Past exam of the numerical solution of differential equations course of the University of Oxford / 2010 / B / 4

Introduce $v=u^{\prime }$ and solve the initial-value systemThe unknown initial slope $s$ is the shooting method parameter. Integrate to one and solve the scalar residual equation $R(s)=u(1;s)-\beta =0$, using a secant iteration or a bracketed root finder when suitable data are available.

For a Newton update, integrate the sensitivity equations along the current shot:Then $R^{\prime }(s)=w(1)$ and the update is $s\leftarrow s-R(s)/w(1)$ when this derivative is nonzero. After a root has been found, the corresponding shot satisfies both endpoint conditions. This formulation does not presume that a nonlinear boundary-value problem necessarily has a unique solution.

At $x_r=rh$, the central finite difference equations arewith $U_0=\alpha$, $U_m=\beta$. For $f=f(x)$ this is a linear system $A_h{U}_h=b_h$, with row coefficientsThe right-hand side starts from $f(x_r)$ and includes $-a_1\alpha$ in its first entry and $-c_{m-1}\beta$ in its last entry. Under $h\max p<2$, both off-diagonal coefficients are negative, so $|a_r|+|c_r|=2h^{-2}<d_r$. Omitting nonexistent endpoint columns can only reduce this sum. Therefore $A_h$ is strictly diagonally dominant. In particular the linear discrete system is nonsingular.

Use the fine matrix $A_h$ and right-hand side from part (b), with diagonal $D_h$. Starting from an iterate $v$, take $\nu$ weighted Jacobi smoothing stepsForm the residual $r_h=b_h-A_{h}v$. For coarse interior indices $j=1,\dots ,m/2-1$, use full-weighting restrictionThe prolongation operator matrix $P$ is linear interpolation of a coarse correction: $(Pe{)}_{2j}=e_j$ and $(Pe{)}_{2j+1}=(e_j+e_{j+1})/2$, with zero endpoint corrections. Thus $R=P^T/2$. Choose the Galerkin coarse-grid operator $A_c=R{A}_{h}P$, solve $A_c{e}_c=R{r}_h$ exactly, and correct $v\leftarrow v+P{e}_c$. The entries of $A_h$, $R$ and $P$ above specify every matrix in the cycle. A rediscretized coarse operator is another standard choice if used consistently.

For a nonsingular coarse operator, the fine-grid iteration matrix of this pre-smoothed two-grid cycle isThe purpose of under-relaxation is to damp fine-grid oscillations, leaving smoother error for coarse-grid correction. For the diffusion-dominated one-dimensional operator, $\omega =2/3$ is the usual effective choice; the high-frequency symbol $1-\omega (1-\cos \xi )$ then has modulus at most $1/3$ for $\pi /2\le |\xi |\le \pi$. Coefficients and convection can change the best parameter, so this is a motivated choice rather than a universal optimum. Under the diagonal-dominance condition, the unweighted Jacobi matrix already has norm less than one in the maximum norm, and its convex combination with the identity remains convergent for $0<\omega <1$.