Strong Stability of Explicit Runge Kutta Time Discretizations...1 STRONG STABILITY OF EXPLICIT RUNGE{KUTTA TIME 2 DISCRETIZATIONS 3 ZHENG SUNyAND CHI-WANG SHUz 4 Abstract. Motivated

STRONG STABILITY OF EXPLICIT RUNGE–KUTTA TIME1

DISCRETIZATIONS∗2

ZHENG SUN† AND CHI-WANG SHU‡3

Abstract. Motivated by studies on fully discrete numerical schemes for linear hyperbolic con-4servation laws, we present a framework on analyzing the strong stability of explicit Runge–Kutta5(RK) time discretizations for semi-negative autonomous linear systems. The analysis is based on the6energy method and can be performed with the aid of a computer. Strong stability of various RK7methods, including a sixteen-stage embedded pair of order nine and eight, has been examined under8this framework. Based on numerous numerical observations, we further characterize the features of9strongly stable schemes. A both necessary and sufficient condition is given for the strong stability of10RK methods of odd linear order.11

Key words. Runge–Kutta methods, strong stability, energy method, hyperbolic problems,12conditional contractivity.13

AMS subject classifications. 65M12, 65M20, 65L0614

1. Introduction. Explicit Runge–Kutta (RK) methods have been commonly15

used for time integration of hyperbolic conservation laws. In this paper, we study16

the strong stability of the methods in such context. We focus on autonomous linear17

ordinary differential equation (ODE) systems18

(1.1)d

dtu = Lu,19

which are obtained from method of lines schemes for linear hyperbolic problems.20

u ∈ RN and L is an N ×N real constant matrix, where N is the degrees of freedom21

for the spatial discretization. If the semidiscrete scheme honors the (weighted) L222

stability of the partial differential equation (PDE), then for certain symmetric and23

positive definite matrix H,24

(1.2) L>H +HL ≤ 025

is a semi-negative definite matrix. Here H can be related with both the symmetrizer of26

the PDE [16] and the mass matrix or quadrature weights of a Galerkin or collocation27

type spatial discretization. If (1.2) holds, then we say L is semi-negative and (1.1)28

satisfies the energy decay law29

(1.3)

d

dt‖u‖2H = 〈 d

dtu, u〉H + 〈u, d

dtu〉H

= 〈Lu,Hu〉+ 〈u,HLu〉 = 〈u, (L>H +HL)u〉 ≤ 0.30

Here 〈·, ·〉H = 〈·, H·〉 with 〈·, ·〉 being the usual l2 inner product in RN and ‖u‖H =31 √〈u, u〉H . We are concerned with whether this property is preserved at the discrete32

∗Submitted to the editors on Nov 25, 2018. This work is mainly done during the symposium heldat ICERM in Providence RI, for celebrating 75 years of Mathematics of Computation. We wouldlike to thank the committee members, speakers and staffs for this inspiring workshop. ZS especiallywants to express his appreciation for the travel and lodging support from ICERM.†Department of Mathematics, The Ohio State University, Columbus, OH 43210

([email protected]).‡Division of Applied Mathematics, Brown University, Providence, RI 02912

([email protected]). Research supported by ARO grant W911NF-16-1-0103 and NSF grantDMS-1719410.

1

mailto:[email protected]

mailto:[email protected]

2 Z. SUN AND C.-W. SHU

level, namely, whether33

(1.4) ‖un+1‖H ≤ ‖un‖H ,34

holds after applying an explicit RK time integrator under a suitably restricted time35

step. The time step constraint is referred to as the Courant–Friedrichs–Lewy (CFL)36

condition for numerical conservation laws. We would say the explicit RK method is37

strongly stable, if (1.4) is satisfied when discretizing (1.1) under the condition (1.2).38

Note this definition of stability is stronger than the usual one, which states that the39

norm of the numerical solution is bounded by the norm of the initial data up to a40

constant (typically dependent on the total time) [16]. We remark that the concept41

of strong stability in our context is also connected with the conditional contractivity42

from the ODE community [28, 18], while our analysis does not rely on the circle43

condition, which is usually assumed in the previous works.44

Solving hyperbolic conservation laws with explicit time discretizations can be45

dated back to early days. They seem to be the most natural choice to avoid the46

inversion of large nonlinear systems resulted from spatial discretizations. Many of47

the first order monotone schemes, which preserves various properties of the contin-48

uous PDE, adopt Euler forward method for time marching [19]. This flavor has49

been pushed further with the development of strong-stability-preserving (or total-50

variation-diminishing) high order time discretizations [25, 26, 14, 15], especially the51

RK methods in this family. The strong-stability-preserving RK (SSPRK) methods52

can be formulated as convex combinations of Euler forward steps, hence automati-53

cally preserve many properties that are achieved by the Euler forward discretization.54

One of the successes is to use them for integrating method of lines scheme obtained55

from discontinuous Galerkin (DG) spatial discretizations for hyperbolic conservation56

laws. The method is referred to as the RKDG method, which was developed by57

Cockburn et al in a series of papers [9, 8, 7, 6, 10]. In the RKDG method, a lim-58

iter can be applied after each Euler forward stage to control total variation. The59

non-increasing total variation semi-norm is then preserved after a full time step with60

SSPRK methods. Another application of SSPRK methods is for designing high order61

positivity-preserving or bound-preserving schemes [43, 45]. The approach, which is62

based on a similar methodology, has received intensive attentions in recent years and63

has been successfully applied to various problems [44, 40, 39, 22, 30, 31]. It should64

also be noted, although progress has been made on preserving positivity with the Eu-65

ler backward method [21], due to the nonexistence of second or higher order methods66

as the positive combinations of Euler backward steps [15], it is a nontrivial task to67

design high order implicit positivity-preserving schemes.68

Along with the popularity of explicit RK methods for hyperbolic problems, a69

growing attention has been paid to the role that the time integrator has played in70

a fully discrete scheme. One of the major issues is on the stability: whether the L271

stability achieved by method of lines schemes would be preserved after the explicit72

time stepping. Indeed, this issue has been raised during the initial development of the73

RKDG method. Even for linear advection equation, it is reported in [3] that under74

the usual CFL condition, the Euler forward time stepping coupled with the linear75

DG method is unstable, even in the sense of the weaker stability. The second order76

RK methods are stable only if the spatial discretization uses at most piecewise linear77

elements [11]. From then on, analyses particularly on fully discrete DG methods78

have been performed, such as [42, 41, 36, 37, 32, 46]. But it seems that an universal79

approach of analyzing the stability is missing.80

STRONG STABILITY OF RK TIME DISCRETIZATIONS 3

One of the attempts is to perform eigenvalue analysis. If the matrix L is normal,81

then it can be diagonalized with an orthogonal matrix. Hence the ODE system is82

transformed into decoupled scalar equations. The method is strongly stable if and83

only if the amplifier of each decoupled equation falls within the absolute stability84

region of the time integrator. However, when L is not normal, which is generic for85

