Variable-Coefficient Diffusion Problems

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the scheme, \[ u_{n+1}+\alpha _1u_n = k\beta _0 f_{n+1}+k\beta _1 f_n, \] where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).

1. (10 pts) Identify the coefficients \(\alpha , \beta \) that yield a scheme of optimal \(k\)-order of accuracy, and identify this order of accuracy.
2. (10 pts) Determine whether or not this scheme is 0-stable and/or A-stable.

(20 pts) Consider the multi-step method, \[ u_{n+1}+\alpha _1u_n+\alpha _2u_{n-1}=k\beta _1 f_n+k\beta _2 f_{n-1}, \] where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).

1. (10 pts) Identify the coefficients \(\alpha , \beta \) that yield a scheme of optimal \(k\)-order of accuracy, and identify this order of accuracy.
2. (10 pts) Determine whether or not this scheme is 0-stable and/or A-stable. Is the scheme convergent?

(20 pts)

1. (10 pts) Compute coefficients \(A\), \(B\), \(C\) in the 3-point one-sided approximation \[ u''(x) \approx Au(x) + Bu(x+h) + Cu(x+2h) \]

   that has optimal \(h\)-order of accuracy. What is this order of accuracy?

2. (10 pts) Compute weights for the following quadrature rule that makes it exact for all polynomials of degree 2. \[ \int _{-1}^1 f(x)\,dx = w_0 f(0) + w_1 f'(0) + w_2 f''(0), \]

(20 pts) Consider the multi-step method, \[ u_{n+1}+\alpha _1 u_n = k\beta _0 f_{n+1}+k\beta _1 f_n, \] where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).

1. (10 pts) Identify the coefficients \(\alpha ,\beta \) that yield a scheme of optimal \(k\)-order of accuracy, and identify this order of accuracy.
2. (10 pts) Determine whether or not this scheme is 0-stable and/or A-stable.

(Numerical Integration). Determine the coefficients \(A_0\), \(A_1\), and \(A_2\) that make the formula \[ \int _0^2 f(x)\,dx \approx A_0 f(0) + A_1 f(1) + A_2 f(2) \] exact for all polynomials of degree 3.

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!

The following function \(f(x)\) [\(x\) is a real variable] is expanded in a series in powers of \((x+3)\). Find the radius of convergence of the series in each case. \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&e^{x^2},\\[1.4ex] \text {(b)}\quad f(x)&=&\left (\dfrac {\sin x-3}{\sin x-2}\right )^2,\\[1.8ex] \text {(c)}\quad f(x)&=&\dfrac {\sin (x+1)}{(x+1)(x+3)}. \end {array} \]

Heat Equation Stability

Consider the variable coefficient diffusion equation \[ v_t=(\beta (x)v_x)_x,\qquad 0<x<1,\quad t>0 \] 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 THE PROBLEM HAS VARIABLE COEFFICIENTS AND THAT THERE ARE BOUNDARY CONDITIONS AT 0 AND 1!

\(F(\omega )\) is the Fourier transform of \(f(x)\) (which is continuous and not identical zero), \(x\) and \(\omega \) are real variables. Explain why it is impossible that both functions \(f(x)\) and \(F(\omega )\) have finite support.

Find the radius of convergence of the Taylor series with the center at the origin (\(x_0=0\)) for the following function

1. \[ f(x)=\frac {\sqrt {x+3}}{x^2+3} \]
2. \[ f(x)=\frac {1}{(\cos x+3)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

Find the radius of convergence of the Taylor series with the center at \(x_0=1\) for the following function

1. \[ f(x)=x^{1/3} \]
2. \[ f(z)=\frac {(\sin x+2)^2}{(\sin x-2)^2} \]

(\(x\) is a real variable; you do not need to find the series themselves).

