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Tutorial_3

# Tutorial_3 - Tutorial 3 Question 1 Given a function =...

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Tutorial 3 Question 1 Given a function, g1878=g1858g4666g1876,g1877g4667, derive a central difference approximation for the second cross-derivative, g3105 g3118 g3033 g3105g3051g3105g3052 , of the highest order possible with the points indicated. Subscript notation for g4666g1876,g1877g4667 grid Simpler notation Question 2 Apply Newton’s method to find all the finite roots of: 1 g1876+2 =g1857 g2879g2873g3051 g3118 Explain what happens with the following initial guesses: (a) g1876=0 , (b) g1876=0.05 , (c) g1876=0.1 , (d) g1876=0.5 , and (e) g1876=1 . Question 3 The flow rate, g1843 , in a system is given by, g1843=15−7.5g466610 g2879g2873 g4667g343580+10.5g1843 g2873 g2871 uni2044 g3439 g2870 . Set up a direct iteration scheme, including relaxation, to solve this equation in the given form. Using an initial guess of g1843=5, investigate what happens with, i) g2033=1 (no relaxation) and ii) g2033=0.25 (under-relaxation). Question 4 Solve the following nonlinear system of equations by Gauss-Seidel iteration with (a) no relaxation, and (b) an under-relaxation constant of g2033=0.8 : (1) g1876 g2870 +g1877+4g1878=15 (2) 4g1876+g1877 g2870 +g1878=11 (3) g1876+4g1877+g1878 g2870 =18 0 0.2 0.4 0.6 0.8 1 1.2 -2 0 2 4 Δy g1877 g1876 g3036g2879g2869 g1876 g3036 g1876 g3036g2878g2869 g1877 g3037g2879g2869

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Tutorial_3 - Tutorial 3 Question 1 Given a function =...

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