Absolute Stability and A-Stability

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the time-integration scheme \[ u_{n+1} = u_n + \frac {k}{4}f(t_n,u_n) + \frac {3k}{4}f\left (t_n+\frac {2}{3}k,\ u_n+\frac {2}{3}kf(t_n,u_n)\right ). \]

1. (5 pts) Identify the Butcher tableau for this scheme.
2. (8 pts) Compute the order of consistency of this scheme.
3. (7 pts) Compute the stability/amplification factor of the scheme.

(20 pts)

1. (10 pts) Use von Neumann stability analysis to determine a stability condition for, \[ D^+u_j^n = D_0u_j^n, \] where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\), \(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and \(D_\pm u_j^n = \pm (u_{j\pm 1}^n-u_j^n)/h\).
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for \[ D^+u_j^n = D_+D_-u_j^n. \]

(20 pts) Consider the PDE, \[ u_t + au_x = 0,\qquad u(x,0)=u_0(x),\qquad a > 0, \] with periodic boundary conditions over the spatial domain \(x \in [0, 1)\). Use the standard notation, \(u_j^n \approx u(x_j,t^n)\) with \(x_j = jh\) and \(t^n = nk\) with \(h = \frac {1}{M} > 0\) for some \(M \in \mathbb {N}\) and \(k > 0\). Consider the following semi-Lagrangian scheme: \[ u_j^{n+1}=p_j(y),\qquad y := X(t^n;x_j,t^{n+1}), \] where \(X(t; x, t_0)\) is the time-\(t\) spatial location of the PDE’s characteristic curve passing through spatial location \(x\) at time \(t_0\), and \(p_j(\cdot )\) is the linear interpolant formed by the data \((x_{j-1},u_{j-1}^n)\) and \((x_j,u_j^n)\).

1. (10 pts) Write this scheme as an explicit function of the \(u_j^n\) values.
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for this scheme.

(20 pts)

1. (10 pts) Compute the stability/amplification factor for forward Euler to solve \(u' = f(t,u)\), and plot the region of the stability.
2. (10 pts) Compute the stability/amplification factor for the trapezoidal rule/Crank- Nicolson scheme and use this to show that the scheme is A-stable.

(20 pts)

1. (10 pts) Use von Neumann stability analysis to determine a stability condition for, \[ D^+u_j^n = D_0u_j^n, \] where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\), \(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and \(D_\pm u_j^n = \pm (u_{j\pm 1}^n-u_j^n)/h\).
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for \[ D^+u_j^n = D_+D_-u_j^n. \]

(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 \(\operatorname {Real}(\lambda ) < 0\).

( Heat Equation and Stability of the Scheme). Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} -10\frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0 = g_m,\qquad m = 0,\pm 1,\pm 2,\ldots ,\qquad n = 0,1,\ldots ,[T/\Delta t]-1, \]

and investigate the stability of the scheme using the von Neumann analysis.

Consider the boundary value problem \[ u''(x) = f(x),\quad \text {for }0 < x < 1, \] \[ u(0) = 0,\quad u(1) = 0, \]

where \(f\) is a smooth function on \([0, 1]\), and the finite difference scheme \[ \frac {U_{j-1}-2U_j+U_{j+1}}{h^2} = f(x_j), \qquad j = 1,2,\ldots ,m, \] for finding approximations \(U_j\) to \(u(x_j)\). Here, \(x_j = jh\) and \((m+1)h = 1\).

Analyze the accuracy, stability, and convergence properties of the scheme.

For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta \), where \(f(u,t)\) is continuous with respect to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme \[ \frac {U^{n+2}-\frac {4}{3}U^{n+1}+\frac {1}{3}U^n}{k} = \frac {2}{3}f(U^{n+2},t_{n+2}), \] where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).

1. Analyze the consistency, zero-stability, and convergence of this scheme.

2. If you apply this scheme to the initial value problem

   \[ u'(t) = -10^{12}(u(t)-\cos (t))-\sin (t),\qquad u(0) = 2, \]

   what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\) must be very small to get a reasonable solution and others in which it does not? Explain your answers.

1. Consider the initial value problem for the constant-coefficient diffusion equation (with \(\beta > 0\)) \[ v_t = \beta v_{xx},\quad t > 0 \]

   with initial data \(v(x,0) = f(x)\). A scheme for this problem is: \[ \frac {u_j^{n+1}-u_j^n}{k} = \frac {\beta }{h^2}\left \{u_{j-1}^{n+1}-2u_j^{n+1}+u_{j+1}^{n+1}\right \}. \]

   Analyze the 2-norm stability of this scheme. For which values of \(k > 0\) and \(h > 0\) is the scheme stable? (Note that there are no boundary conditions here.)

