Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 40C (Numerical Analysis)

Let $U_m^n=u(mh,nk)$ be the sampled exact solution and let $A_h$ denote the update matrix of the Forward Euler diffusion scheme. For $0\le \mu \le 1/2$,is a convex combination, with the homogeneous boundary values included. ThereforeThis is the parabolic discrete maximum principle and proves max-norm stability directly.

The Taylor theorem and the second-order central difference give the exact-grid residualfor a sufficiently smooth solution on a fixed interval $0\le t\le T$. If $e^n=U^n-u^n$, thenIteration for $nk\le T$ yieldsThus $\mu \le 1/2$ implies convergence when the initial grid values converge to the initial data.

Let $L=[1,-2,1]$ be the symmetric Dirichlet second-difference matrix on the $M$ interior points. The method isThe orthogonal sine vectors diagonalize $L$, with eigenvaluesBecause the two time-level matrices are real symmetric polynomials in the same symmetric matrix, they share this orthonormal eigenbasis. Whenever the left matrix is invertible, the amplification matrix is therefore normal, so its Euclidean operator norm is the largest modulus of its eigenvalues. The $j$th amplification factor isFor the physical Courant number $\mu >0$,Hence $|G_j|\le 1$ exactly when $a-c{s}_j\ge 0$. Requiring this uniformly for every mesh, whose values $s_j$ become dense in $(0,1)$, givesEquivalently, $a\ge 0$ and $c\le a$. Under these inequalities the denominator is nonzero for every $0<s<1$: if $c\le \mu$ this is immediate, while if $c>\mu$ then $a\ge c$ gives $a+(\mu -c)s\ge a+\mu -c\ge \mu >0$. Thus the necessary and sufficient mesh-uniform stability conditions areThis is the stability of a two-parameter implicit-explicit diffusion scheme. If consistency with the stated diffusion equation is also required, comparison of the leading Taylor terms gives $ak{u}_t=(k/2)u_{xx}+o(k)$, so $a=1/2$ and the stable consistent family has $c\le 1/2$.