Summation by parts, projections, and stability. II

Abstract

Abstract.In this paper we prove strict stability of high-order finite difference approximations of parabolic and symmetric hyperbolic systems of partial differential equations on bounded, curvilinear domains in two space dimensions. The boundary need not be smooth. We also show how to derive strict stability estimates for inhomogeneous boundary conditions. Strict stability in several space dimensionsIn [2] we proved stability for high-order finite difference approximations of hyperbolic and parabolic systems using certain projections and difference operators satisfying a summation-by-parts rule. In one space dimension we showed how to obtain strict stability. The aim of this paper is to prove strict stability in several space dimensions. Furthermore, it will also be demonstrated how to handle inhomogeneous boundary conditions. We limit ourselves to two space dimensions for convenience.The purpose of strict stability is to ensure the same growth rate of the discrete and analytic solutions. If the analytic problem is defined on a curvilinear domain Q with boundary Y (cf. Fig. 1 on next page), then there must exist a diffeomorphism Ç = Ç(x) of Q, onto the unit square (0, 1) x (0, 1) in order for the finite difference method to be well defined. Consequently, a constantcoefficient problem in the original domain may be transformed to a variablecoefficient problem on the unit square, which may account for a nonphysical growth in the discrete estimate.

Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

“…), boundary accuracy r = s, and global accuracy p = s + 1. There are SBP operators that have boundary accuracy r = 2s − 1 and global accuracy p = 2s, but using these makes stability proofs difficult for problems with variable coefficients, coordinate transforms, and nonlinear boundary/interface conditions (Nordström and Carpenter, 2001;Olsson, 1995;Nordström, 2006;Kozdon et al, 2011).…”

Scalar field confinement as a model for accreting systems

“…We use this metric to evaluate the equation of motion for φ, equation (31). To solve this PDE we use the following ansatz that yields separation of variables [35],…”

“…In this work we adopt: (i) second order accuracy by implementing second-order derivative operators satisfying summation by parts [22,23,24,25,26]; (ii) a third-order Runge-Kutta operator for the time integration through the method of lines [27]; (iii) a KreissOliger [28] style dissipative algorithm to control the high frequency modes of the solution [26,29,30] and (iv) maximally dissipative boundary conditions setting all incoming modes to zero [29,31].…”

Do unbounded bubbles ultimately become fenced inside a black hole?

“…and apply Olsson's projection method [11] in order to incorporate the Sommerfeld-type conditions (25). This method consists in applying the projector P = 1 2 1 −e λB −e −λB 1 to the two-vectors (π A ,ψ A ) N and (π C ,ψ C ) N .…”