Math 6610 • Analysis of Numerical Methods I

Linear Multistep Methods Consider the initial value problem \[ y'=f(x,y),\qquad y(a)=y_0 \] Let \(h\) be some stepsize, \(x_n=a+nh\), \(y_n\approx y(x_n)\), and \(f_n=f(x_n,y_n)\), for \(n=0,1,2,3,\ldots \). Let \(k\) be some step number, and ignore the question of obtaining starting values \(y_1,y_2,\ldots ,y_{k-1}\). Suppose the approximations \(y_n\), \(n=k,k+1,\ldots \) are obtained by the Linear Multistep Method \[ \sum _{j=0}^k \alpha _j y_{n+j} =h\sum _{j=0}^k \beta _j f_{n+j}. \tag {4} \] Define what is meant by the local truncation error and the order of the linear multistep method (4). Compute the order and the local truncation error of Euler’s Method \[ y_{n+1}-y_n=hf_n. \tag {5} \]