On pseudo-spectral time discretizations in summation-by-parts form

doi:10.1016/j.jcp.2018.01.043

Journal of Computational Physics

Volume 360, 1 May 2018, Pages 192-201

https://doi.org/10.1016/j.jcp.2018.01.043 Get rights and content

Highlights

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Pseudo-spectral SBP–SAT time approximations for IBVP are studied.
•
Such approximations lead to invertible fully-discrete problems.
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The eigenvalues of the resulting system have strictly positive real parts.
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These results do not depend on the number of time nodes or the quadrature.

Abstract

Fully-implicit discrete formulations in summation-by-parts form for initial-boundary value problems must be invertible in order to provide well functioning procedures. We prove that, under mild assumptions, pseudo-spectral collocation methods for the time derivative lead to invertible discrete systems when energy-stable spatial discretizations are used.

Introduction

Well-posedness of partial differential equations is a key concept in mathematical modeling. A well-posed problem has a unique solution which is bounded by the data of the problem [1], [2]. Once initial and boundary conditions that lead to well-posedness are determined, a discrete approximation of the continuous problem can be formulated and solved, if it is stable. An additional requirement for fully-implicit discrete problems is the invertibility of the resulting system of equations. If this condition is not met, an iterative solver applied to such problems may converge to an erroneous solution [3].

Stable and high-order accurate discretizations for well-posed linear problems can be achieved in a straightforward way by combining Summation-By-Parts (SBP) operators [4], [5] and weak boundary and initial procedures using Simultaneous-Approximation-Terms (SATs) [6], [7], [8]. For initial value problems, the dual-consistent [9] SBP–SAT formulations are A- and L-stable implicit Runge–Kutta schemes [10], [11]. However, the invertibility of fully-discrete stable approximations have so far only been conjectured in this setting [7], [11] (it does not follow from any known stability property). For further reading on these topics, see for example [12], [13].

In this work, we focus on pseudo-spectral collocation methods on finite domains [14] for initial value problems. We prove that these discretizations, which can be recast as SBP–SAT schemes with diagonal norms [15], [16], have eigenvalues with strictly positive real parts for specific choices of the penalty parameter. As a natural consequence, pseudo-spectral time discretizations combined with energy stable spatial approximations of initial-boundary value problems lead directly to invertible fully discrete approximations.

This paper is organized as follows. In section 2, the pseudo-spectral methods in SBP form are introduced for initial value problems. We also reformulate the invertibility issue in terms of the eigenvalues of the time discretization. In section 3, we show that pseudo-spectral SBP–SAT approximations yield discrete operators which can be easily studied through a coordinate transformation to Legendre polynomials. The eigenvalues of these matrices are analyzed in section 4, where the main result of the paper is stated and proved. Next, the properties of fully discrete systems with pseudo-spectral time discretizations are exemplified in section 5. Section 6 contains our conclusions.

Section snippets

Pseudo-spectral SBP–SAT time discretizations

The aim of this section is two-fold: firstly, we introduce the pseudo-spectral SBP–SAT discretizations for initial value problems [7]; secondly, we outline the main theoretical issue in this paper.

SBP operators on modal basis

In this section, we show that, for pseudo-spectral methods, $D$ is similar to a matrix with a particular structure suitable for analysis. This result, which can be obtained through a coordinate transformation to Legendre polynomials, is based on a previous study in [19] and will enable the eigenvalue analysis in the next section.

Invertibility of pseudo-spectral first derivative discretizations

Before the invertibility for discretizations of (5) is discussed, we recall the following theorem [23], [24]

Theorem 4.1

Given a matrix $A \in R^{n \times n}$ , suppose that $H = \frac{1}{2} (G A + A^{T} G)$ and $S = \frac{1}{2} (G A - A^{T} G)$ for some positive definite matrix G and some positive semidefinite matrix H. Then A has eigenvalues with strictly positive real parts if, and only if, no eigenvector of $G^{- 1} S$ lies in the null space of H.

Moreover, we will make use of the following

Lemma 4.2

The determinant of ${\hat{D}}_{κ}$ is equal to $- (2 n - 3)!!$ $σ / κ$ .

Proof

The matrix ${\hat{D}}_{κ}$ can be

An illustrative example

Consider the linearized one dimensional symmetrized form of the compressible Euler equations [25] $\begin{matrix} U_{t} + A U_{x} & = & F, & 0 < x < 1, & 0 < t < T, \\ U (x, 0) & = & f, & 0 < x < 1, \end{matrix}$ with subsonic characteristic boundary conditions [26]. In (18) we have $U = {[\begin{matrix} \frac{\overline{c}}{\sqrt{γ} \overline{ρ}} ρ, u, \frac{1}{\overline{c} \sqrt{γ (γ - 1)}} θ \end{matrix}]}^{T}, A = [\begin{matrix} \overline{u} & \frac{\overline{c}}{\sqrt{γ}} & 0 \\ \frac{\overline{c}}{\sqrt{γ}} & \overline{u} & \sqrt{\frac{γ - 1}{γ}} \overline{c} \\ 0 & \sqrt{\frac{γ - 1}{γ}} \overline{c} & \overline{u} \end{matrix}]$ with $\overline{u} = 1$ , $\overline{c} = 2$ , $\overline{ρ} = 1$ and $γ = 1.4$ . The variables ρ, u and θ are the density, velocity and temperature perturbations of the fluid, respectively.

A fully-discrete approximation of (18) with a pseudo-spectral collocation approximation in time

Conclusions

The invertibility of fully discrete approximations for initial boundary value problems has been discussed. We have proved that the SBP–SAT pseudo-spectral methods in time lead to invertible formulations for energy-stable spatial discretizations.

We have also shown that the eigenvalues of these fully discrete operators have strictly positive real parts, allowing for the use of iterative techniques such as dual time-stepping. These results hold in general, irrespectively of the number of grid

Acknowledgments

This work was supported by VINNOVA, the Swedish Governmental Agency for Innovation Systems, under contract number 2013-01209.

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