STRONG-STABILITY-PRESERVING 4-STAGE HERMITE-BIRKHOFF TIME … · a higher-order time-discretization (RK, multistep or HB) method that maintains strong stability for the same norm,

CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 19, Number 1, Spring 2011

STRONG-STABILITY-PRESERVING 4-STAGE

HERMITE-BIRKHOFF TIME-DISCRETIZATION

METHODS

TRUONG NGUYEN-BA, HUONG NGUYEN-THUTHIERRY GIORDANO AND REMI VAILLANCOURT

ABSTRACT. Strong-stability-preserving (SSP) time-dis-cretization methods have a nonlinear stability property thatmakes them particularly suitable for the integration of hyper-bolic conservation laws. A collection of 4-stage explicit SSPHermite-Birkhoff methods of orders 4 to 8 with nonnegativecoefficients are constructed as k-step analogues of fourth-orderRunge-Kutta methods with three off-step points. Generally,they have high-stage orders and hence are less susceptible thanRK methods to order reduction from source terms or nonhomo-geneous boundary conditions. The new methods generally havelarger effective SSP coefficients and larger maximum effectiveCFL numbers than Huang’s hybrid methods of the same orderon Burgers’ equations, independently of the number of steps.This is more so when the number of steps is small.

1 Introduction The method of lines is a popular semidiscretiza-tion method for the solution of time-dependent partial differential equa-tions (PDEs). A suitable discretization of the spatial variables (e.g., byfinite differences, finite volumes, finite elements, or spectral methods)yields a set of ordinary differential equations (ODEs) with respect totime. Then, these ODEs can be integrated using standard time-steppingtechniques such as Hermite-Birkhoff (HB) general linear multistep, lin-ear multistep or Runge-Kutta (RK) methods. A relevant question hereconcerns stability. For problems with smooth solutions, usually a linear

This work was supported in part by the Vietnam Ministry of Education andTraining, the Natural Sciences and Engineering Research Council of Canada and theCentre de recherches mathematiques of the Universite de Montreal.

AMS subject classification: Primary: 65L06; Secondary: 65M20.Keywords: Strong stability preserving, Hermite-Birkhoff method, SSP coeffi-

cient, time discretization, method of lines, comparison with other SSP methods.Copyright c©Applied Mathematics Institute, University of Alberta.

79

80 T. NGUYEN-BA ET AL.

stability analysis is adequate. But for problems with discontinuous so-lutions, such as solutions to hyperbolic problems, a stronger measure ofstability is usually required.

In this paper, we develop a class of high-order strong-stability-preserv-ing (SSP) time-discretization methods to solve PDE’s by the method oflines. These methods were called TVD (total variation diminishing) timediscretizations in [35, 37]. This class of methods was further developedin [10]. One assumes that the first-order forward Euler (FE) time dis-cretization of the method of lines is strongly stable under a certain normwhen the time step, ∆t, is suitably restricted, and then one tries to finda higher-order time-discretization (RK, multistep or HB) method thatmaintains strong stability for the same norm, but perhaps under a dif-ferent time step restriction. For this purpose, the total variation normwas used in [35, 37]. As FE was assumed to be TVD, then the classof high-order time discretization developed there was termed TVD time

discretization. However, following [11], in this paper we refer to themas strong-stability-preserving (SSP) methods.

We briefly review the developments of SSP methods. Shu and Osher[37] constructed a series of second- to fifth-order SSP RK methods. Shu[35] found a class of first-order SSP RK methods with very large SSPcoefficients, as well as a class of high-order SSP linear multistep meth-ods. Gottlieb and Shu [10] derived optimal s-stage SSP RK methodsof order s for s = 2, 3, and proved that for s = 4 there is no such SSPmethod with nonnegative coefficients. Spiteri and Ruuth [40, 41] stud-ied optimal s-stage SSP RK methods of order p with s > p for p ≤ 4.They proved the nonexistence of fifth-order SSP RK methods with non-negative coefficients [31] and constructed some fifth-order methods ofseven to nine stages with downwind-biased spatial discretization [32].A 10-stage method of order 5 was given in [29]. Hundsdorfer, Ruuthand Spiteri [16] proved that the implicit Euler method can uncondition-ally preserve the strong stability of the FE method (see also [13]) andstudied multistep methods with specific starting procedures.