Find the radius of convergence of the Taylor series for the function \[ \frac {1}{2+\cos x} \qquad (x\text { is a real variable}) \] around the origin (\(x_0=0\)). {You do not need to find the Taylor series itself.}

Heat Equation Consider the variable coefficient diffusion equation \[ v_t=(\beta v_x)_x,\qquad 0<x<1,\quad t>0 \] with Dirichlet boundary conditions \[ v(0,t)=0.0,\qquad v(1,t)=0.0 \] and initial data \(v(x,0)=f(x)\). Assume that \(\beta (x)\ge \beta _0>0\), and that \(\beta (x)\) is smooth. For any fixed value of \(\theta \in [0,1]\), consider the \(\theta \)-scheme for this problem: \[ \begin {aligned} \frac {u_j^{n+1}-u_j^n}{k} =\frac {1}{h^2}\{& (1-\theta )(\beta _{j-1/2}u_{j-1}^{n+1} -(\beta _{j-1/2}+\beta _{j+1/2})u_j^{n+1} +\beta _{j+1/2}u_{j+1}^{n+1})\\ &+\theta (\beta _{j-1/2}u_{j-1}^n -(\beta _{j-1/2}+\beta _{j+1/2})u_j^n +\beta _{j+1/2}u_{j+1}^n)\}. \end {aligned} \] with appropriate treatment of the boundary conditions in the relevant equations. Here, \(h\) is the spacestep, \((J+1)h=1\), \(k\) is the timestep, and \(\beta _{j+1/2}=\beta (x_{j+1/2})\), where \(x_{j+1/2}=(j+1/2)h\). Let \(M\) be the symmetric \(J\times J\) matrix defined by \[ \setlength {\arraycolsep }{3pt} M= \begin {pmatrix} -(\beta _{1/2}+\beta _{3/2}) & \beta _{3/2} & \cdots & \cdots & \cdots & \cdots \\ \beta _{3/2} & -(\beta _{3/2}+\beta _{5/2}) & \beta _{5/2} & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \beta _{J-3/2} & -(\beta _{J-3/2}+\beta _{J-1/2}) & \beta _{J-1/2}\\ \cdots & \cdots & \cdots & \cdots & \beta _{J-1/2} & -(\beta _{J-1/2}+\beta _{J+1/2}) \end {pmatrix}. \] Analyze the stability of this scheme in terms of \(\theta \), \(\frac {k}{h^2}\), and the eigenvalues of \(M\). Are there values of \(\theta \) for which the scheme is stable for all choices of \(\frac {k}{h^2}>0\)? (Note that answering this question should not require lots of writing.)

Bernstein-Bezier Form Let \(T\) be a non-degenerate triangle with vertices \(V_1\), \(V_2\), and \(V_3\). The barycentric coordinates \(b_1\), \(b_2\), and \(b_3\) of a point \(x\in \mathbb {R}^2\) with respect to \(T\) are defined by \[ x=\sum _{i=1}^3 b_i V_i \qquad \text {where}\qquad \sum _{i=1}^3 b_i=1. \tag {6} \] A polynomial \(p\) of degree \(d\) can be written uniquely in its Bernstein-Bezier form as \[ p(x)=\sum _{i+j+k=d}\frac {d!}{i!j!k!}c_{ijk}b_1^i b_2^j b_3^k. \tag {7} \] Assuming you know the coefficients \(c_{ijk}\) of \(p\) as a polynomial of degree \(d\), show how to write \(p\) as a polynomial of degree \(d+1\) in Bernstein-Bezier form. What are the coefficients of \(p\) if \(p(x)=1\)?

Newton’s Method Describe how (the two-variable version of) Newton’s Method can be used to solve the nonlinear system \[ \begin {aligned} x^2+y^2 &= 1,\\ e^x-y &= 0 \end {aligned} \tag {1} \] Discuss the selection of a starting point and the advantages and disadvantages of Newton’s method.

Suppose \(f(\theta )\) is a \(2\pi \) periodic function. Define its Hilbert Transform as follows: Let \(u(r,\theta )\) be the solution of Laplace’s equation on the unit circle satisfying Neumann data \(u_r(1,\theta )=f(\theta )\). The Hilbert transform of \(f\) is \(H(f)=u_\theta (1,\theta )\).

Use separation of variables to find the Hilbert Transform of \(f\) in terms of its Fourier coefficients.

Interpolation. Suppose you are given an arbitrarily smooth function \(f\) of a single variable \(x\), and points \[ x_0<x_1<\cdots <x_n. \] Let \(p\) be the interpolating polynomial of degree \(n\) satisfying \[ p(x_i)=f(x_i),\qquad i=0,1,\ldots ,n. \] Derive an expression for the error \[ f(x)-p(x) \] for a general value of \(x\), in terms of the derivatives of \(f\).