On an eigenvalue property of Summation-By-Parts operators

01/04/2022
by   Viktor Linders, et al.
0

Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property.

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1 Introduction

The Summation-By-Parts (SBP) methodology fernandez2014review; svard2014review and its extensions (generalized SBP fernandez2014generalized and upwind SBP mattsson2017diagonal

) constitute an algebraic framework for designing provably stable discretizations of partial differential equations. SBP operators can be designed within essentially every family of spatial discretizations, including finite difference methods

kreiss1974finite; strand1994summation, finite volume methods nordstrom2001finite, finite element methods abgrall2020analysis; abgrall2021analysis, pseudospectral methods carpenter1996spectral and the related discontinuous Galerkin gassner2013skew and Flux Reconstruction methods ranocha2016summation, as well as more specialized discretizations such as WENO yamaleev2009systematic; fisher2011boundary and DRP schemes linders2016summation; linders2017summation. Further, they can be used as time marching schemes akin to implicit Runge-Kutta methods nordstrom2013summation; lundquist2014sbp; boom2015high, in which case they satisfy an assortment of desirable stability and convergence properties linders2020properties; nordstrom2018well. All such methods share certain algebraic similarities that can be exploited to obtain stable and accurate discretizations.

This paper addresses a difficulty pertaining to an eigenvalue property that frequently arises in SBP theory. Much of what is known about the convergence of SBP methods relies on this property. It is known that there are methods that lack the property, essentially due to the existence of SBP operators with ”bad” null-spaces. Here, we consider SBP operators that do not suffer from bad nullspaces; so called nullspace consistent methods.

Three new results will be presented: After introducing nullspace consistent SBP operators in Section 2, it is shown that not every such operator satisfies the eigenvalue property. In Section 3 it is established that for each SBP operator that lacks the property, another SBP operator of the same order can be found that possesses it, and that differs from the first one by an arbitrarily small perturbation. Section 4 discusses pseudospectral methods, all of which are shown to satisfy the eigenvalue property. This generalizes an earlier result on the topic ruggiu2018pseudo. A summary is given in Section 5.

2 SBP operators and the eigenvalue property

This section introduces the notion of nullspace consistent SBP operators as well as the eigenvalue property. Let be an interval with .

Definition 1.

The matrices , are said to form a pair of SBP operators of order on the interval if there exist matrices ,

and vectors

, , such that the following relations hold:

  1. ,

  2. ,

  3. ,

  4. ,

  5. .

Here and elsewhere the notation should be understood as the elementwise exponentiation of . The convention is used throughout. Definition 1 incorporates classical (, , ), generalized () and upwind SBP methods (, , ) as special cases.

From (A) it is seen that and approximate derivative operators and that and interpolate grid functions to the domain boundaries. Further, defines a quadrature rule hicken2013summation; linders2018order. The role of is somewhat obfuscated in Definition 1, which warrants a comment: Subtracting (D) from (C), transposing and multiplying by shows that . From (A) and (B) it can then be deduced that

(1)

Together with positive semi-definiteness, this reveals that contributes with artificial dissipation to the operator; see e.g. mattsson2017diagonal; mattsson2004stable.

SBP operators are frequently used in combination with simultaneous approximation terms (SATs) that weakly impose initial, boundary or interface conditions carpenter1994time. Suppose that an inflow-outflow problem on is described by the the scalar differential equation

(2)

where and are given. An SBP-SAT discretization takes the form

(3)

where the second term on the right-hand side is the SAT. In (3), is a scalar parameter that is chosen to ensure stability. The choice is almost exclusively used in practice, hence we focus on this case here.

Remark 1.

If the flow in (2) is reversed, then is used and the boundary condition is imposed at using and .

Collecting terms that multiply the solution , (3) may be rewritten as

(4)

A necessary condition for the existence of a unique solution to (4) is that the matrix is invertible. SBP operators that have this property can be characterized in terms of nullspace consistency, a concept introduced in svard2019convergence:

Definition 2.

An SBP operator is said to be nullspace consistent if

For a proof of the following lemma, see (linders2020properties, Lemma 2).

Lemma 1.

For any SBP operator , the matrix is invertible if and only if is null-space consistent.

Unfortunately, nullspace consistency does not follow from Definition 1; counterexamples are given in ranocha2019some; linders2020properties. Yet, an even stronger condition is frequently needed:

Definition 3.

An SBP operator is said to have the eigenvalue property if each eigenvalue of has positive real part.