In addition, some SSP methods for special purposes have also beeninvestigated. For example, low storage RK methods were constructed in[10, 11, 29, 32, 40]. SSP methods for constant-coefficient linear sys-tems were studied in [7, 11]. For a description of the state-of-the-art, werefer the reader to the review papers by Gottlieb, Shu and Tadmor [11],Shu [36], Gottlieb [6], and Gottlieb, Ketcheson and Shu [8] and theirrecent monograph [9].

Previous studies have investigated optimally-contractive, one-step,multistage methods [7, 21, 44] and one-stage, multistep methods [24,

SSP 4-STAGE HB METHODS 81

25]. The equivalence between the SSP coefficient and the radius of abso-lute monotonicity for multistep, multistage, and general linear methodsis discussed in [14, 19, 39]. It is interesting that recent research by Fer-racina and Spijker [4, 5] and Higueras [13, 14] established a connectionbetween SSP and contractivity studies. The latter is a classic topic andhas been developed independently of the former (see, e.g., [22, 24, 38]).As a consequence, some optimal SSP methods agree with optimal con-tractive methods and the SSP coefficient of a method is also related tothe radius of absolute monotonicity. This discovery led to the develop-ment of new optimal and efficient SSP methods [18, 20]. Huang [15]explored hybrid methods (HM) based on linear multistep methods.

Spijker extended the theory of absolutely monotone methods to thelarger class of general linear methods [39] which combine multistep andRK methods by evaluating functions at multiple steps and multiplestages.

The explicit 4-stage HB methods introduced in this paper are a com-bination of k-step methods and 4-stage RK methods of order 4 withfunction evaluations at three off-step points. For a given accuracy, theexplicit 4-stage HB methods need fewer function evaluations than RKmethods, and fewer steps than linear k-step and hybrid methods [15].Several new explicit 4-stage SSP HB methods with nonnegative coeffi-cients have been found by computer search and are presented here.

These new methods have larger effective SSP coefficients than knownthird- to seventh-order SSP hybrid methods of the same order and thesame number of steps to go from tn to tn+1, especially when the stepnumber k is small. In particular, no counterparts of HB25, HB36 andHB47 have been found among hybrid and general linear multistep meth-ods in the literature. Moreover, we have new HB methods of order 8.Some of these proposed general linear methods can attain high-stageorders, a property that alleviates the order-reduction phenomenon en-countered in the classic explicit RK schemes due to nonhomogeneousboundary/source terms (see [3]).

We remark that multistep methods require some sort of starting con-ditions. So there are two approaches to analyze their SSP property inthe literature, dependent on whether starting conditions are taken intoaccount or not. For example, the SSP property of linear multistep meth-ods which depend on starting conditions has been analyzed in [16]. Inthis paper, we only consider the SSP property independent of startingconditions.

Section 2 introduces 4-stage SSP HB methods. Order conditions arelisted in Section 3. Section 4 derives the Shu-Osher representation of


k-step 4-stage HB methods of order p. Several new SSP HB methodsare constructed in Section 5. Section 6 presents numerical results forseveral methods applied to Burgers’ equations.

2 Four-stage SSP HB methods We study high-order SSP time-discretization methods for ODE initial value problems of the form

(1) y′ = f(t, y), y(t0) = y0, where ′ =d

dt,

obtained by applying the method of lines to a time dependent PDEspatially discretized by finite difference or finite element (see, for in-stance, [2, 12, 17, 28, 42, 43]). Throughout the paper, we assumethat there exists a maximum step size ∆tFE > 0 such that the forwardEuler method is strongly stable, that is,

(2) ‖v + ∆tFEf(t, v)‖ ≤ ‖v‖,

where ‖ ·‖ is a given (semi)norm. Natural choices are the total variationnorm or the maximum norm.

Notation 1. We shall denote the k-step SSP methods of order p usedin this paper as follows:

• GLkp: general linear method,• HBkp: Hermite-Birkhoff,• HMkp: hybrid method,

The s-stage SSP Runge-Kutta method of order p will be denoted byRKsp.

All the methods considered in this work are SSP. except for the non-SSP classic 4-stage Runge-Kutta method of order 4, RK44. Thereforethe denomination SSP will often be omitted in what follows.