2. Consider the variable coefficient diffusion equation \[ v_t = (\beta (x)v_x)_x,\quad 0 < x < 1,\quad t > 0 \]

   with Dirichlet boundary conditions \[ v(0,t) = 0,\qquad v(1,t) = 0 \]

   and initial data \(v(x,0) = f(x)\). Assume that \(\beta (x) \ge \beta _0 > 0\), and that \(\beta (x)\) is smooth. Let \(\beta _{j+1/2} = \beta (x_{j+1/2})\). A scheme for this problem is: \[ \frac {u_j^{n+1}-u_j^n}{k} = \frac {1}{h^2}\left \{\beta _{j-1/2}u_{j-1}^{n+1} -(\beta _{j-1/2}+\beta _{j+1/2})u_j^{n+1} +\beta _{j+1/2}u_{j+1}^{n+1}\right \}. \]

   Analyze the 2-norm stability of this scheme for solving this initial boundary value problem. DO NOT NEGLECT THE FACT THAT THERE ARE BOUNDARY CONDITIONS!

For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta \), where \(f(u,t)\) is continuous with respect to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme \[ \frac {U^{n+2}-\frac {4}{3}U^{n+1}+\frac {1}{3}U^n}{k} = \frac {2}{3}f(U^{n+2},t_{n+2}), \] where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).

1. Analyze the consistency, zero-stability, and convergence of this scheme.

2. If you apply this scheme to the initial value problem

   \[ u'(t) = -10^{12}(u(t)-\cos (t))-\sin (t),\qquad u(0) = 2, \]

   what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\) must be very small to get a reasonable solution and others in which it does not? Explain your answers.

 Heat Equation and Stability of the Scheme

Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - 4\frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1, \] and investigate the stability of the scheme using the von Neumann analysis.

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.

Trapezoid rule

For the ODE \(y'=f(t,y)\), the Trapezoid rule is \[ \frac {y_{n+1}-y_n}{\Delta t} = \frac {1}{2}\left (f(t_n,y_n)+f(t_{n+1},y_{n+1})\right ). \] Compute the region of absolute stability for the Trapezoid rule. Is this method \(A\)-stable?

Unconditionally stable scheme

For the one-dimensional heat equation, \[ u_t=u_{xx}, \] consider the Crank-Nicolson, central difference scheme \[ \frac {u_j^{n+1}-u_j^n}{\Delta t} = \frac {1}{2} \left ( \frac {u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{h^2} + \frac {u_{j+1}^{n}-2u_j^{n}+u_{j-1}^{n}}{h^2} \right ) \] where \(\Delta t\) and \(h\) are temporal and spatial stepsizes, respectively, and where \(u_j^n\) is the numerical approximation to \(u(x=jh,t=n\Delta t)\), with \(x_0=0\) and the initial time \(t=0\). Use von Neumann stability analysis to show that this scheme is unconditionally stable.

Central scheme

Consider the one-dimensional advection equation, \[ u_t=cu_x, \] where \(c\in \mathbb {R}\) is the wavespeed. Use von Neumann stability analysis to determine the values of \(c\) for which the following the central difference, forward Euler scheme is stable: \[ \frac {u_j^{n+1}-u_j^n}{\Delta t} = c\frac {u_{j+1}^{n}-u_{j-1}^{n}}{2h}. \]

Heat Equation and Stability of the Scheme

Consider the implicit in time, Backward Euler method for the solution of the heat equation: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_{m+1}^{n+1}-2u_m^{n+1}+u_{m-1}^{n+1}}{h^2} = f_m^{n+1}, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \] and investigate the stability of the scheme using the von Neumann analysis.

1. Prove: If a function is analytic then its real and imaginary parts are harmonic.
2. Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function \(\phi (z)=e^{f(z)}\) has absolute value \(\lvert \phi (z)\rvert =e^u\).]

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.

Upwind Scheme

Consider the advection equation \[ u_t-u_x=0,\qquad x_L<x<x_R,\qquad 0<t\le T, \] where \(u(x,0)=g(x)\), and \(u(x_R,t)=u_R(t)\) for \(t>0\).

a) Write the Upwind Scheme for this problem.

b) What is the stencil of the scheme? What is the CFL condition for this method?

c) Investigate the stability of the method using Von Neumann Stability Analysis.

