Linear Multistep Methods

(Unstable Multistep Method). Consider the numerical method \[ y_{k+1} = 3y_k - 2y_{k-1} + \frac {h}{2}\left (f(x_k,y_k)-3f(x_{k-1},y_{k-1})\right ), \qquad k \ge 1. \] Illustrate with an example of a simple initial value problem that the above scheme is unstable.

(Linear Multistep Methods).

1. Define linear multistep method (give formula). Give definition of the region of absolute stability.
2. Show that the region of absolute stability for the trapezoidal method is the set of all complex \(h\lambda \) with \(\operatorname {Real}(\lambda ) < 0\).

Unstable Multistep Method

Consider the numerical method \[ y_{k+1}=3y_k-2y_{k-1} +\frac {h}{2}\left (f(x_k,y_k)-3f(x_{k-1},y_{k-1})\right ), \qquad k\ge 1 \] Illustrate with an example of a simple initial value problem that the above scheme is unstable.

Region of Absolute Stability and Example of Linear One-Step Method

1. Define a region of absolute stability for the linear multistep method.

2. Find the region of absolute stability for the Backward Euler Method.

Unstable Multistep Method

Consider the numerical method \[ y_{p+1}=3y_p-2y_{p-1} +\frac {h}{2}\left (f(x_p,y_p)-3f(x_{p-1},y_{p-1})\right ), \qquad p\ge 1, \]

for the solution of the initial value problem \[ y'=f(x,y),\qquad y(x_0)=y_0. \] Illustrate with an example of a simple initial value problem that the above numerical scheme is unstable.

Linear Multistep Methods

Give the definition of linear multistep method (give formula).

a) State necessary and sufficient condition for linear multistep method to be consistent.

b) State root condition for linear multistep method.

c) Construct an example of a consistent but not stable linear multistep method. Justify your answer.

Region of Absolute Stability and Example of Linear One-Step Method

1. Define a region of absolute stability for Linear Multistep Method.

2. Find the region of absolute stability for the Forward Euler Method.

Linear Multistep Methods

Define linear multistep method (give formula).

a) State necessary and sufficient condition for linear multistep method to be consistent.

b) State root condition for linear multistep method.

c) What can you say about consistency and stability properties of the method: \[ y_{n+1}-3y_n+2y_{n-1}=-h f(y_{n-1})? \]

Example of A-stable Linear Multistep Method

Give an example of A-stable linear multistep method. Justify your answer.

Define what is meant by a linear multistep method and its region of absolute stability. Show that a convergent explicit linear multistep method cannot have an infinite region of absolute stability.

Linear Multistep Methods

Construct an example of:

a) a consistent but not stable (not zero-stable) linear multistep method

b) a stable (zero-stable) but not consistent linear multistep method

What kind of behavior do you expect from the numerical solution produced by the methods in a) and in b)?

Convergence of Linear Multistep Method

Consider the method \[ y_{n+2}-2y_{n+1}+y_n = \frac {h}{2}\left (f(t_{n+2},y_{n+2})-f(t_n,y_n)\right ) \] Apply the method to the scalar IVP \(y'=y\), \(y(0)=1\) and solve exactly the resulting difference equation, considering the starting values to be \(y_0=y_1=1\). Show theoretically that the numerical solution does not converge as \(h\to 0\) and \(n\to \infty \).

Explicit Methods Define what we mean by Linear Multistep Method and what we mean by its region of absolute stability. Show that a convergent explicit linear multistep method cannot have an unbounded region of absolute stability.

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 \[ y_{n+1}-y_n=hf_n. \tag {5} \]

Absolute Stability Consider solving the initial value problem \[ y'=f(x,y),\qquad y(a)=y_0 \tag {4} \] by the linear multistep method (LMM) \[ \sum _{j=0}^k \alpha _j y_{n+j} = h\sum _{j=0}^k \beta _j f_{n+j}. \tag {5} \]

where we use the standard notation \[ x_n=a+nh,\qquad y_n\approx y(x_n),\qquad f_n=f(x_n,y_n). \tag {6} \] Define what it means that the LMM is explicit. Define its region of absolute stability. Show that an explicit linear multistep method cannot have an infinite region of absolute stability.

Numerical ODEs Define the terms linear multistep method (LMM), order, consistency, zero-stability, local truncation error, and region of absolute stability of an LMM. What is the highest order implicit 1 step method and discuss its region of absolute stability?

Numerical Methods for ODEs Consider the Linear Multistep Method \[ y_{n+2}-\frac {4}{3}y_{n+1}+\frac {1}{3}y_n=\frac {2}{3}hf_{n+2} \] for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta \). You may assume that \(f\) is Lipschitz continuous with respect to \(y\) for all \(x\).

a) Analyze the consistency, stability, accuracy, and convergence properties of this method.

b) Would it be more reasonable to use this method or Euler’s method for the initial value problem: \[ y'=-10^8[y-\cos (x)]-\sin (x),\qquad y(0)=1? \] Justify your answer.

Numerical ODEs Define the terms linear multistep method (LMM), order, consistency, zero-stability, local truncation error, and region of absolute stability of an LMM. Derive the highest order implicit 2 step method and discuss its region of absolute stability.

Numerical Methods for ODEs Consider the Linear Multistep Method \[ y_{n+1}-y_n=\frac {1}{2}h(f_{n+1}+f_n) \] for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta \). You may assume that \(f\) is Lipschitz continuous with respect to \(y\) for all \(x\).

a) Analyze the consistency, stability, accuracy, and convergence properties of this method.

b) Would it be more reasonable to use this method or Euler’s method for the initial value problem: \[ y'=-10^8[y-\cos (x)]-\sin (x),\qquad y(0)=1? \] Justify your answer.

Numerical Methods for ODEs Consider the Linear Multistep Method \[ y_{n+2}-\frac {4}{3}y_{n+1}+\frac {1}{3}y_n=\frac {2}{3}hf_{n+2} \] for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta \). You may assume that \(f\) is Lipschitz continuous with respect to \(y\) for all \(x\).

a) Analyze the consistency, stability, accuracy, and convergence properties of this method.

b) Would it be more reasonable to use this method or Euler’s method for the initial value problem: \[ y'=-10^8[y-\cos (x)]-\sin (x),\qquad y(0)=1? \] Justify your answer.

Numerical Methods for ODEs Consider the Linear Multistep Method \[ y_{n+2}-\frac {4}{3}y_{n+1}+\frac {1}{3}y_n=\frac {2}{3}hf_{n+2} \] for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta \). You may assume that \(f\) is Lipschitz continuous with respect to \(y\) for all \(x\). a) Analyze the consistency, stability, accuracy, and convergence properties of this method. b) Would this method be a reasonable one to use for the initial value problem: \[ y'=-10^8[y-\cos (x)]-\sin (x),\qquad y(0)=1? \] Justify your answer.