L obtained from method of lines schemes, eigenvalue analysis gives only necessary86

but not sufficient conditions on strong stability. In such situation, eigenvalue analysis87

should be avoided for the following reasons. Firstly, if L can be diagonalized, the88

analysis ensures the strong stability of the transformed system. But in practice, the89

diagonalizing matrix can be ill-conditioned, the resulting stability would have very90

weak controls on the actual solution after the backward transformation [20]. Secondly,91

to ensure stability, we need the matrix norm of the amplifier to be properly controlled,92

but the eigenvalue analysis only examines the spectral radius, which is strictly smaller93

than the matrix norm for non-normal cases. Hence the actual time step constraint94

should be stricter than that from the eigenvalue analysis. The result solely relying on95

eigenvalue analysis can be misleading. We refer to [17] for a particular example.96

A more sound way of analyzing stability is to use the energy method. The study97

of (1.1) through this approach initiates from problems with coercive matrices L>H +98

HL ≤ −ηL>HL, where η is a positive constant. In [20], Levy and Tadmor proved that99

for coercive matrices, the classic third order and fourth order explicit RK methods100

are strongly stable, under the time step constraint τ ≤ λη with λ = 350 for the third101

order method and λ = 162 for the fourth order method. Later, a simpler proof was102

discovered based on the strong-stability-preserving RK formulation and the fact that103

Euler forward method is strongly stable for the coercive problems [15]. This approach104

gives a significantly relaxed time step τ ≤ η and extends the result to linear RK105

methods of arbitrary order. These analyses coincide with the earlier contractivity106

analysis in [28] and [18]. In their study of contractivity, or strong stability in our107

context, a circle condition is assumed, which is essentially equivalent to the strong108

stability assumption for the Euler forward steps.109

The coercivity condition typically arises from diffusive problems, but is uncommon110

in numerical approximations of hyperbolic problems. This motivates us to remove111

this assumption and consider the general semi-negative case. The difficulty is that112

the operator becomes less dissipative and the Euler forward step is no longer strongly113

stable. In [34], Tadmor proved the third order RK method is strongly stable for semi-114

negative L. Then in [33], we found a counter example showing that the fourth order115

RK method can not preserve the strong stability, although the fourth order method116

satisfies the necessary condition in the eigenvalue analysis and is hence strongly stable117

if L is normal. Furthermore, in the same paper, the fourth order RK method is proven118

to be strongly stable in two steps. In other words, successively applying the method119

for two steps yields a strongly stable method with eight stages. Recently in [23], it is120

shown that the low storage SSPRK method of order four with ten stages also admits121

strong stability.122

As one can see from [33] and [23], although the computation for analyzing RK123

methods with many stages looks complicated at the first glance, it only involves124

elementary algebraic manipulations. A single technique, which corresponds to inte-125

gration by parts for the PDE, has been repeatedly used. Inspired by the proofs in126

[33] and [23], as well as many previous analyses on DG methods, we develop a unified127

framework on analyzing the strong stability of explicit RK methods. The main idea128

is based on an induction procedure. With the aid of a computer, we can easily obtain129

the energy equality, which equates the energy at the next time steps with that at the130


current time level plus terms of specific forms. Then a sufficient condition, generalized131

from Lemma 2.4 in [33], is used to justify the strong stability. A necessary condition132

is also provided to exclude some methods that are not strongly stable. With this133

framework, we can easily examine strong stability of various RK methods, including134

a ninth order and eighth order embedded pair with sixteen stages. Finally, base on135

numerous observations, we summarize patterns in the energy equality to further char-136

acterize the strongly stable RK methods. In particular, we give a both necessary and137

sufficient condition for the RK methods of odd linear order to be strongly stable. We138

remark that the framework has its limitation, for example, it can not determine if the139

classic fourth order method is strongly stable.140

The rest of the paper is organized as follows. In section 2, we present our frame-141

work on analyzing the strong stability of explicit RK methods of any order with142

arbitrary stages. In section 3, the strong stability of various methods has been ex-143

amined using the framework, including linear RK methods, the classic fourth order144

methods (strong stability in multiple steps), several SSPRK methods and the em-145

bedded pairs used in the commercial software Mathematica. Then in section 4, we146

further investigate the energy equality to characterize the structure of strongly stable147

methods. Numerical dissipation of these methods for energy conserving systems has148

also been discussed. Finally in section 5, conclusions are given.149

2. Stability analysis: a framework. Let us drop the superscript n in un.150

Consider an explicit RK time discretization for the linear autonomous system (1.1).151

The scheme is of the form152

(2.1) un+1 = Rsu,153

where154

(2.2) Rs =

s∑k=0

αs(τL)k, α0 = 1, αs 6= 0.155

Here τ is the time step and s is the number of stages. The coefficients {αk}sk=0 depend156

solely on the scheme itself. For an s-stage method, it is of linear order p if and only if157

the first p+ 1 terms in the summation (2.2) coincide with the truncated Taylor series158

of eτL. In particular, p ≤ s [2]. We would like to examine the strong stability of (2.1)159

under the usual CFL condition: if there exists a constant λ, such that160

(2.3) ‖Rsu‖2H ≤ ‖u‖2H ,161

for all τ‖L‖H ≤ λ and all inputs u. This is equivalent to162

(2.4) ‖Rs‖H ≤ 1,163

under the prescribed condition and ‖Rs‖H is the matrix norm of Rs.164

A natural attempt is to adopt the following expansion to compare ‖Rsu‖2H with165

‖u‖2H .166

(2.5)

‖Rsu‖2H =

s∑i,j=0

αiαjτi+j〈Liu, Lju〉H = ‖u‖2H +

∑1≤max{i,j}≤s

αiαjτi+j〈Liu, Lju〉H .167

However, each term 〈Liu, Lju〉H may not necessarily have a sign. The idea for over-168

coming the difficulty is to convert 〈Liu, Lju〉H into linear combinations of terms of169


the form ‖Lku‖2H , JLkuK2H and [Liu, Lju]H . Here170

(2.6) [v, w]H = −〈v, (L>H +HL)w〉, v, w ∈ RN171

is a semi inner product and172

(2.7) JvKH =√

[v, v]H173

defines the induced semi-norm. Indeed, this can be achieved through the following174

induction procedure.175

Proposition 2.1 (Integration by parts). Suppose j ≥ i, then176

(2.8) 〈Liu, Lju〉H =

‖Liu‖2H , j = i,− 1

2JLiuK2H , j = i+ 1,−〈Li+1u, Lj−1u〉H − [Liu, Lj−1u]H , otherwise.

