Chapter 3 - Numerical Approximation Methods PDF

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This course was instructed by Melissa McGuirl, a Doctoral Candidate in Applied Mathematics at Brown University, during the summer of 2018. These are the notes that were supplied to her students. ...

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APMA 0350 Applied Ordinary Differential Equations Summer 2018 Instructor: Melissa McGuirl

Course Notes by Bj¨orn Sandstede

3

Numerical Methods

Goals • overview of selected methods for solving ODEs numerically • intuition behind derivation of algorithms • exploration of methods in Matlab • order of numerical methods Setting Assume that x(t) solves x˙ = f (x) for 0 ≤ t ≤ T . We know some methods to solve for x(t) but f (x) might be too complicated for us to solve for x(t) analytically. Instead, we want to approximate the solution x(t) for 0 ≤ t ≤ T by a sequence of points xn that are close to x(tn ) for 0 ≤ tn ≤ T , where n = 0, . . . , N . Goal Given an initial condition x(0) = x0 , find an approximation for the solution x(t) at a sequence of points t0 , . . . , tn in the interval 0 ≤ t ≤ T .

x(T ) xn x(tn )

x1 x(t1 )

x(tn + 1) xn+1

x0 x(0)

xN

want max ||x(tn ) − xn || 0 such that max |x(nh) − xn | ≤ chK

0≤n≤N

for 0 < h ≪ 1, where x(nh) is the true solution evaluated at the grid point t = nh and xn is the approximated solution at the grid point t = nh, and h is the step size.

x(T ) xn x(tn )

x1 x(t1 )

x(tn + 1) xn+1

x0 x(0)

xN

want max ||x(tn ) − xn ||...