Notation 2. The abscissa vector c = [c1, c2, c3, c4]T , 0 ≤ cj ≤ 1, de-

fines the off-step points tn + ∆tcj .

A general 4-stage HBkp method requires the following four formulaeto perform integration from tn to tn+1, where, for simplicity, c1 = 0 isused in the summations.


Let Fj := f(tn + cj , Yj) be the jth stage derivative and set Y1 = yn.For i = 2, 3, 4, an HB polynomial of degree 2k+ i−3 is used as predictorPi to obtain the ith stage value Yi,

(3) Yi =

k−1∑

j=0

αijyn−j + ∆t

[i−1∑

j=1

aijFj +

k−1∑

j=1

βijfn−j

]

, i = 2, 3, 4.

An HB polynomial of degree 2k + 2 is used as integration formula toobtain yn+1:

(4) yn+1 =

k−1∑

j=0

αjyn−j + ∆t

[ 4∑

j=1

bjFj +

k−1∑

j=1

βjfn−j

]

.

3 Order conditions of general 4-stage HBkp To derive theorder conditions of general 4-stage HBkp we shall use the following ex-pressions coming from the backsteps of the methods:

(5) Bi(j) =

k−1∑

`=1

αi`

(−`)j

j!+

k−1∑

`=1

βi`

(−`)j−1

(j − 1)!,

{

i = 2, 3, 4,

j = 1, 2, . . . , p.

As in the construction of RK methods, we impose the following sim-plifying conditions on the abscissa vector c = [c1, c2, c3, c4]

T :

(6) ci =

i−1∑

j=1

aij + Bi(1), i = 2, 3, 4.

Forcing an expansion of the numerical solution produced by formulae(3)–(4) to agree with a Taylor expansion of the true solution, we obtainmultistep- and RK-type order conditions that must be satisfied by gen-eral 4-stage HBkp methods. These order conditions are simply RK orderconditions with backstep parts Bi(j) defined in (5) and B(j) defined in(15). To reduce the large number of these RK order conditions withbackstep parts to the five order conditions (10)–(14) below, we imposethe following simplifying assumptions, as in similar searches for ODEsolvers [27]:

i−1∑

j=1

aijckj + k!Bi(k + 1) =

1

k + 1ck+1i ,

{

i = 2, 3, 4,

k = 1, 2, . . . , p − 4.(7)


Note that (7) with k = 0 is (6). There remain seven sets of equations tobe solved:

k−1∑

j=0

αij = 1, i = 2, 3, 4,(8)

k−1∑

i=0

αi = 1,(9)

4∑

i=1

bicki + k!B(k + 1) =

1

k + 1, k = 0, 1, . . . , p − 1,(10)

4∑

i=2

bi

[i−1∑

j=1

aij

cp−3j

(p − 3)!+ Bi(p − 2)

]

+ B(p − 1) =1

(p − 1)!,(11)

4∑

i=2

bi

ci

p − 1

[i−1∑

j=1

aij

cp−3j

(p − 3)!+ Bi(p − 2)

]

+ B(p) =1

p!,(12)

4∑

i=2

bi

[i−1∑

j=1

aij

cp−2j

(p − 2)!+ Bi(p − 1)

]

+ B(p) =1

p!,(13)

4∑

i=2

bi

[i−1∑

j=1

aij

[j−1∑

k=1

ajk

cp−3k

(p − 3)!+ Bj(p − 2)

]

(14)

+ Bi(p − 1)

]

+ B(p) =1

p!,

where the backstep parts, B(j), are defined by

(15) B(j) =k−1∑

i=1

αi

(−i)j

j!+

k−1∑

i=1

βi

(−i)j−1

(j − 1)!, j = 1, . . . , p + 1.

4 Shu-Osher representation of four-stage HBkp In order toget conditions for SSP, we rewrite (3) with i = 3, 4 and (4) as follows.From the convex combinations

(16) λki ≥ 0,k−1∑

i=1

λki = 1, k = 3, 4, λi ≥ 0,4

∑

i=1

λi = 1,


we have

(17) Yi =

[i−1∑

`=1

λi`

]

αi0yn +k−1∑

j=1

αijyn−j

+ ∆t

[i−1∑

j=1

aijFj +k−1∑

j=1

βijfn−j

]

, i = 3, 4,

and

(18) yn+1 =

[ 4∑

`=1

λ`

]

α0yn +

k−1∑

j=1

αjyn−j + ∆t

[ 4∑

j=1

bjFj +

k−1∑

j=1

βjfn−j

]

.