177

Proof. The case j = i can be justified with the definition of ‖·‖H . When j = i+1,178

(2.9)〈Liu, Li+1u〉H =

1

2〈Liu, (HL)Liu〉+

1

2〈L>HLiu, Liu〉

=1

2〈Liu, (L>H +HL)Liu〉 = −1

2JLiuK2H .

179

When j > i+ 1,180

(2.10)

〈Liu, Lju〉H = 〈Liu, (HL)Lj−1u〉= −〈Liu, L>HLj−1u〉+ 〈Liu, (L>H +HL)Lj−1u〉= −〈Li+1u, Lj−1u〉H − [Liu, Lj−1u]H .

181

In the context of approximating the spatial derivative ∂x for periodic functions182

with L, Proposition 2.1 is the discrete version of integration by parts. Since L may183

not preserve the exact anti-symmetry of ∂x, namely L>H + HL 6= 0, an extra term184

[Liu, Lj−1u]H is produced. Furthermore, − 12JLiuK2H is usually the numerical dissipa-185

tion from the spatial discretization. In particular, J·KH is the jump semi-norm in the186

DG method.187

Furthermore, one can repeat the induction procedure to obtain the following188

expansion.189

Corollary 2.2. For j ≥ i,190

(2.11) 〈Liu, Lju〉H = ζi,j −b j−i

2 c−1∑k=0

(−1)k[Li+ku, Lj−1−ku]H ,191

where192

(2.12) ζi,j =

{(−1)

j−i+12

12JL

i+j−12 uK2H , i+ j odd,

(−1)j−i2 ‖L

i+j2 u‖2H , i+ j even.

193

Based on Corollary 2.2, we have the following energy equality.194

Lemma 2.3 (Energy equality). Given H and Rs =∑sk=0 αk(τL)k with α0 = 1.195

There exists a unique set of coefficients {βk}sk=0 ∪ {γi,j}s−1i,j=0, such that for all u and196

L satisfying L>H +HL ≤ 0,197

(2.13) ‖Rsu‖2H =

s∑k=0

βkτ2k‖Lku‖2H +

s−1∑i,j=0

γi,jτi+j+1[Liu, Lju]H , γi,j = γj,i.198


Proof. The existence of such expansion can be justified by (2.5) and Corollary 2.2.199

It remains to show that the set of coefficients is unique. Each term in the summation200

should be considered as a polynomial of τ , elements in L and elements in u. We are201

going to show these polynomials are linearly independent.202

(i) It suffices to analyze the case H being the identity matrix, otherwise we can203

consider L =√HL√H−1

and u =√Hu instead.204

(ii) It suffices to show {τ i+j〈Liu, Lju〉}0≤i≤j≤s are linearly independent. Since205

elements in the set can be expressed as linear combinations of {τ2k‖Lku‖2}sk=0 ∪206

{τ i+j [Liu, Lju]}0≤i≤j≤s−1 due to Corollary 2.2. Since the two sets have the same207

cardinality, the linear independence of the previous set implies that of the latter one.208

Particularly, we take L =

(cos θ − sin θsin θ cos θ

)with θ ∈ (π2 ,

3π2 ) (hence L> + L ≤ 0)209

and u =

(10

). Noting that L is a rotation matrix, which is orthogonal, one can obtain210

(2.14)

0 =∑

0≤i≤j≤s

αi,jτi+j〈Liu, Lju〉 =

2s∑m=0

bm2 c∑k=max{0,m−s}

αk,m−k〈Lku, Lm−ku〉

τm

=

2s∑m=0

bm2 c∑k=max{0,m−s}

αk,m−k cos((m− 2k)θ)τm.

211

Note {m − 2k}bm2 c

k=max{0,m−s} are distinct non-negative integers. Due to linear inde-212

pendence of cos((m−2k)θ)τm, {αk,m−k}bm2 ck=max{0,m−s} are all zeros for each m. Hence213

{τ i+j〈Liu, Lju〉}0≤i≤j≤s are linearly independent.214

Remark 2.4. The uniqueness is not used in the framework. But it facilitates our215

analysis in section 4. Note that the uniqueness is nontrivial. For example, if we216

restrict ourselves to a small subset {L : L>H + HL = −ηL>HL, 0 > η ∈ R}, then217

one can certainly obtain different linear combinations.218

To facilitate our discussion, we introduce the following definitions.219

Definition 2.5. The leading index of Rs, denoted as k∗, is the positive integer220

such that βk∗ 6= 0 and βk < 0 for all 1 ≤ k < k∗. The coefficient βk∗ is called the221

leading coefficient. The k∗-th order principal submatrix Γ∗ = (γi,j)0≤i,j≤k∗−1 is called222

the leading submatrix.223

Note that k∗ is well-defined since βs = α2s 6= 0, which implies k∗ ≤ s. For small224

τ‖L‖H , βk∗τ2k∗‖Lku‖2H and

∑k∗−1i,j≥0 γi,jτ

i+j+1[Liu, Lju]H become dominant terms in225

the energy equality. Hence the strong stability would be closely related with the226

negativity of βk∗ and Γ∗. In particular, we have the following necessary condition and227

sufficient condition.228

Theorem 2.6 (Necessary condition). The method is not strongly stable if βk∗ >229

0. More specifically, if βk∗ > 0, then there exists a constant λ, such that ‖Rs‖H > 1230

if 0 < τ‖L‖H ≤ λ and L>H +HL = 0.231

Proof. With L>H+HL = 0, the latter summation of terms [Liu, Lju]H in (2.13)232


is zero. Hence233

(2.15)

‖Rsu‖2H = ‖u‖2H + βk∗τ2k∗‖Lk

∗u‖2H +

s∑k=k∗+1

βkτ2k‖Lku‖2H

≥ ‖u‖2H +

(βk∗ −

s∑k=k∗+1

|βk|(τ‖L‖H)2(k−k∗)

)τ2k

∗‖Lk

∗u‖2H .

