Stability analysis of inverse Lax–Wendroff boundary treatment of high order compact difference schemes for parabolic equations

Abstract

In this paper, we study the stability of a numerical boundary treatment of high order compact finite difference methods for parabolic equations. The compact finite difference schemes could achieve very high order accuracy with relatively small stencils. To match the convergence order of the compact schemes in the interior domain, we take the simplified inverse Lax–Wendroff (SILW) procedure (Tan et al., 2012; Li et al., 2017) as our numerical boundary treatment. The third order total variation diminishing (TVD) Runge–Kutta method (Shu and Osher, 1988) is taken as our time-stepping method in the fully-discrete case. Two analysis techniques are adopted to check the algorithm’s stability, one is based on the Godunov–Ryabenkii theory, and the other is the eigenvalue spectrum visualization method (Vilar and Shu, 2015). Both the semi-discrete and fully-discrete cases are investigated, and these two different analysis techniques yield consistent results. Several numerical experimental results are shown to validate the theoretical results.