Since from formula (3) with i = 2, 3, 4, yn can be expressed as a functionof Yi, respectively, that is,

yn =1

α20

{

Y2 −

k−1∑

j=1

α2jyn−j − ∆t

[

a21fn +

k−1∑

j=1

β2jfn−j

]}

,(19)

yn =1

α30

{

Y3 −

k−1∑

j=1

α3jyn−j − ∆t

[ 2∑

`=1

a3`F` +

k−1∑

j=1

β3jfn−j

]}

,(20)

yn =1

α40

{

Y4 −k−1∑

j=1

α4jyn−j − ∆t

[ 3∑

`=1

a4`F` +k−1∑

j=1

β4jfn−j

]}

,(21)

we replace the variable yn in the term λ32α30yn in (17) by the right-hand side of (19). Similarly, yn in the terms λ42α0yn and λ43α0yn in(17) with i = 4 can be replaced by the right-hand sides of (19) and(20), respectively. Also, yn in the terms λ2α0yn, λ3α0yn and λ4α0yn

in (18) can be replaced by the right-hand sides of (19), (20) and (21),respectively.

Now, we rewrite (3) with i = 2 and (17)–(18) in the Shu-Osher equiv-alent form:

Y2 =

k−1∑

j=0

A2jyn−j + ∆t

k−1∑

j=0

B2jfn−j ,(22)

Y3 =

k−1∑

j=0

A3jyn−j + ∆t

k−1∑

j=0

B3jfn−j + e32Y2 + ∆t g32F2,(23)


Y4 =

k−1∑

j=0

A4jyn−j + ∆t

k−1∑

j=0

B4jfn−j +

3∑

`=2

(e4`Y` + ∆tg4`F`),(24)

yn+1 =

k−1∑

j=0

Ajyn−j + ∆t

k−1∑

j=0

Bjfn−j +

4∑

`=2

(e`Y` + ∆tg`F`).(25)

Thus, by

e32 = λ32α30/α20, e42 = λ42α40/α20, e43 = λ43α40/α30,

e2 = λ2α0/α20, e3 = λ3α0/α30, e4 = λ4α0/α40,

a21 = β20, a31 = β30, a41 = β40, b1 = β0,

we have

Aij = αi,j −

i−1∑

`=2

ei`α`j , j = 0, 1, . . . , k − 1, i = 2, 3, 4,

Aj = αj −

s∑

`=2

e`α`j , j = 0, 1, . . . , k − 1,

Bij = βi,j −i−1∑

`=2

ei`β`j , j = 0, 1, . . . , k − 1, i = 2, 3, 4,

Bj = βj −

s∑

`=2

e`β`j , j = 0, 1, . . . , k − 1,

gij = aij −i−1∑

`=j+1

ei`a`j , i = 3, 4, j = 2, 3, . . . , i − 1,

gj = bj −

4∑

`=j+1

e`a`j , j = 2, 3, 4.

The representation (22)–(25) of HBkp allows us to derive the SSP ofthe methods. In fact, using the nonnegativity of the coefficients of (22)–(25), one finds that the following functions are convex combinations offorward Euler steps:

• Y2 with step sizesB2j

A2j∆t, j = 0, 1, . . . , k − 1,



A3j∆t, j = 0, . . . , k − 1, and g32

e32∆t,


A4j∆t, j = 0, . . . , k − 1, g42

e42∆t, and g43

e43∆t,

• yn+1 with step sizesBj

Aj∆t, j = 0, . . . , k−1, g2

e2∆t, g3

e3∆t and g4

e4∆t.

This immediately gives the following result, which is a straightforwardextension of the corresponding result presented in [11, 15].

Theorem 1. If the forward Euler method is strongly stable under the

Courant-Friedrichs-Lewy condition ∆t ≤ ∆tFE, then the k-step 4-stageHB methods (22)–(25) are SSP provided

∆t ≤ c(Aij , Bij , eij , gij , Aj , Bj , ej , gj)∆tFE,

where c(Aij , Bij , eij , gij , Aj , Bj , ej , gj) is the coefficient taken to be

c(Aij , Bij , eij , gij , Aj , Bj , ej , gj)

= min

{

minj=0,k−1

{

A2j

B2j

}

, minj=0,k−1

{

A3j

B3j

}

,e32

g32

, minj=0,k−1

{

A4j

B4j

}

,

minj=2,3

{

e4j

g4j

}

, minj=0,k−1

{

Aj

Bj

}

, minj=2,4

{

ej

gj

} }

,

(26)

with the convention that a/0 = +∞.