234

Therefore,235

(2.16) ‖Rs‖2H ≥ 1 + (βk∗ − β)τ2k‖Lk∗‖2H > 1,236

if β =∑sk=k∗+1 |βk|λ2(k−k

∗) < βk∗ .237

Theorem 2.7 (Sufficient condition). If βk∗ < 0 and Γ∗ is negative definite, then238

there exists a constant λ such that ‖Rs‖H ≤ 1 if τ‖L‖H ≤ λ.239

Proof. Let −ε to be the largest eigenvalue of Γ∗. Then Γ∗ + εI is negative semi-240

definite. From Lemma 2.3 in [33],∑k∗−1i,j=0(γi,j + εδi,j)τ

i+j+1[Liu, Lju]H ≤ 0, where241

δi,j is the Kronecker delta function. Hence242

(2.17)

‖Rsu‖2H ≤‖u‖2H + βk∗τ2k∗‖Lk

∗u‖2H +

s∑k=k∗+1

βkτ2k‖Lku‖2H

− εk∗−1∑k=0

τ2k+1JLkuK2H +∑

k∗≤max{i,j}≤s−1

γi,jτi+j+1[Liu, Lju]H .

243

Note that ‖Lku‖H ≤ ‖L‖k−k∗

H ‖Lk∗u‖H and τ‖L‖H ≤ λ. Hence we have244

(2.18)

s∑k=k∗+1

βkτ2k‖Lku‖2H ≤

(s∑

k=k∗+1

|βk|λ2(k−k∗)

)τ2k

∗‖Lk

∗u‖2H .245

Using the fact246

(2.19)

[Liu, Lju]H ≤ JLiuKHJLjuKH , τJLjuK2H ≤ 2λ‖Lju‖2H ≤ 2λ‖L‖2(j−k∗)

H ‖Lk∗u‖2H ,247

together with the arithmetic-geometric mean inequality, one can obtain248

(2.20)

τ i+j+1[Liu, Lju]H ≤

{ετ2i+1

2βJLiuK2H + βλ2(j−k∗)+1

ε τ2k∗‖Lk∗u‖2H , i < k∗, j ≥ k∗,

2λi+j+1−2k∗τ2k∗‖Lk∗u‖2H , i, j ≥ k∗.

249

Therefore,250

(2.21)

‖Rsu‖2H ≤‖u‖2H +

βk∗ +

s∑k=k∗+1

|βk|λ2(k−k∗) + 2

s∑i,j≥k∗

|γi,j |λi+j+1−2k∗

+2β

ε

s∑j=k∗

k∗−1∑i=1

|γi,j |λ2(j−k∗)+1

τ2k∗‖Lk

∗u‖2H

− εk∗−1∑k=0

(1−

∑sj=k∗ |γk,j |β

)τ2k+1JLkuK2H .

251


It suffices to take β = max{∑sj=k∗ |γk,j |}

k∗−1k=0 and then choose λ sufficiently small so252

that the second coefficient on the right is negative.253

Given an s-stage RK scheme Rs, we expand ‖Rsu‖2H with Lemma 2.3 and then254

use the necessary condition in Theorem 2.6 and sufficient condition in Theorem 2.7 to255

examine its strong stability. Note that {βk}sk=0 and {γi,j}s−1i,j=0 can be obtained from256

Algorithm 2.1, which is based on Proposition 2.1. Since Γ∗ is symmetric, one only257

needs to check its largest eigenvalue to determine if Γ∗ is negative definite.258

Algorithm 2.1 Obtain coefficients in Lemma 2.3

Input: β = (β0, · · · , βs) = 0, Γ = (γi,j)s−1i,j=0 = 0

for i← 0 to s doβi ← βi + α2

i

for j ← i+ 1 to s doα← 2αiαjk ← il← jwhile l > k do

switch l docase k do

βk ← βk + αendcase k + 1 do

γk,k ← γk,k − α/2endotherwise do

γk,l−1 ← γk,l−1 − αα← −α

end

endk ← k + 1l← l − 1

end

end

end

3. Applications. In this section, we examine strong stability of various RK259

methods using the framework in the previous section. In tables that will be provided260

later, the last column “SS” refers to the strong stability property. A question mark261

will be put into the entry if strong stability of the corresponding scheme can not be262

determined. “no*” means particular counter examples can be constructed.263

3.1. Linear RK methods. For general nonlinear systems, to admit accuracy264

order higher than four, RK methods must have more stages than its order [2]. How-265

ever, for autonomous linear systems, the desired order of accuracy can be achieved266

with the same number of stages. All such methods would be equivalent to the Taylor267

series method268

(3.1) Rp = Pp =

p∑k=0

(τL)k

k!.269


In Table 1 and Table 2, we document the leading indexes and coefficients of linear270

RK methods from first order to twelfth order. The leading submatrices for low order271

RK methods are also given in Table 1. We report that the third order, seventh order272

and eleventh order methods are strongly stable. The fourth order, eighth order and273

twelfth order methods can not be judged with the framework. All other methods are274

not strongly stable.

p k∗ βk∗ Γ∗ λ(Γ∗) SS

1 1 1 −(

1)

−1.00000 no

2 2 14

−(

1 12

12

12

)−1.30902−1.90983 × 10−1 no

3 2 − 112

−(

1 12

12

13

)−1.26759−6.57415 × 10−2 yes

4 3 − 172

−

1 12

16

12

13

18

16

18

124

−1.30128−7.93266 × 10−2

+5.60618 × 10−3no*

5 3 1360

−

1 12

16

12

13

18

16

18

120

−1.30150−8.07336 × 10−2

−1.10151 × 10−3no

6 4 12880

−

1 1

216

124

12

13

18

130

16

18

120

172

124

130

172

1240

−1.30375−8.21871 × 10−2

−1.40529 × 10−3

−1.60133 × 10−4

no

7 4 − 120160

−

1 1

216

124

12

13

18

130

16

18

120

172

124

130

172

1252

−1.30375−8.21836 × 10−2

−1.36301 × 10−3

−7.86229 × 10−6

yes

8 5 − 1201600

−

1 1

216

124

1120

12

13

18

130

1144

16

18

120

172

1336

124

130

172

1252

11152

1120

1144

1336

11152

23120960

−1.30384−8.22588 × 10−2

−1.38580 × 10−3

−9.32706 × 10−6

+2.24989 × 10−6

?

Table 1Linear RK methods: from first order to eighth order.

