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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 40 Dec 5, 2011 Prof. Kamesh Madduri Class Overview Backward Euler method Implicit Trapezoid method Slides from textbook follow. 2 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Implicit Methods Eulers method is explicit in that it uses only information at time t k to advance solution to time t k +1 This may seem desirable, but Eulers method has rather limited stability region Larger stability region can be obtained by using information at time t k +1 , which makes method implicit Simplest example is backward Euler method y k +1 = y k + h k f ( t k +1 , y k +1 ) Method is implicit because we must evaluate f with argument y k +1 before we know its value Michael T. Heath Scientific Computing 43 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Implicit Methods, continued This means that we must solve algebraic equation to determine y k +1 Typically, we use iterative method such as Newtons method or fixedpoint iteration to solve for y k +1 Good starting guess for iteration can be obtained from explicit method, such as Eulers method, or from solution at previous time step < interactive example > Michael T. Heath Scientific Computing 44 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods Eulers Method Accuracy and Stability Implicit Methods and Stiffness Example: Backward Euler Method Consider nonlinear scalar ODE y = y 3 with initial condition y (0) = 1 Using backward Euler method with step size h = 0 . 5 , we obtain implicit equation y 1 = y + hf ( t 1 , y 1 ) = 1 . 5 y 3 1 for solution value at next step This nonlinear equation for y 1 could be solved by fixedpoint iteration or Newtons method To obtain starting guess for y 1 , we could use previous solution value, y = 1 , or we could use explicit method, such as Eulers method, which gives y 1 = y . 5 y 3 = 0 . 5 Iterations eventually converge to final value y 1 . 7709 Michael T. Heath Scientific Computing 45 / 84 Ordinary Differential Equations Numerical Solution of ODEs Additional Numerical Methods...
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.
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