5 Construction of 4-stage SSP HB methods Since there aremany free parameters in HBkp when the number of steps, k, is suffi-ciently large, we use the Matlab Optimization Toolbox to search for themethods with largest cHB for different k.

To optimize HBkp, we maximize c(Aij , Bij , eij , gij , Aj , Bj , ej , gj) ofTheorem 1 to obtain the SSP coefficient cHB by solving the nonlinearprogramming problem

(27) cHB = maxAij ,Bij ,eij ,gij ,Aj ,Bj ,ej ,gj

c(Aij , Bij , eij , gij , Aj , Bj , ej , gj),

where all the numbers in all pairs

{Aj , Bj} , j = 0, 1, . . . , k − 1, {Aij , Bij} ,

{

i = 2, 3, . . . , s,

j = 0, 1, . . . , k − 1,

{ej , gj} , j = 2, 3, . . . , s, {eij , gij} ,

{

i = 3, 4, . . . , s,

j = 2, 3, . . . , i − 1,


are nonnegative, excluding null pairs {0, 0}, and the objective func-tion (27) is subject to

• the convex combinations constraints (16),• the simplifying assumptions (6) and (7) for HBkp,• the order conditions (8) to (14) for HBkp.

Definition 1. (See [32]) The effective SSP coefficients of SSP HB andother methods are

(28) cHB,eff =cHB

l, Ceff =

C

l,

respectively, where l is the number of function evaluations per time stepand C is the SSP coefficient of RK methods, or hybrid methods, or linearmultistep methods or general linear multistep methods.

The number C is defined in [3, 11, 15, 40]. Gottlieb [6] pointed outthat one looks for high-order SSP methods with C as large as possible,taking their computational costs and orders into account.

In this paper, l = 4 for HB methods and l = 2 for hybrid methods.The effective SSP coefficients provide a fair comparison between methodsof the same order.

5.1 Fourth-order methods Spiteri and Ruuth [40] found a five-stageSSP RK method of order 4, called RK54, with cRK = 1.508 and cRK,eff =0.302. Other fourth-order SSP RK methods with more stages can befound in [41].

Gottlieb, Shu and Tadmor [11] proved that there are no two-step HMmethods of order four with nonnegative coefficients. Huang [15] found

• 3-step HM34 with cHM = 0.494 and cHM,eff = 0.247,• 4-step HM44 with cHM = 0.682 and cHM,eff = 0.341,• 5-step HM54 with cHM = 0.793 and cHM,eff = 0.396.

We present SSP HB methods of order 4 with 2, 3, 4, and 5 steps.

HB24. Here cHB = 1.593, cHB,eff = 0.398, andc = [0, 0.41402182360075634, 0.51006789128486130, 0.95950236308462156]T .

Y2 = 1.3137364256438025 e-01yn−1 + 8.6862635743561967 e-01yn

+ 5.4539546616513657 e-01∆tfn,

Y3 = 3.7611628586354845 e-01yn−1 + 2.3615691062661964 e-01∆tfn−1


+ 6.2388371413645161 e-01Y2 + 3.9172579348020337 e-01∆tF2,

Y4 = 6.0575318286657687 e-02yn−1 + 4.3008189202816044 e-02yn

+ 8.9641649251052635 e-01Y3 + 5.6284441132346308 e-01∆tF3,

yn+1 = 4.4524211403575709 e-02yn−1 + 3.8295387371954609 e-01yn

+ 2.4045011377921155 e-01∆tfn + 1.6774221781036905 e-01Y2

+ 9.2083912086026398 e-02∆tF2 + 4.0477969706650918 e-01Y4

+ 2.5415417076166058 e-01∆tF4.