275

3.2. The classic fourth order method. The classic fourth order method with276

four stages, which is widely used in practice due to its stage and order optimality, is277

unfortunately not covered under the framework. In [33], we found a counter example278

to show that the method is not strongly stable, but successively applying the method279

for two steps yields a strongly stable method with eight stages.280

Proposition 3.1 (Sun and Shu, 2018). The fourth order RK method with four281

stages is not strongly stable. More specifically, for H = I and L = −

1 2 20 1 20 0 1

,282

we have ‖P4‖ > 1, if τ‖L‖H > 0 is sufficiently small.283

Theorem 3.2 (Sun and Shu, 2018). The fourth order RK method with four stages284

is strongly stable in two steps. In other words, there exists a constant λ, such that285

‖(P4)2‖H ≤ 1 if τ‖L‖H ≤ λ.286

Here we examine multi-step strong stability of the fourth order method using our287

framework. Note the derivation using this framework is slightly different from that288


p k∗ βk∗ λ(Γ∗) SS

9 5 11814400

−1.30384−8.22588 × 10−2

−1.38585 × 10−3

−9.75366 × 10−6

−3.11800 × 10−8

no

10 6 1221772800

−1.30384−8.22613 × 10−2

−1.38688 × 10−3

−9.91006 × 10−6

−4.70638 × 10−8

−1.63872 × 10−8

no

11 6 − 1239500800

−1.30384−8.22613 × 10−2

−1.38688 × 10−3

−9.90966 × 10−6

−3.87351 × 10−8

−7.87018 × 10−11

yes

12 7 − 13353011200

−1.30384−8.22614 × 10−2

−1.38691 × 10−3

−9.91617 × 10−6

−3.93334 × 10−8

+1.45458 × 10−10

−8.54170 × 10−11

?

Table 2Linear RK methods: from ninth order to twelfth order.

in [33]. The relevant quantities for strong stability are given in Table 3. Note that289

the method is both two-step and three-step strongly stable (with the same time step290

size), which means the norm of the solution after the first step is always bounded by291

the initial data, if sufficiently small uniform time steps are used.

(P4)m k∗ βk∗ Γ∗ λ(Γ∗) SS

(P4)2 3 − 136

−

2 2 43

2 83

243

2 1912

−5.73797−4.99093 × 10−1

−1.29329 × 10−2yes

(P4)3 3 − 124

−

3 92

92

92

9 818

92

818

978

−2.28380 × 101

−1.21069−7.62892 × 10−2

yes

Table 3The classic fourth order method: multi-step strong stability.

292

Theorem 3.3. The four-stage fourth order RK method has the following property.293

With uniform time steps such that τ‖L‖H ≤ λ for sufficiently small λ, ‖un‖H ≤294

‖u0‖H for all n > 1.295

3.3. SSPRK methods. In this section, we study the strong stability of several296

SSPRK methods. The (explicit) SSPRK methods are a class of RK methods that can297

be formulated as combinations of Euler forward steps. The second order method with298

two stages, and the third order method with three stages are equivalent to the linear299


RK methods for autonomous linear systems, which have been discussed. For fourth300

order methods, in order to avoid backward-in-time steps and negative coefficients,301

at least five stages should be used [14, 18], which are denoted as SSPRK(5,4). We302

specially consider the method that is independently discovered in [18] and [29]. When303

applied on (1.1), the method takes the form304

(3.2) SSPRK(5,4) = P4 + 4.477718303076007× 10−3(τL)5.305

We will study its strong stability (in multiple steps) using our framework. Besides306

SSPRK(5,4), we will also consider two commonly used low storage SSPRK methods,307

a third order method with four stages SSPRK(4,3) and a fourth order method with308

ten stages SSPRK(10,4) [18, 29]. The two methods are309

(3.3) SSPRK(4,3) = P3 +1

48(τL)4,310

and311

(3.4)SSPRK(10,4) =P4 +

17

2160(τL)5 +

7

6480(τL)6 +

1

9720(τL)7

+1

155520(τL)8 +

1

4199040(τL)9 +

1

251942400(τL)10,

312

when applied on (1.1). We remark that the strong stability of SSPRK(10,4) has been313

proved by Ranocha and Offner in [23]. This is a reexamination using our framework.314

From Table 4, we are able to conclude the follow results.

SSPRK k∗ βk∗ Γ∗ λ(Γ∗) SS

(4,3) 2 − 124

−(

1 12

12

13

)−1.26759−6.57415 × 10−2 yes

(10,4) 3 − 13240

−

1 12

16

12

13

18

16

18

1072160

−1.30149−8.06493 × 10−2

−7.35115 × 10−4yes

(5,4) 3 −4.93345 × 10−3 −

1 12

16

12

13

18

16

18

124

−1.30140−8.00541 × 10−2

+1.97309 × 10−3no*

(5,4)2 3 −9.86690 × 10−3 −

2 2 43

2 83

243

2 1.5923

−5.74021−5.01739 × 10−1

−1.70056 × 10−2yes

(5,4)3 3 −1.48004 × 10−2 −

3 92

92

92

9 818

92

818

12.138

−2.28450 × 101

−1.21415−7.93174 × 10−2

yes

Table 4SSPRK methods: strong stability and multi-step strong stability.

315

Theorem 3.4. SSPRK(4,3) and SSPRK(10,4) are strongly stable.316

Theorem 3.5. The property stated in Theorem 3.3 also holds for SSPRK(5,4).317

The behavior of SSPRK(5,4) is very similar to that of the classic fourth order318

method, since it is almost the four-stage method except for a small fifth order per-319

turbation. Although the method can not be judged within this framework, one can320

indeed use the same counter example in Proposition 3.1 to disprove its strong stability.321

The proof would be similar to that in [33].322


3.4. Embedded RK methods in NDSolve. Finally, we consider embedded323

RK pairs that are used in NDSolve, a function for numerically solving differentiable324

equations in the commercial software Mathematica. Embedded RK methods are pairs325

of RK methods sharing the same stages. The notation p(p) is commonly used, if two326

methods in the pair are of order p and order p respectively. Their Butcher tableau327

has the following form.328

(3.5)

0c2 a2,1c3 a3,1 a3,2...

......

. . .

cs as,1 as,2 · · · as,s−1b1 b2 · · · bs−1 bsb1 b2 · · · bs−1 bs

329

For ddtu = f(t, u), the tableau gives two solutions330

(3.6) un+1 = un + τ

s∑i=1

biki, un+1 = un + τ

s∑i=1

biki,331

where332

(3.7) ki = f(t+ ciτ, un + τ

s∑j=1

ai,jkj).333

Then the difference un+1 − un+1 can be used for local error estimates for time step334

adaption.335

We examine strong stability of all such pairs used in Mathematica from order336

2(1) to order 9(8). These methods are chosen with several desired properties being337

considered, including the FSAL (First Same As Last) strategy and stiffness detection338

capability [38]. Tableaux of 2(1), 3(2) and 4(3) pairs [27] are given in Table 5, Table 6339

and Table 7. For the 5(4) pair [1, 24] and higher order pairs [35], the tableaux can

01 11 1

212

12

12 0

1 − 16

16

Table 5Tableau of embedded RK 2(1).