ODEs Consider the initial value problem for a system of ODEs \(y'=Ay\) where \(A\) has eigenvalues that are widely spread in the left half plane. Which of the following schemes would be the best choice for solving this problem? Justify your answer in terms of stability, accuracy, and efficiency. \[ \begin {aligned} (1)\quad y^{n+1} &= y^n+kAy^n,\\ (2)\quad y^{n+1} &= y^n+kAy^{n+1},\\ (3)\quad y^{n+1} &= y^n+\frac {k}{2}A\left (y^n+y^{n+1}\right ). \end {aligned} \]

Heat Equation Consider the following heat equation problem \[ u_t=u_{xx},\qquad t>0,\qquad 0<x<1, \] \[ u(0,t)=u(1,t)=0,\qquad t>0, \] \[ u(x,0)=u_0(x),\qquad 0\le x\le 1. \] Using the forward or backward difference for the time derivative and central difference for the space derivative, derive the explicit and implicit schemes. What’s the stability condition for each case? Perform a Fourier analysis on both the explicit and implicit methods and use the von Neumann condition to verify these stability conditions.

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.

Stability of the Scheme

Using the von Neumann method investigate the stability of the implicit downwind scheme: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_m^{n+1}-u_{m-1}^{n+1}}{h} = f_m^n, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \]

Stability of the Scheme

Using the von Neumann method investigate the stability of the implicit upwind scheme: \[ \frac {u_m^{n+1}-u_m^n}{\Delta t} - \frac {u_{m+1}^{n+1}-u_m^{n+1}}{h} = f_m^n, \] \[ u_m^0=g_m,\qquad m=0,\pm 1,\pm 2,\ldots ,\qquad n=0,1,\ldots ,[T/\Delta t]-1. \] Comment on the CFL condition for this scheme.

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.

Crank Nicolson Method Describe the Crank-Nicolson Method for the solution of the equation \[ u_t=u_{xx} \tag {6} \] and use the von Neumann method to examine its stability.

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.

ODE Consider the initial value problem for a system of ODEs \[ y'=Ay \] where \(A\) has eigenvalues ranging from \(-10^4\) to \(-1\). Which of the following schemes would be the best choice for solving this problem? Justify your answer in terms of stability, accuracy, and efficiency. \[ y^{n+1}=y^n+hAy^n, \tag {1} \] \[ y^{n+1}=y^n+hAy^{n+1}, \tag {2} \] \[ y^{n+1}=y^n+\frac {h}{2}(Ay^n+Ay^{n+1}). \tag {3} \]

Wave Equation For the PDE \(v_t+cv_x=0\), derive the Lax-Wendroff scheme \[ u_j^{n+1}=u_j^n -\frac {1}{2}\left (\frac {c\Delta t}{\Delta x}\right ) (u_{j+1}^n-u_{j-1}^n) +\frac {1}{2}\left (\frac {c\Delta t}{\Delta x}\right )^2 (u_{j-1}^n-2u_j^n+u_{j+1}^n). \] Analyze this scheme in terms of accuracy and stability. Does this scheme satisfy a maximum principle? If it does, explain why. If it does not, give an example to show this.

Convection-Diffusion Equation For the convection-diffusion equation \[ v_t+av_x=bv_{xx}, \] where \(a>0\) and \(b>0\), a natural Crank-Nicolson scheme is \[ \begin {aligned} \frac {u_j^{n+1}-u_j^n}{\Delta t} &+\frac {a}{2}\left ( \frac {u_{j+1}^n-u_{j-1}^n}{2\Delta x} +\frac {u_{j+1}^{n+1}-u_{j-1}^{n+1}}{2\Delta x} \right ) \\ &= \frac {b}{2}\left ( \frac {u_{j+1}^n-2u_j^n+u_{j-1}^n}{(\Delta x)^2} +\frac {u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}}{(\Delta x)^2} \right ). \end {aligned} \] Let \(\nu =b\Delta t/(\Delta x)^2\) and \(\mu =a\Delta t/(2\Delta x)\). Show that the scheme is stable in the maximum norm if \(\mu \le \nu \le 1\). Also use Fourier analysis to derive the stability condition in the \(\ell _2\) norm and show that the scheme is always stable in the \(\ell _2\) norm. Suppose \(u\) stands for some volume fraction of a certain species, which norm is more appropriate? Explain why this scheme is so restrictive that it is not a good choice for many applications.

