To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem

B. V. Pal'tsev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 4, pp. 538-574

Résumé

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by $O(\tau^2)$, where $\tau$ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left(\tau^2\ln\frac T\tau\right)$, where $T$ is the length of the time interval.