340be obtained through the Mathematica command341

(3.8) NDSolve`EmbeddedExplicitRungeKuttaCoefficients[p, Infinity].342

The output takes the form343

(3.9) {A, b, c, b− b}344


012

12

1 -1 21 1

623

16

16

23

16 0

22−√82

72

√82+1436

√82−4144

16−√82

48


025

25

35 − 3

2034

1 1944 − 15

441011

1 1172

2572

2572

1172

1172

2572

2572

1172 0

12515158970912

37101058970912

25196958970912

611058970912

119041747576


and the corresponding tableau is345

(3.10)

c Ab

b

.346

The stability results are documented in Table 8 and Table 9.347

4. Characterization of strongly stable methods. By numerically examining348

various RK methods, we recognize certain patterns of the coefficients in Lemma 2.3,349

which will be proved in this section.350

To characterize the coefficients, we would like to split the RK operator as a351

truncated exponential and a high order perturbation.352

(4.1) Rs = Pp + (τL)p+1Qs−(p+1),353

where354

(4.2) Pp =

p∑k=0

(τL)k

k!, Qs−(p+1) =

s−(p+1)∑k=0

αk+p+1(τL)k, αp+1 6=1

(p+ 1)!.355

Note that an RK method of order p for general nonlinear system may achieve higher356

order accuracy when applied to linear autonomous problems. Without special clarifi-357

cation, the order p in this section refers to the linear order.358

4.1. Coefficients in the energy equality. Coefficients in Lemma 2.3 have the359

following pattern.360

Lemma 4.1. For an s-stage RK method of linear order p, β0 = 1. Furthermore,361

(i) if p is odd, then362

(4.3) k∗ =p+ 1

2, βk∗ = (−1)k

∗2

(αp+1 −

1

(p+ 1)!

),363


Methods s p/p k∗ βk∗ λ(Γ∗) SS

2(1) 3 2 2 14

−1.30902−1.90983 × 10−1 no

1 1 1 −1.00000 no

3(2) 4 3 2 − 112

−1.26759−6.57415 × 10−2 yes

2 2 112

−1.28130−1.11257 × 10−1 no

4(3) 5 4 3 − 172

−1.30128−7.93266 × 10−2

+5.60618 × 10−3no*

3 2 − 1190414485456

−1.26759−6.57415 × 10−2 yes

5(4) 8 5 3 − 436209280

−1.3015−8.07336 × 10−2

−1.10151 × 10−3yes

4 3 51767367590960

−1.30150−8.07430 × 10−2

−1.14174 × 10−3no

6(5) 9 6 4 790072560896000

−1.30375−8.21839 × 10−2

−1.36689 × 10−3

−2.38718 × 10−5

no

5 3 12334679027158400

−1.30150−8.07336 × 10−2

−1.10151 × 10−3no

7(6) 10 7 4 2961560506338967665360400000

−1.30375−8.21836 × 10−2

−1.36301 × 10−3

−7.86229 × 10−6

no

6 4 − 202029199011855603112400000

−1.30375−8.21833 × 10−2

−1.35985 × 10−3

+5.49402 × 10−6

?

*The fourth order method in the 4(3) pair is exactly the classic four-stage fourth order method for

autonomous linear systems.Table 8

Embedded RK pairs: from 2(1) to 7(6) pairs.

364

(4.4) γi,j = − 1

i!j!(i+ j + 1), ∀0 ≤ i, j ≤ k∗ − 1;365

(ii) if p is even, then366

367

(4.5) k∗ ≥ p

2+ 1, β p

2+1 = (−1)p2+12

(αp+2 − αp+1 +

1

p!(p+ 2)

),368

369

(4.6) γi,j = − 1

i!j!(i+ j + 1)+ ιi,j,p ∀0 ≤ i, j ≤ p

2,370


Methods s p k∗ βk∗ λ(Γ∗) SS

8(7) 13 8 5 −3.21308 × 10−7

−1.30384−8.22588 × 10−2

−1.38584 × 10−3

−9.71236 × 10−6

+1.43671 × 10−7

?

7 4 −2.39706 × 10−6

−1.30375−8.21836 × 10−2

−1.36301 × 10−3

−7.86229 × 10−6

yes

9(8) 16 9 5 −8.95352 × 10−9

−1.30384−8.22588 × 10−2

−1.38585 × 10−3

−9.75366 × 10−6

−3.11800 × 10−8

yes

8 5 −5.46447 × 10−7

−1.30384−8.22588 × 10−2

−1.38585 × 10−3

−9.78641 × 10−6

−1.64476 × 10−7

yes

Table 9Embedded RK pairs: 8(7) pair and 9(8) pair.

where371

(4.7) ιi,j,p =

{(−1)

p2+1

(αp+1 − 1

(p+1)!

), i = j = p

2 ,

0, otherwise.372

The proof of this lemma is postponed to the end of the section.373

4.2. Criteria for strong stability. With Lemma 4.1, one can obtain the fol-374

lowing theorem regarding strong stability of linear RK methods with Rs = Pp. Note375

that the case p ≡ 0 (mod 4) is not covered. The difficulty for analyzing this class of376

methods has already been recognized in [33].377

Theorem 4.2. Consider a linear RK method of order p with p stages.378

(i) The method is not strongly stable if p ≡ 1 (mod 4) or p ≡ 2 (mod 4).379

(ii) The method is strongly stable if p ≡ 3 (mod 4).380

Proof. Note αp+1 = αp+2 = 0. βk∗ > 0 if p ≡ 1 (mod 4) or p ≡ 2 (mod 4). The381

method can not preserve strong stability due to Theorem 2.6. On the other hand,382

for p ≡ 3 (mod 4), the leading submatrix Γ∗ can be written as −ΛMΛ, where Λ =383

diag(1/0!, 1/1!, 1/2!, · · · , 1/(k∗ − 1)!) and M is the Hilbert matrix of order k∗. Γ∗ is384

negative definite since the Hilbert matrix is positive definite. Also note βk∗ = − 2(p+1)!385

in such cases. The strong stability can be proved using Theorem 2.7.386

Remark 4.3. This modulo pattern with periodicity 4 is closely related with the387

fact that i4 = 1, in which i is the imaginary unit. One can get a flavor by considering388

the scalar ODE ddtu = (iω)u, ω ∈ R.389

For a method with non-zero Qs−(p+1), noting that only limited number of stages390

would affect the leading coefficient and submatrix, one can conclude the following391

criteria for strong stability. We highlight that the condition for methods of odd linear392

order is both necessary and sufficient.393


Theorem 4.4. An RK method of odd linear order p is strongly stable if and only394

if395

(4.8) (−1)p+12

(αp+1 −

1

(p+ 1)!