Y2 = 1.8162651927653470 e-01yn−1 + 8.1837348072346527 e-01yn

+ 4.3925422184221165 e-02∆tfn−1 + 4.4392923901936121 e-01∆tfn,

Y3 = 2.2153848827415687 e-01yn−1 + 1.2017424174852812 e-01∆tfn−1

+ 7.7846151172584321 e-01Y2 + 4.2227886734649689 e-01∆tF2,

Y4 = 6.7530828539879564 e-04yn−2 + 1.3756312556575984 e-01yn−1

+ 7.4621545159975711 e-02∆tfn−1 + 8.6176156614884136 e-01Y3

+ 4.6746524085603702 e-01∆tF3,

yn+1 = 1.0945244030080964 e-02yn−2 + 4.4070641867034777 e-01yn

+ 2.3906256700586392 e-01∆tfn + 5.4834833729957133 e-01Y4

+ 2.9745326043523995 e-01∆tF4.


Y2 = 4.7524774508095377 e-02yn−2 + 1.2685090981790423 e-01yn−1

+ 8.2562431567400041 e-01yn + 2.4589659756032358 e-02∆tfn−2

+ 6.5633572056908576 e-02∆tfn−1 + 4.2718395234621287 e-01∆tfn,

Y3 = 1.9918300748893264 e-01yn−1 + 1.0305871903720029 e-01∆tfn−1

+ 8.0081699251106719 e-01Y2 + 4.1434846512195339 e-01∆tF2,

Y4 = 1.2681476034277145 e-01yn−1 + 6.5614868058771342 e-02∆tfn−1

+ 8.7318523965722883 e-01Y3 + 4.5179231610038256 e-01∆tF3,


yn+1 = 3.2258872667719520 e-03yn−2 + 1.4711725267828712 e-02yn−1

+ 4.0081143477759190 e-01yn + 7.6119523449503096 e-03∆tfn−1

+ 2.0738271584706414 e-01∆tfn + 5.8125095268780758 e-01Y4

+ 3.0074341872002397 e-01∆tF4.


Y2 = 4.0893858040237707 e-02yn−2 + 1.4166030901112323 e-01yn−1

+ 8.1744583294857931 e-01yn + 2.0662801764908201 e-02∆tfn−2

+ 7.1577958728480565 e-02∆tfn−1 + 4.1303809445288953 e-01∆tfn,

Y3 = 1.9239967708901987 e-01yn−1 + 9.7215488531579436 e-02∆tfn−1

+ 8.0760032291097994 e-01Y2 + 4.0806336641472890 e-01∆tF2,

Y4 = 1.2261927162438579 e-01yn−1 + 6.1956925160720017 e-02∆tfn−1

+ 8.7738072837561432 e-01Y3 + 4.4332192978558854 e-01∆tF3,

yn+1 = 1.1841942363290039 e-03yn−4 + 2.5436996045346688 e-02yn−1

+ 3.6010914521170023 e-01yn + 1.2852776235066601 e-02∆tfn−1

+ 1.8195553654826183 e-01∆tfn + 6.1326966450662379 e-01Y4

+ 3.0987219385521370 e-01∆tF4.

Note that the new HB24 requires one step less than HM34 while both arefourth order. Also, the 4-stage HB24 uses one fewer function evaluations thanthe 5-stage RK54, both being of order 4.

5.2 Fifth-order methods Ruuth and Spiteri [31] proved that there areno fifth-order SSP Runge-Kutta methods with nonnegative coefficients (theyrecently considered fifth-order methods with negative coefficients in [29, 32]).In case of linear multistep methods, Ruuth and Hundsdorfer [30] pointed outthat fifth-order methods of this type require at least k = 7 steps. Huang[15] constructed a hybrid four-step fifth-order SSP method with nonnegativecoefficients, called HM45, with cHM = 0.371 and cHM,eff = 0.185.



Y2 = 2.9087136529909646 e-01yn−1 + 7.0912863470090359 e-01yn


+ 8.3019628155931935 e-01∆tfn,

Y3 = 1.7386041575857100 e-01yn−1 + 4.4327424061899495 e-01yn

+ 1.7154130384111310 e-01∆tfn−1 + 3.8286534362243407 e-01Y2

+ 4.4823092603973058 e-01∆tF2,

Y4 = 1.0738138250177699 e-01yn−1 + 6.4165836289914413 e-01yn

+ 4.2428460618156771 e-01∆tfn + 6.9789004846984226 e-03Y2

+ 2.4398135411438043 e-01Y3 + 2.8563564217231024 e-01∆tF3,

yn+1 = 2.0843560497871033 e-02yn−1 + 3.7256984232963808 e-01yn

+ 2.4401278765485472 e-02∆tfn−1 + 9.9464375557570234 e-02∆tfn

+ 1.9865525045683016 e-01Y2 + 4.0793134671566084 e-01Y4

+ 4.7757638121279422 e-01∆tF4.


