6620-2023-08

(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.

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

( 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.