quiz_1_sol - 2.29: Numerical Fluid Mechanics Solution of...

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2.29: Numerical Fluid Mechanics Solution of Quiz 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS— SPRING 2007 Solution of Quiz 1 Time 1 hour and 15 min, Totally 25 points Thursday 11 a.m. 03/22/07, Focused on Lecture 1 to 11 Problem 1 (6 points): State which of the following statements are true and which are false. You do not have to justify your answer. 1. The number of significant digits achievable by a specific floating point representation 2. If , then the relative error of f is more sensitive to relative error of x, compared to relative error of y. 3. Bi-section method is capable of predicting the maximum number of iterations required for a specific error level ahead in time. 4. If the Newton-Raphson’s method converges for a root finding problem, then the absolute error in each step will be less than the square of absolute error in previous step. 5. The Jacobi iterative method for a linear problem will always converge for a positive definite matrix. 6. The numerical stability of Gaussian elimination is guaranteed provided that we do full pivoting and equilibration. Solution: 1. True . Indeed if the length of mantissa is “t”, then it can distinguish up to log 10 2 t = t log 10 2 2 t states and the number of significant digits will be in decimal base. 2. False . The relative error in multiplication is dependent on absolute value of power. The sign of power only determines the sign of error. is not dependent on exponent length. f = ax 2 y ! 2 f = ax 2 y ! 2 log f = log a + 2log x ! 2log y df f = 2 dx x ! 2 dy y " f = 2 x ! 2 y , : Relative Error
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2.29: Numerical Fluid Mechanics Solution of Quiz 1 3. True . If the first guesses are x l , x u , then the maximum error at the beginning is x l ! x u . Also in each step, the maximum error is divided by 2. Consequently for a specific level of absolute error (e), we can find the maximum number of iterations by x l ! x u 2 n " e . 4. False e n + 1 ! a ( x r ) e n 2 . Quadratic convergence means that the error decays such that the relation holds for a constant “ a = a ( x r ) ”. In general “a” is not equal to “1” and the statement will be false. 5. False . However, the Gauss-Seidel iterative method for a linear problem will always converge for a positive definite matrix. 6. False . Refer to notes and recall that pivoting and equilibration will improve the solution accuracy. However if we do not have enough significant digits, compared to matrix condition number the solution will be unstable and inaccurate. Problem 2 (3 points):
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quiz_1_sol - 2.29: Numerical Fluid Mechanics Solution of...

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