)< 0.396

Proof. Since Γ∗ = −ΛMΛ is always negative definite, the method is strongly397

stable if and only if βk∗ < 0, which corresponds to the prescribed condition in the398

theorem.399

For RK methods with even linear order, we can only obtain a sufficient condition,400

which is given in Theorem 4.5. A similar condition has also been discussed in [23] for401

p = 4.402

Theorem 4.5. An RK method of even linear order p is strongly stable if403

(4.9) (−1)p2+1

(αp+2 − αp+1 +

1

p!(p+ 2)

)< 0,404

and405

(4.10) (−1)p2+1

(p2

!)2(

αp+1 −1

(p+ 1)!

)< ε.406

Here ε is the smallest eigenvalue of the Hilbert matrix of order p2 + 1.407

Proof. From Lemma 4.1, (4.9) implies k∗ = p2 +1 and βk∗ < 0. Note Γ∗ is negative408

definite with (4.10). Strong stability then follows from Theorem 2.7.409

We remark that, constraints in Theorem 4.4 and Theorem 4.5 can be used with410

order conditions for designing strongly stable RK methods.411

4.3. Regarding energy conserving systems. Problems for wave propaga-412

tions are usually featured with a conserved L2 energy. This conservation is also413

expected numerically to maintain the accurate shape and phase of the waves in long414

time simulations. Suppose an energy-conserving spatial discretization is used, for ex-415

ample [5, 4, 12, 13], the resulting method of lines scheme would satisfy L>H+HL = 0.416

Hence ddt‖u‖

2H = 0. While a strongly stable RK time discretization may not preserve417

this equality, we would like know how the total energy is dissipated. The following418

interpretation can be obtained based on Lemma 2.3. While one can also perform419

eigenvalue analysis alternatively, since L is normal in 〈·, ·〉H for energy conserving420

systems.421

With L>H +HL = 0, [·, ·]H = 0. The energy equality in Lemma 2.3 would then422

become423

(4.11) ‖Rsu‖2H = ‖u‖2H +

s∑k=k∗

βkτ2k‖Lku‖2H .424

Since βk∗ < 0 for strongly stable RK methods, we have dissipative energy unless425

L = 0.426

Proposition 4.6. With a suitably restricted time step, a strongly stable explicit427

RK method is energy conserving, if and only if L = 0.428

At the final time T = nτ , ‖un‖2H = ‖u0‖2H + O(τ2k∗−1). The total numerical429

dissipation due to the time integrator would then be of order 2k∗ − 1. We refer to430

2k∗ − 1 as the energy accuracy of the RK time integrator. With k∗ determined in431

Lemma 4.1, we have the following proposition.432


Proposition 4.7. Consider an RK method of linear order p applied on an energy433

conserving system.434

(i) The order of energy accuracy is p if p is odd.435

(ii) The order of energy accuracy is at least p + 1 if p is even. It achieves higher436

energy accuracy if and only if αp+2 = αp+1 − 1p!(p+2) .437

One can see from Proposition 4.7, the linearly even order RK methods achieve438

at least one degree higher order of energy accuracy than we usually expect. Hence439

compared with the odd order methods, they may be more suitable for integrating sys-440

tems modeling wave propagations. This also reveals the fact that even order methods441

introduce less numerical dissipation, and explains why it is harder to achieve strong442

stability.443

4.4. Proof of Lemma 4.1. We now prove Lemma 4.1. Instead of using the444

mathematical induction, we provide a more motivated proof below. Let Es(τ) =445

‖Rs(τ)‖2H . The idea is to use dm

dτm E(0) to determine coefficients in the energy equality.446

For clearness of the presentation, we first consider the case Qs−(p+1) = 0 and then447

move on to general RK methods. As a convention, matrices and coefficients with448

negative or fractional indexes are considered as 0.449

Proof. Step 1: (Rs = Pp.)450

It is easy to check β0 = α20 = 1. For m > 0, using the fact that451

(4.12)dm

dτm〈v, w〉H =

m∑k=0

(m

k

)〈 d

k

dτkv,

dm−k

dτm−kw〉H452

and453

(4.13)d

dτPp = Pp−1L,454

we have455

(4.14)dm

dτmEp(τ) =

m∑k=0

(m

k

)〈Pp−kLku, Pp−(m−k)Lm−ku〉H .456

Noting that Pp−k(0) = I if k ≤ p, for 1 ≤ m ≤ p+ 2, one can obtain457

(4.15)dm

dτmEp(0) = −2µ+

m∑k=0

(m

k

)〈Lku, Lm−ku〉H ,458

where459

(4.16) µ =

0, 1 ≤ m ≤ p,〈u, Lp+1u〉H , m = p+ 1,

〈u, Lp+2u〉H + (p+ 2)〈Lu,Lp+1u〉H , m = p+ 2.460


Furthermore, since461

(4.17)

m∑k=0

(m

k

)〈Lku, Lm−ku〉H

=

m∑k=1

(m− 1

k − 1

)〈Lku, Lm−ku〉H +

m−1∑k=0

(m− 1

k

)〈Lku, Lm−ku〉H

=

m−1∑k=0

(m− 1

k

)(〈Lk+1u, Lm−(k+1)u〉H + 〈Lku, Lm−ku〉H

)=−

m−1∑k=0

(m− 1

k

)[Lku, Lm−1−ku]H ,

462

(4.15) can be written as463

(4.18)dm

dτmEp(0) = −2µ−

m−1∑k=0

(m− 1

k

)[Lku, Lm−1−ku]H , 1 ≤ m ≤ p+ 2.464

On the other hand, by differentiating the expansion in Lemma 2.3, we obtain465

(4.19)dm

dτmEp(0) = βm

2m!‖Lm

2 u‖2H +

p−1∑k=0

γk,m−1−km![Lku, Lm−1−ku]H .466

Due to the uniqueness of the expansion, coefficients in (4.18) and (4.19) must be the467

same. With m = 1, · · · , p, one can get468

(4.20) βm2

= 0, ∀1 ≤ m ≤ p469

and470

(4.21) γi,j = − 1

(i+ j + 1)!

(i+ j

i

)= − 1

i!j!(i+ j + 1), i+ j ≤ p− 1, i, j ≥ 0.471

Case I: (p is odd.) If p is odd, then472

(4.22) βk = 0, k = 1, 2, · · · , p− 1

2.473

We need to compute dp+1

dτp+1 Ep(0) to determine β p+12

. With m = p+1, (4.18) and (4.19)474

imply the following identity.475

(4.23)

dp+1

dτp+1Ep(0) = β p+1

2(p+ 1)!‖L

p+12 u‖2H +

p−1∑k=0

γk,p−k(p+ 1)![Lku, Lp−ku]H

= −2〈u, Lp+1u〉H −p∑k=0

(p

k

)[Lku, Lp−ku]H .

