am205_quiz3_sol.md

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AM205 Quiz 3. Numerical calculus. Solution	Petr Karnakov

AM205 Quiz 3. Numerical calculus. Solution

Q1

Newton-Cotes formulas are quadrature rules that

• are obtained by integrating a polynomial interpolant
• use Newton's method to find the quadrature weights

Q2

Consider a quadrature rule $Q(f)=\sum_{k=0}^n w_k f(k)$ to approximate the integral $\int_0^2 f(x)dx$. This rule uses $n+1$ function values at integer points $0,\ldots, n$. However, only the points $0,1,2$ belong to the integration range $[0,2]$. Suppose that the constant weights $w_0,\ldots,w_n$ are chosen to maximize the degree of polynomials on which this quadrature is exact. This implies that the quadrature rule is exact on all polynomials of degree (choose the highest)

• $2$
• $3$
• $n$
• $n+1$

Answer: Newton-Cotes formulas using an interpolating polynomial will be exact on polynomials of degree $n$ regardless of the integration range. The interpolating polynomial coincides with the integrand.

Q3

Gauss quadrature using $n+1$ points is exact on all polynomials of degree (choose the highest)

• $n$
• $2n+1$

Q4

The centered difference approximation $\frac{f(x+h)-f(x-h)}{2h}$ to $f'(x)$ is exact on all polynomials $f(x)$ of degree (choose the highest)

• 0
• 1
• 2
• 3
• 4

Q5

Compare two methods for solving a system of linear ODEs: forward Euler (explicit) and backward Euler (implicit). The backward Euler method

• has a larger stability region
• has a higher order of accuracy
• requires solving a linear system at every time step
• none of the above

Q6

Richardson extrapolation applies to an existing numerical method and can be used to

• increase its order of accuracy
• estimate its order of accuracy
• estimate its absolute error
• none of the above

Q7

Compare one-step and multistep methods for solving ODEs. To achieve the same order of accuracy, multistep methods require

• fewer function evaluations
• more function evaluations

Q8

Recall the $\theta$-method for the heat equation. The method is fully explicit with $\theta=0$ and fully implicit with $\theta=1$. Which of the following methods are unconditionally stable?

• Crank-Nicolson
• $\theta$-method with $\theta=0.25$
• $\theta$-method with $\theta=0.5$
• $\theta$-method with $\theta=1$
• none of the above

Q9

Examples of hyperbolic equations are

• advection equation
• heat equation
• Poisson equation
• wave equation
• none of the above

Q10

Recall the central difference method $\frac{U_j^{n+1} - U_j^n}{\Delta t} + c \frac{U_{j+1}^{n} - U_{j-1}^n}{2 \Delta x} = 0$ for the advection equation. Even if $c\Delta t/\Delta x$ is small, the method cannot be used in practice for the following reasons:

• does not satisfy the CFL condition
• unconditionally unstable
• requires two boundary conditions