Eigenvalue Analysis for Summation-by-Parts Finite Difference Time Discretizations

Abstract

Diagonal norm finite difference based time integration methods in summation-byparts form are investigated. The second, fourth, and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully discrete approximations of initial boundary value problems. Our findings also allow us to conclude that the Runge-Kutta methods based on second, fourth, and sixth order summation-by-parts finite difference time discretizations automatically satisfy previously unreported stability properties. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

Multigrid Schemes for High Order Discretizations of Hyperbolic Problems

“…The invertibility of L 1 can be shown for pseudo-spectral approximations [9], but not in general for discretizations based on finite-difference methods [11]. The 1st and 2nd order upwind SBP operators lead to matrices L 1 with triangular and block-triangular structure, respectively, for which invertibility follows in a straightforward way.…”

“…Quantities on the fine and coarse grid are indicated with superscripts 1, 2, respectively. In (11) we have introduced:…”

“…In (11), the coarse grid evolution step converges to the steady-state solution L −1 2 F 2 . This vector can be recast as Fig.…”

“…This vector can be recast as Fig. 3 The interpolation operators used in the two-level algorithm (11)…”

Some notes on summation by parts time integration methods

“…Turning to SBP methods in time [2,16,21], a class of linearly and nonlinearly stable SBP schemes has been constructed and studied in this context, see also [15,36,37]. If the underlying quadrature is chosen as Radau or Lobatto quadrature, these Runge-Kutta schemes are exactly the classical Radau IA, Radau IIA, and Lobatto IIIC methods [26].…”