CME306 / CS205B Homework 6Essentially Non-Oscillatory SchemesGiven the following data for φn, write down the interpolating polynomial that third order HJ ENO wouldconstruct in order to compute φn+1iin approximating the equation φt+ φx= 0.φni−3= 5, φni−2= 5, φni−1= 4, φni= 5, φni+1= 1, φni+2= −2, φni+3= 01Weighted ENOIf we consider an upwind discretization of φx, we have three possible third-order interpolating polynomials,given byφ1x=v13−7v26+11v36φ2x= −v26+5v36+v43φ3x=v33+5v46−v56Where vj= D∗φi+j−3, and D∗φ is the first-order upwind discretization of φx.However, the philosophy of picking exactly one of the three candidate stencils is overkill in smooth regionsof φ where φ is well-behaved. Instead, we can take a convex sum of the three stencils,φx= ω1φ1x+ ω2φ2x+ ω3φ3x(1)Where 0 ≤ ωi≤ 1, ω1+ ω2+ ω3= 1. It has been shown that we can pick ω1= .1, ω2= .6, ω3= .3 andachieve a 5thorder accurate approximation of φx.1. Show that if we perturb ω by O(∆x2) we still get a 5thorder approximation to φx.2. Why is this a bad idea in non-smooth areas of the flow? In order to demonstrate this, considerφt+ φx= 0 for a heaviside step function, with initial data given by:φni−3= 0, φni−2= 0, φni−1= 0, φni= 1, φni+1= 1, φni+2= 1, φni+3=
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