Ordinary Differential Equations Consider the initial value problem for a system of two ODEs \[ y'=Ay \] where \(A\) has eigenvalues \(-10^4\) and \(1\). Which of the following schemes would be the best choice for solving this problem? Justify your answer in terms of stability, accuracy, and efficiency. \[ y^{n+1}=y^n+hAy^n, \tag {1} \] \[ y^{n+1}=y^n+hAy^{n+1}, \tag {2} \] \[ y^{n+1}=y^n+\frac {h}{2}(Ay^n+Ay^{n+1}). \tag {3} \]

Wave Equation Consider the Lax-Friedrichs scheme below for the PDE \(v_t+cv_x=0\), where you may assume that \(c>0\): \[ u_j^{n+1}=\frac {1}{2}(u_{j-1}^n+u_{j+1}^n) -\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n), \] where \(h\) is the spacestep and \(k\) is the timestep. Note that the scheme can also be written \[ u_j^{n+1}=u_j^n-\frac {ck}{2h}(u_{j+1}^n-u_{j-1}^n) +\frac {1}{2}(u_{j-1}^n-2u_j^n+u_{j+1}^n). \]

a) Analyze the stability of this scheme in terms of \(\alpha =\frac {ck}{h}\).

b) Recall that a modified equation for a scheme is the PDE satisfied, to a certain order, by a smooth solution \(u(x,t)\) of the difference scheme. To zeroth order in \(k\) and \(h\), the modified equation for the Lax-Friedrichs scheme is \(u_t+cu_x=0\). Derive the modified equation through terms of first order in \(k\) and \(h\). You may assume that \(\alpha \) is constant. Does this modified equation predict the same stability condition as your analysis in part a)? Explain your answer. What does the modified equation suggest about the numerical solution that will be obtained for the step-function initial data \(u(x,0)=1\) for \(x\le 0\) and \(u(x,0)=0\) for \(x>0\)?

Numerical ODEs

a) (8%) Consider the general system of ODEs for \(y(x)\in \mathbb {R}^n\), \(x\in \mathbb {R}\), and \[ f=[f_1(x,y),\ldots ,f_n(x,y)]^T: \qquad y'=f(x,y), \] where we assume \(f\) is continuously differentiable in both \(x\) and \(y\). To study local stability, we compare two solutions \(y_1(x)\) and \(y_2(x)\) where \(y_1(x_0)\approx y_2(x_0)\). Derive the linearized system for \(e(x)=y_1(x)-y_2(x)\) near \(x=x_0\) to show that \[ e'\approx Ae \] and find this square matrix \(A\). What is the most important property of \(A\) we need to find that will affect the choice of numerical schemes and step sizes?

b) (12%) Now consider the initial value problem \[ y''+2\mu y'+(1+\mu ^2)y=0, \] \[ y(0)=0,\qquad y'(0)=1, \] where \(\mu >0\) is a given constant. If we use a simple explicit Euler’s method, determine the maximum step size \(h\) that is allowed for the solution to be stable.

Convection-Diffusion Equation For the convection-diffusion equation \[ u_t+au_x=bu_{xx}, \] where \(a>0\) and \(b>0\), a natural Crank-Nicolson scheme is \[ \begin {aligned} \frac {U_j^{n+1}-U_j^n}{\Delta t} &+\frac {a}{2}\left ( \frac {U_{j+1}^n-U_{j-1}^n}{2\Delta x} + \frac {U_{j+1}^{n+1}-U_{j-1}^{n+1}}{2\Delta x} \right )\\ &= \frac {b}{2}\left ( \frac {U_{j+1}^n-2U_j^n+U_{j-1}^n}{(\Delta x)^2} + \frac {U_{j+1}^{n+1}-2U_j^{n+1}+U_{j-1}^{n+1}}{(\Delta x)^2} \right ). \end {aligned} \] Denote \(\nu =b\Delta t/(\Delta x)^2\) and \(\mu =a\Delta t/(2\Delta x)\); show that the scheme is stable in the max norm if \(\mu \le \nu \le 1\). Also use Fourier analysis to derive the stability condition in the \(\ell _2\) norm and show that the scheme is always stable in the \(\ell _2\) norm. Suppose \(u\) stands for some chemical concentration, which norm is more appropriate? Explain why this scheme is so restrictive that it is not often used.

