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NumericalAnalysis-684-2010aug

# NumericalAnalysis-684-2010aug - (d Can one obtain a larger...

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MAT 684 Numerical Analysis August, 2010 NAME: 1. Let f : [0 , 1] R be a given continuous function. Consider the following boundary value problem with Dirichlet boundary conditions - u 00 ( x ) + u ( x ) = f ( x ) , 0 < x < 1 u (0) = 0 , u (1) = 0 . (a) Derive the variational equation of the boundary value problem and specify the appropriate Sobolev space. (b) Define a finite element scheme by using a piecewise linear finite element space for the above equation. 2. Consider the two stage Runge-Kutta method for solving y 0 = f ( y ), y * = y n + a Δ tf ( y n ) y n +1 = y n + Δ t [ b 1 f ( y n ) + b 2 f ( y * )] . (a) Derive an expression for the leading terms of the truncation error. (b) Give the values of coefficients a , b 1 , b 2 so that the method is second order accurate. (c) Derive the interval of absolute stability for the numerical method.
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Unformatted text preview: (d) Can one obtain a larger interval of absolute stability by relaxing the restriction on accuracy (e.g., accepting ﬁrst order accuracy)? Justify your answer. 3. Consider the equation u tt = u xx to be solved for ≤ x ≤ 1 , t > with initial data u ( x, 0) = u ( x ) u t ( x, 0) = u 1 ( x ) where u ,u 1 are smooth and vanishing near x = 0, x = 1. (a) Give boundary conditions at x = 0 and x = 1 to make this a well-posed problem. (b) Give a stable, convergent numerical approximation to this initial boundary value problem. 1...
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