The computation of equilibrium solutions of forced hyperbolic partial differential equations
This thesis investigates the convergence of numerical schemes for the computation of equilibrium solutions. These are solutions of evolutionary PDEs that arise from (bounded, non-decaying) boundary forcing after the dissipation of any (initial data dependent) transients. A rigorous definition of the term 'equilibrium solution' is given. Classes of evolutionary PDEs for which equilibrium solutions exist uniquely are identified. The uniform well-posedness of equilibrium problems is also investigated. Equilibrium solutions may be approximated by evolutionary initialization: that is, by finding the solution of an initial boundary value problem, with arbitrary initial data, over a period of time t ϵ [0,T]. If T is chosen large enough, the analytic transient will be small, and the analytic solution over t ϵ [T, T + T0] will be a good approximation to the analytic equilibrium solution. However, in numerical computations, T must be chosen so that the analytical transient is small in comparison with the numerical error Eh, which depends on the fineness of the grid h. Thus T = Th, and, in general, Th→∞ as h→0. Convergence is required over t ϵ [Th,Th + T0]. The existing Lax-Richtmyer and GKS convergence theories cannot ensure convergence over such increasing periods of time. Furthermore, neither of these theories apply when the forcing does not decay. Consequently, these theories are of little help in predicting the convergence of finite difference methods for the computation of equilibrium solutions. For these reasons, a new definition of stability - uniform stability — is proposed. Uniformly stable, consistent, finite difference schemes, for uniformly well posed problems, converge uniformly over t ≥ 0. Uniformly convergent schemes converge for bounded and nondecaying forcing. Finite difference schemes for hyperbolic PDEs may admit waves of zero group velocity, even when the underlying analytic problem does not. Such schemes may be GKS convergent, provided that the boundary conditions exclude these waves. The deficiency of the GKS theory for equilibrium computations is traced to this fact. However, uniform stability finds schemes that admit waves of zero group velocity to be (weakly) unstable, regardless of the boundary conditions. It is also shown that weak uniform instabilities are the result of time-dependent analogues of the 'spurious modes' that occur in steady-state calculations. In addition, uniform stability theory sheds new light on the phenomenon of spurious modes.