Convergence of Runge-Kutta approximations for parabolic problems with Neumann boundary conditions (Q1361792)

The author examines the stability and convergence of Runge-Kutta/Galerkin schemes for inhomogeneous linear parabolic equations, where coefficients and inhomogeneous term are time dependent, with Neumann boundary constraints. A typical result is that if the time step satisfies \(k\leq C_0h\), where \(h\) is the size of the spatial Galerkin mesh, then \[ \max|u(t^n)-u^n_k|_{L^2(\Omega)}\leq C(h^r+k^{p+\frac{3}{2}}). \] Here \(u\) is the analytic solution, \(u^n_k\) is the numerical solution at level \(nk\), \(\Omega\) is a bounded domain, \(r\) is the optimal spatial rate of approximation in Sobolev norm, and \(p\) is the stage order of the chosen Runge-Kutta method, which is assumed to satisfy a number of standard hypotheses. Consistency and stability are proved for the proposed schemes, which yield estimates of the error as the one above. The main tool to obtain these results is a new consistency technique based on the construction at the intermediate time levels of suitable smooth approximations to the solution. This consistency technique has been introduced by \textit{O. Karakashian, G. D. Akrivis} and \textit{V. A. Dougalis} [SIAM J. Numer. Anal. 30, No. 2, 377-400 (1993; Zbl 0774.65091)] in a recent paper dealing with the nonlinear Schrödinger equation. It is important to stress that, although not necessarily with unconditional stability, the results in the paper extend well established estimates to the numerical solution of problems with time dependent coefficients.