476

From Corollary 2.2, we have477

(4.24) 〈u, Lp+1u〉H = (−1)p+12 ‖L

p+12 u‖2H −

p−12∑

k=0

(−1)k[Lku, Lp−ku]H .478


Hence (4.23) together with (4.24) gives479

(4.25) β p+12

= −(−1)p+12

2

(p+ 1)!6= 0,480

which implies k∗ = p+12 . For 0 ≤ i, j ≤ k∗−1, we have i+ j ≤ 2k∗−2 = p−1. Hence481

the leading submatrix Γ∗ is completely determined by (4.21).482

Case II: (p is even). For even p,483

(4.26) βk = 0, k = 1, 2, · · · , p2.484

Note β p2+1 appears in the expansion of dp+2

dτp+2 Ep(0). With m = p + 2, one can derive485

from (4.18) and (4.19) that486

(4.27)

dp+2

dτp+2Ep(0)

=β p2+1(p+ 2)!‖L

p2+1u‖2H +

p−1∑k=0

γk,p+1−k(p+ 2)![Lku, Lp+1−ku]H

=− 2〈u, Lp+2u〉H − 2(p+ 2)〈Lu,Lp+1u〉H −p+1∑k=0

(p+ 1

k

)[Lku, Lp+1−ku]H .

487

From Proposition 2.1, we have488

(4.28) 〈u, Lp+2u〉H = (−1)p2+1‖L

p2+1u‖2H −

p2∑

k=0

(−1)k[Lku, Lp+1−ku]H489

and490

(4.29) 〈Lu,Lp+1u〉H = (−1)p2 ‖L

p2+1u‖2H −

p2−1∑k=0

(−1)k[Lk+1u, Lp−ku]H .491

After identifying the coefficients of ‖Lp2+1u‖2H in (4.27) with (4.28) and (4.29), one492

can get493

(4.30) β p2+1 = (−1)

p2+1 2

p!(p+ 2)6= 0.494

Hence k∗ = p2 + 1.495

Except for γ p2 ,

p2, all other γi,j with i + j ≤ p − 1 are clarified in (4.21). To496

determine γ p2 ,

p2, we need to consider dp+1

dτp+1 Ep(0). Note that (4.23) holds regardless of497

the parity of p. Furthermore, with p being even,498

(4.31) 〈u, Lp+1u〉H = (−1)p2+1 1

2JL

p2 uK2H −

p2−1∑k=0

(−1)k[Lku, Lp−ku]H .499

Hence by comparing the coefficient of JLp2 uK2H in (4.23) with that in (4.31), we get500

(4.32) γ p2 ,

p2

= − 1

(p2 )!(p2 )!(p+ 1)− (−1)

p2+1 1

(p+ 1)!.501


Now we have proved Lemma 4.1 for Rs = Pp.502

Step 2: Rs = Pp + (τL)p+1Qs−(p+1).503

The proof of the general case is based on the fact that not all high order stages504

will contribute to k∗ and Γ∗. Note that505

(4.33) ‖Rsu‖2H = ‖Ppu‖2H +∑

max(i,j)>p

αiαjτi+j〈Liu, Lju〉H .506

If p is odd, from Corollary 2.2, only 2αp+1τp+1〈L0u, Lp+1u〉H (since α0 = 1) in the507

second term would affect β p+12

. With this additional term being considered, instead508

of (4.23), we obtain509

(4.34)

dp+1

dτp+1Es(0) = β p+1

2(p+ 1)!‖L

p+12 u‖2H +

p−1∑k=0

γk,p−k(p+ 1)![Lku, Lp−ku]H

= (2αp+1(p+ 1)!− 2)〈u, Lp+1u〉H −p∑k=0

(p

k

)[Lku, Lp−ku]H .

510

After expanding 〈u, Lp+1u〉 with (4.24), (4.34) implies511

(4.35) β p+12

= (−1)p+12 2

(αp+1 −

1

(p+ 1)!

).512

Since αp+1 6= 1(p+1)! , we have β p+1

26= 0 and k∗ = p+1

2 . Also note that 〈Liu, Lju〉H513

can only produce [Li′u, Lj

′u]H terms with i′ + j′ = i+ j − 1. Hence if max{i, j} > p,514

〈Liu, Lju〉H can not affect values of γi′,j′ with i′ + j′ ≤ 2(k∗ − 1) = p − 1. In other515

words, the leading submatrix Γ∗ is unchanged.516

Similarly, for even p, we still have β1 = · · · = β p2

= 0 after adding extra high517

order terms, which implies k∗ ≥ p2 + 1. When computing dp+2

dτp+2 E(0) to obtain β p2+1,518

an extra term 2τp+2(αp+2〈L0u, Lp+2〉H + αp+1〈L1u, Lp+1u〉H

)should be considered.519

Then we have520

(4.36)

dp+2

dτp+2Es(0)

=β p2+1(p+ 2)!‖L

p2+1u‖2H +

p−1∑k=0

γk,p+1−k(p+ 2)![Lku, Lp+1−ku]H

=2 (αp+2(p+ 2)!− 1) 〈u, Lp+2u〉H + 2 (αp+1(p+ 2)!− (p+ 2)) 〈Lu,Lp+1u〉H

−p+1∑k=0

(p+ 1

k

)[Lku, Lp+1−ku]H .

521

After substituting (4.28) and (4.29) into (4.36), we get522

(4.37) β p2+1 = (−1)

p2+12

(αp+2 − αp+1 +

1

p!(p+ 2)

).523

As for the corresponding principle submatrix, only γ p2 ,

p2

will be changed due to524

2αp+1τp+1〈L0u, Lp+1u〉H . Then with (4.34) and (4.31), one can obtain525

(4.38) γ p2 ,

p2

= − 1

(p2 )!(p2 )!(p+ 1)+ (−1)

p2+1

(αp+1 −

1

(p+ 1)!

),526


which completes the proof.527

Remark 4.8 (Uniqueness in Lemma 2.3). Without showing the uniqueness in528

Lemma 2.3, one can still prove Lemma 4.1 with induction and then derive other529

results in section 4. The reason for justifying the uniqueness, is to exclude the possi-530

bility that the leading submatrices for even order RK methods are negative definite531

under another expansion, which may result in an if and only if condition, as that for532

the methods of odd linear order in Theorem 4.4.533

5. Conclusions. In this paper, we present a framework on analyzing the strong534

stability of explicit RK methods for solving semi-negative linear autonomous systems,535

which are typically obtained from stable method of lines schemes for hyperbolic prob-536

lems. With this framework, strong stability in one step or multiple steps of various537

RK methods are examined. Finally, we analyze the coefficients in the energy equality,538

based on which, corollaries regarding the criteria of strong stability are derived.539

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