Stability analysis of Galerkin/Runge-Kutta Navier-Stokes discretisations on unstructured grids

Abstract

This paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the Navier-Stokes equations on a 3D domain with periodic boundary conditions. Using a suitable de nition of the`perturbation energy' it is shown that the energy is monotonically decreasing for both the original p.d.e. and the semi-discrete system of o.d.e.'s arising from a Galerkin discretisation on a tetrahedral grid. Using recent theoretical results concerning algebraic and generalised stability, su cient stability limits are obtained for both global and local timesteps for fully discrete algorithms using Runge-Kutta time integration. Subject classi cations: AMS(MOS): 65M10, 65M20, 65M60, 76-08, 76N10

New optimized fourth-order compact finite difference schemes for wave propagation phenomena

“…Other important parameters are r a = r( π 2 ), which is the length of the positive imaginary axis segment within S, and r d = r(π ), which is the length of the negative real axis segment within S. Their numerical values are r a = 3.819 and r d = 4.017, respectively. For instance, we notice that for the classic fourth-order four-stage RK44 scheme, we have r c = 2.616, r a = 2.828, and r d = 2.785 [13].…”

“…Additional schemes on, and near the boundaries, at j = 0, 1, and 2, (and at j = N, N − 1, and N − 2) where the stencil protrudes the domain, are required. In this work, we use the fourth-order (second-order at j = 0 for numerical stability) optimized pentadiagonal schemes proposed by Kim and Lee [25], represented as follows: (12) (13) (14) for j = 0 (boundary node), 1, and 2, respectively. The coefficients in (12)-(14) are obtained, as described in [24], by minimizing over an optimization range of wave numbers, dispersion and, for the non-centered structure of (12)-(14) also dissipation errors.…”

On the stability and convergence of discretizations of initial value p.d.e.s

“…cjyj r+1 : To prove the result for z 2 ? (2) n , note that since (V \D c ) int(S), it is possible to choose n 1 n 0 such that (V n 1 \D c ) int(S). The constant b can then be de ned by e ?b = sup Proof We start with the standard Cauchy integral formula,…”

“…For z 2 ? (2) n , using the last lemma, n?1 X j=0 ' j (z) For z 2 ? (1) n , we rst note that z 6 =0 and '(z)6 =0 and so n?1 X j=0 ' j (z) ?…”