ODEs Consider the initial value problem of ODEs: \[ y'=f(x,y),\qquad y(a)=y_0. \tag {10} \] Describe the Trapezoidal Rule applied to this problem and analyze its local accuracy and its region of absolute stability.

PDEs Consider the one-dimensional Heat Equation \[ u_t=u_{xx},\qquad t\ge 0,\qquad x\in [0,1],\qquad u(x,0)=f(x),\qquad u(0,t)=u(1,t)=0. \tag {11} \] Describe how to discretize this problem by applying the Method of Lines and the Backward Euler Method. Analyze the stability and local accuracy of your method.

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 PDEs Consider the Initial-Boundary Value Problem \[ u_t=u_{xx},\qquad x\in [0,1],\qquad t\ge 0, \qquad u\text { given for }x=0,\ x=1,\text { and }t=0. \tag {4} \] Describe the Crank-Nicolson Method, and analyze its local truncation error and stability.

Numerical ODEs

(a) Consider the general system of ODEs for \(y(x)\in \mathbb {R}^n\), \(x\in \mathbb {R}\) and \(f=[f_1,\ldots ,f_n]^T\): \[ y'=f(x,y), \] where we assume \(f\) to be Lipschitz continuous. To study the numerical stability, we linearize \(f\) near \(x=x_0,y=y_0\) to obtain: \[ y'\approx f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+\nabla _y f(x_0,y_0)(y-y_0). \] If we denote \(A=\nabla _y f(x_0,y_0)\), then we claim that the local stability of the above system can be analyzed by studying the following constant coefficient system \[ y'=Ay. \] Justify this claim by analyzing the simple scalar equation \[ y'=a+bx+cy, \] where \(a,b,c\) are constants, in the explicit Euler’s method, to show that the stability is determined by how we solve \(y'=cy\). What is the most important property of \(A\) we need to find that will affect the choice of numerical schemes and step sizes?

(b) Now consider the initial value problem \[ y''+2\mu y'+(1+\mu ^2)y=0, \] \[ y(0)=0,\qquad y'(0)=1, \] where \(\mu >0\) is a given constant. If we use a simple explicit Euler’s method, determine the maximum step size \(h\) that is allowed for the solution to be stable.

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.

Heat Equation Consider the following heat equation problem \[ u_t=u_{xx},\qquad t>0,\qquad 0<x<1, \] \[ u(0,t)=u(1,t)=0,\qquad t>0, \] \[ u(x,0)=u_0(x),\qquad 0\le x\le 1. \] Using the forward or backward difference for the time derivative and central difference for the space derivative, derive the explicit and implicit schemes. What’s the stability condition for each case? Perform a Fourier analysis on both the explicit and implicit methods and use the von Neumann condition to verify these stability conditions.

1. Find the Fourier transform of \(\dfrac {\sin x}{x}\).
2. Suppose that the Fourier transform of the function \(f(t)\) is 0 for frequencies whose absolute value is greater than \(\pi /h\). Find the representation of \(f(t)\) in terms of sinc functions.

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 PDEs Consider the PDE \[ u_t+u_x=0 \tag {5} \] and the usual discretization \[ U_m^n\approx u(x_m,t_n) \qquad \text {where}\qquad x_m=mh\quad \text {and}\quad t_n=nk. \tag {6} \] The Leap Frog Method is given by \[ U_m^{n+1}=U_m^{n-1}-\frac {k}{h}\left (U_{m+1}^n-U_{m-1}^n\right ). \tag {7} \] Compute the local truncation error and discuss the stability of this method.

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.

Stability. What does stability of a numerical scheme mean? Give a specific example.

Convergence is equivalent to stability and convergence. Describe and prove a specific example for this fundamental result.

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.

a) Derive a multistep method to approximate \(y'=f(t,y)\) of the form \[ y_{n+1}=y_n+h(b_{-1}y'_{n+1}+b_0y'_n+b_1y'_{n-1}) \] with \(y'_j=f(t_j,y_j)\), \(t_j=t_0+jh\) by interpolating \(f(s,y(s))\) at \(t_{n-1}\), \(t_n\), \(t_{n+1}\).

b) Analyze the local truncation error of this method and the consequences regarding convergence. What would be an appropriate predictor for this method?

c) Discuss the region of absolute stability of this method, and the predictor-corrector method described in part b).