Many important results in the theory of SBP operators rely on the eigenvalue property. The following list is by no means complete.

  • The proof that spatial discretizations using (classical) finite difference SBP methods converge with order for hyperbolic problems and order for parabolic problems assumes the eigenvalue property svard2006order.

  • The proof that functional estimates from SBP methods in two or more dimensions are superconvergent requires the eigenvalue property

    hicken2011superconvergent.

  • SBP methods used for time discretization of partial differential equations need the eigenvalue property to ensure uniqueness of solutions nordstrom2013summation. Proofs of convergence for linear and certain nonlinear problems also rely on the property; see linders2020properties for an overview.

The eigenvalue property is evidently very important and it is therefore of interest to know which SBP operators possess it. Nullspace consistency is of course necessary. This bids the question: Is nullspace consistency sufficient for the eigenvalue property? Unfortunately, this is not the case.

Theorem 1.

There are nullspace consistent SBP operators that lack the eigenvalue property.

Proof.

Consider

defined on the nodes . This is an SBP operator of order with , , and . The matrix is invertible, hence the operator is nullspace consistent by Lemma 1. However, two of the eigenvalues of

and their corresponding eigenvectors are

The real parts of these eigenvalues are non-positive, hence the SBP operator does not have the eigenvalue property. ∎

3 Perturbed SBP operators

In this section we consider an SBP operator that is nullspace consistent but lack the eigenvalue property, i.e. it satisfies Definition 2 but not Definition 3. It will be shown that the eigenvalue property can be reclaimed by adding an appropriately designed perturbation to . More precisely, the eigenvalue property is enforced by constructing a new operator , which adheres to Definition 1 so long as is symmetric positive semi-definite and satisfies (1). In this case, is replaced by in Definition 1. The purpose of is to push problematic eigenvalues into the right half-plane while leaving the remaining eigenvalues untouched. Furthermore, the matrix and the vectors and are left unperturbed. To achieve this, it is first necessary to establish certain properties of the problematic eigenvalues and eigenvectors.

The starting point is the following simple result. For a proof, see e.g. (linders2020properties, Lemma 1):

Lemma 2.

Consider the SBP operator and let be an eigenpair of , i.e. . Then with equality if and only if and .

Lemma 2 shows that has an imaginary eigenvalue. By Lemma 1 this eigenvalue is non-zero since is nullspace consistent by assumption. In fact, has an even number of non-zero imaginary eigenvalues since the conjugate is also an eigenpair of .

At this point, two lemmas will be established that are key to the construction of . Recall that an eigenvalue of is normal if

  1. every eigenvector of corresponding to is orthogonal to every eigenvector of corresponding to each eigenvalue different from , and

  2. the algebraic and geometric multiplicities of are equal.

Remark 2.

Herein, the inner product of two vectors and are taken to be , where the superscript indicates conjugate transposition. Orthogonality should thus be understood with this inner product in mind. The norm will also be used.

Lemma 3.

Let be an eigenpair of with . Then is a normal eigenvalue with respect to the inner product .

Proof.

The two properties (a) and (b) must be established. For (a), let be any eigenpair of with and note that . Using (C) from Definition 1 and the definition of in (4) it follows that

(5)

Thus,

In the final equality, has been used, which holds since is imaginary. Consequently, , however since by assumption, the sought orthogonality follows.

For (b), consider the following problem: Find a symmetric positive definite matrix such that . From (5) it is seen that solves this problem. However, the existence of a solution implies that all elementary divisors of imaginary eigenvalues of are linear (carlson1963inertia, Corollary 2), which is equivalent to the stated assertion on the algebraic and geometric multiplicities (lancaster1985theory, Chapter 7). ∎

Lemma 4.

Let be an eigenpair of with . Then for .

Proof.

Consider first the case , for which we have , and note that . From Definition 1 and (4) it follows that

(6)

Thus,

Since is nullspace consistent by assumption, Lemma 1 ensures that . It follows that , hence the claim holds when .

Next, suppose that the claim holds for some . Then, similarly,

where the induction hypothesis implies the final equality. From nullspace consistency it follows that , and the claim holds by induction. ∎

Using Lemmas 3 and 4, the main result of this section can be proven:

Theorem 2.

For any SBP operator of order that satisfies Definition 2 but not Definition 3, there is another SBP operator of order that satisfies both definitions. Further, can be constructed such that for any , where the norm is arbitrary.

