Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 80 / 2

For fixed real $\kappa$, consider initial data $u_0\in L^2(0,1)$ and a solution continuous into $L^2$ at time zero, with homogeneous Dirichlet boundary conditions for positive times. Existence, uniqueness and continuous dependence in this norm give Hadamard well-posedness.

To construct a solution, put $v(x,t)=e^{\kappa x}u(x,t)$. Substitution givesLet $b_j=2{\int }_0^1{e}^{\kappa x}{u}_0(x)\sin (j\pi x)dx$. The Fourier sine series solution isMultiplication by $e^{\pm \kappa x}$ is bounded on $L^2(0,1)$. Completeness of the sine basis gives convergence to $u_0$ in $L^2$ at time zero, while the exponential decay gives smoothness for positive times and the required equation and boundary values. For a classical solution continuous up to the endpoints at time zero, the usual initial-boundary compatibility conditions are additionally required.

For the energy method, integration by parts givesBoth boundary terms vanish. The same identity applies to the difference of two solutions, proving uniqueness and continuous dependence. The Poincaré inequality further giveswith the identical estimate for differences of initial data. This Dirichlet convection-diffusion contraction holds for every real $\kappa$, and its energy constant is independent of the drift strength.

Write $h=\Delta x$, impose $u_0=u_{M+1}=0$, and use the discrete inner product $⟨u,v{⟩}_h=h{\sum }_{m=1}^M{\overline{u}}_m{v}_m$. Discrete integration by parts for the second difference givesFor the centered drift, reindexing with the zero endpoints givessince the two sums are complex conjugates. This is the fact that centered discrete advection is skew-adjoint. The semidiscrete energy estimate is thereforeThe finite-dimensional linear system has a unique solution for all times, and $\Vert u(t){\Vert }_h\le \Vert u(0){\Vert }_h$, uniformly in the mesh and in real $\kappa$. Differences of two semidiscrete solutions obey the same bound, which is the required stability.

One may also use the smallest eigenvalue ${\lambda }_h=4h^{-2}{\sin }^2(\pi /[2(M+1)])$ of the negative second-difference operator to obtain $\Vert u(t){\Vert }_h\le e^{-{\lambda }_{h}t}\Vert u(0){\Vert }_h$. This conclusion concerns the method of lines; a later choice of time integrator has its own stability conditions.