Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 66 / 1

Take real $\kappa$ and use the L2 norm on the spatial interval. Existence, uniqueness and continuous dependence are the three requirements of Hadamard well-posedness. An energy method supplies the decisive estimate. For a smooth solution with homogeneous Dirichlet boundary conditions, integration by parts givesThe drift contributes only a boundary term, which vanishes. The Poincare inequality $\Vert u_x{\Vert }_2^2\ge {\pi }^2\Vert u{\Vert }_2^2$ further givesApply the same argument to the difference of two solutions to obtain uniqueness and continuous dependence on the initial data.

For existence, use the Dirichlet gauge transform for constant drift: $w=e^{\kappa x/2}u$ satisfies $w_t=w_{xx}-{\kappa }^{2}w/4$ with zero boundary values. Expanding $w_0$ in its Fourier sine series givesFor $u_0\in L^2(0,1)$ the series defines a solution continuous in $L^2$ down to $t=0$ and smooth for positive time; multiplication by the fixed bounded exponentials preserves this interpretation. Its energy estimate follows by approximation with smooth initial data. For a classical solution at the initial corners, require the usual smoothness and boundary compatibility instead. The problem is well posed in $L^2$, with a contraction estimate independent of the initial data.

Let $d=\Delta x$, impose $u_0=u_{M+1}=0$, and write the method of lines system as $U^{\prime }=LU$, whereThus $D_2$ is a negative definite symmetric matrix and $D_1$ is a skew-symmetric matrix. Use the mesh-weighted Euclidean norm $\Vert U{\Vert }_d^2=d{\sum }_{m=1}^M|u_m{|}^2$. Discrete summation by parts yields the centered Dirichlet drift-diffusion energy identityConsequentlyThe same estimate controls perturbations and is uniform in the number of grid points and in the fixed drift coefficient. Finite-dimensional linear ODE theory guarantees existence, so this proves stability of a numerical method for the semidiscretization.

The factor $d^{1/2}$ simply rescales the vector norm and does not change the induced matrix norm. Equivalently the symmetric part is $(L+L^{\ast })/2=D_2$, whose largest eigenvalue is $-4d^{-2}{\sin }^2[\pi /(2(M+1))]<0$. This is the Euclidean logarithmic norm, rather than generally the spectral abscissa of a nonnormal matrix. No periodic Fourier mode assumption has been made: the zero endpoint terms are part of the proof. In particular positivity of both off-diagonal coefficients is not needed for this $L^2$ stability result.

Put $k=\Delta t>0$ and retain the same matrix $L$. The Crank--Nicolson method isTake the real mesh-weighted inner product with $U^{n+1}+U^n$. The cross terms cancel, and the identity from part (b) givesThe implicit system is uniquely solvable: if $(I-kL/2)V=0$, thenso $V=0$. Therefore its dissipative Cayley-transform contraction satisfiesIterating proves unconditional stability for every $\mu =k/d^2>0$; the perturbation bound is one and does not depend on the mesh or time-step ratio.

The printed hint's exponential estimate is valid with the logarithmic norm, but its proposed bound by $r(k\alpha [L])$ is not a general inheritance principle. For the trapezoidal stability function $r(z)=(1+z/2)/(1-z/2)$, that real number can even be negative when $k\alpha [L]<-2$, whereas a norm is nonnegative. The direct energy proof above establishes the required result without that assertion.