Proof.

Suppose that has precisely imaginary eigenvalues, including multiplicity, and denote these . Select corresponding eigenvectors . By Lemma 3

each of these eigenvectors are orthogonal to any eigenvector corresponding to another eigenvalue. Since the algebraic and geometric multiplicities equal for each imaginary eigenvalue, their respective eigenspaces are complete. An orthogonal basis can be found for each space and these basis vectors can be taken as one of the listed eigenvectors. Thus, mutual orthogonality can be ensured among the eigenvectors

even if some of them correspond to the same eigenvalue.

Choose positive numbers and construct the matrix

Then is real since . Further, is symmetric positive semi-definite.

At this point, construct . Note that

for each . Here, Lemma 4 has been used in the final equality. Consequently, satisfies (1) such that is an SBP operator of order .

Consider an eigenvalue of . If , then pick any eigenvector corresponding to and note that by the orthogonality property in Lemma 3. Thus, the eigenpair is unaltered by whenever . However, if so that for some , then pick the appropriate eigenvector and note that

where the second equality follows from the orthogonality of the eigenvectors . Thus, has an eigenpair , where . By repeating this procedure for each and , every eigenvalue of is accounted for. Consequently, the SBP operator has the eigenvalue property.

Finally, note that the eigenvectors and may be normalized as desired and that the constants can be chosen arbitrarily small. Thus, for any norm it is possible to find such that for any ,

4 Pseudospectral methods

An important family of SBP operators consists of the pseudospectral methods. They all satisfy Definition 1 with order ; see carpenter1996spectral; fernandez2014generalized. These operators are also used in discontinuous Galerkin and flux reconstruction methods gassner2013skew; ranocha2016summation. Given , the pseudospectral SBP operator is uniquely defined. To see this, note that the accuracy conditions (A) in Definition 1 imply that the elements in the th row of satisfy

Note that is a Vandermonde matrix and that the numbers are distinct. Thus, is invertible and are unique. The same of course holds for each row , hence is uniquely defined.

Remark 3.

This argument does not imply uniqueness of , , and .

The goal of this section is to demonstrate that pseudospectral methods defined on arbitrary grids satisfy the eigenvalue property. In ruggiu2018pseudo it was shown that this is the case if defines a quadrature rule that is exact for polynomials of degree or higher, i.e. if for . This holds in particular if the grid is chosen to be the Legendre-Gauss, Legendre-Gauss-Radau or Legendre-Gauss-Lobatto nodes (mapped to the interval ) and is a diagonal matrix containing the corresponding Gaussian quadrature weights. However, for general grids , the resulting quadrature rule is exact only for polynomials of degree fernandez2014generalized. A different approach is thus necessary to generalize the result from ruggiu2018pseudo.

The starting point here is to show that pseudospectral methods on arbitrary grids are nullspace consistent.

Lemma 5.

Every pseudospectral SBP operator is null-space consistent.

Proof.

The columns of the Vandermonde matrix form a basis for . Thus, if and is expanded in terms of this basis as , then

By linear independence of the basis vectors it follows that for and consequently that . Thus, and is consequently null-space consistent. ∎

With Lemma 5 in place, the results from Section 3 can be used to establish the eigenvalue property.

Theorem 3.

Every pseudospectral SBP operator has the eigenvalue property.

Proof.

Suppose that there is a pseudospectral method that does not have the eigenvalue property. Then, by Theorem 2, another operator of the same order of accuracy can be found that operates on the same grid and that is distinctly different from . However, this violates the uniqueness of pseudospectral methods.

Alternatively, suppose that there is a pseudospectral method that does not have the eigenvalue property and denote by one of the eigenvectors of that corresponds to an imaginary eigenvalue. It follows from Lemma 4 that . But both and are invertible, so and can therefore not be an eigenvector, which contradicts its definition. ∎

5 Summary

Three results on the eigenvalues of SBP operators have been proven. Firstly, the eigenvalue property, which is essential in many important accuracy and convergence proofs, does not hold for all nullspace consistent SBP operators. Secondly, this problem can be addressed by carefully constructing an arbitrarily small artificial dissipation term that pushes each problematic eigenvalue into the right half-plane without affecting the other eigenvalues or the order of accuracy of the method. Thirdly, all pseudospectral methods satisfy the eigenvalue property. Thus, SBP operators used within the discontinuous Galerkin and flux reconstruction frameworks are free from the nuisance of problematic eigenvalues.

References