depend on u>0 or <0 (and much more) to decide a polynomial approximation p to φ then use p x to estimate φ x . • ENO chooses p as an interpolating polynomia x φ ENO chooses p as an interpolating polynomial but its points of interpolation depend on the function value nearby. 11/17/2010 2 Divided Differences at Grid Points • Define i i D φ φ = • First divided differences: • Second divided differences: x D D D i i i Δ − = + + φ φ φ 1 1 2 / 1 x D D D i i i Δ − = − + 2 1 2 / 1 1 2 / 1 2 φ φ φ • Third divided differences: x D D D i i i Δ − = + + 3 2 2 1 3 2 / 1 φ φ φ Interpolating Polynomial • At a given grid point x i , we need to decide the points of interpolation so that the associated interpolating polynomial has the least oscillation....
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Hamilton-Jacobi WENO - 1 Hamilton ‐ Jacobi ENO WENO,…...

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