Y2 = 6.7487162492609845 e-02yn−2 + 5.1463622662637439 e-02yn−1

+ 8.8104921484475274 e-01yn + 3.7666033438306788 e-02∆tfn−1

+ 6.4483663353200249 e-01∆tfn,

Y3 = 5.0435237298896751 e-01yn−1 + 4.6382691456699904 e-02yn

+ 3.6913362027045182 e-01∆tfn−1 + 3.3947318843433938 e-02∆tfn

+ 4.4926493555433245 e-01Y2 + 3.2881533031940274 e-01∆tF2,

Y4 = 3.4089035733282883 e-02yn−2 + 1.1966629872952983 e-01yn−1

+ 8.7583317617032308 e-02∆tfn−1 + 8.4624466553718747 e-01Y3

+ 6.1936331373448850 e-01∆tF3,

yn+1 = 1.7929870801754462 e-02yn−2 + 2.1701226168846510 e-07yn−1

+ 4.2335763047475578 e-01yn + 1.3122805551224935 e-02∆tfn−2

+ 1.5883046480813512 e-07∆tfn−1 + 1.8492270511436729 e-01∆tfn

+ 1.7607534416799414 e-01Y2 + 2.4724918308539948 e-03Y3

+ 3.8016444571237995 e-01Y4 + 1.2886888753573725 e-01∆tF2

+ 2.7824093958806490 e-01∆tF4.



Y2 = 4.2142284811009928 e-03yn−3 + 3.2715727283842166 e-01yn−1

+ 6.6862849868047747 e-01yn + 1.5201041151078148 e-01∆tfn−1

+ 4.3490188184705292 e-01∆tfn,

Y3 = 8.4798329800832174 e-03yn−3 + 2.8216308590334344 e-01yn−1

+ 1.8352980360440418 e-01∆tfn−1

+ 7.0935708111657336 e-01Y2 + 4.6139332991033022 e-01∆tF2,

Y4 = 7.5522769122803070 e-03yn−3 + 1.8534236113702979 e-01yn−1

+ 4.9122935200378633 e-03∆tfn−3 + 1.2055385285482725 e-01∆tfn−1

+ 8.0710536195068983 e-01Y3 + 5.2497259906497395 e-01∆tF3,

yn+1 = 1.0180337593023980 e-02yn−3 + 8.6301767145259885 e-03yn−2

+ 4.2511998654296279 e-02yn−1 + 4.2490789599385109 e-01yn

+ 6.6216860121605485 e-03∆tfn−3 + 5.6134013151203964 e-03∆tfn−2

+ 2.7651451070840605 e-02∆tfn−1 + 2.7637655879772210 e-01∆tfn

+ 5.1376959104430264 e-01Y4 + 3.3417564824399548 e-01∆tF4.


Y2 = 9.7061474501702416 e-04yn−4 + 3.0978781320890014 e-01yn−1

+ 6.8924157204608283 e-01yn + 1.2962869645032163 e-01∆tfn−1

+ 4.4106347408298863 e-01∆tfn,

Y3 = 6.3156160214467997 e-03yn−4 + 2.7396035366074556 e-01yn−1

+ 4.0415257238825743 e-03∆tfn−4 + 1.7531430233946144 e-01∆tfn−1

+ 7.1972403031780774 e-01Y2 + 4.6056998600739968 e-01∆tF2,

Y4 = 2.6272423021883179 e-03yn−4 + 1.9321016726372053 e-01yn−1

+ 1.6812401689889544 e-03∆tfn−4 + 1.2364017357298093 e-01∆tfn−1

+ 8.0416259043409111 e-01Y3 + 5.1460440032877375 e-01∆tF3,

yn+1 = 6.5956149353558462 e-03yn−3


+ 2.0214560228139670 e-02yn−2 + 2.0405925954591175 e-02yn−1

+ 4.3971729806361420 e-01yn

+ 4.2207042568047446 e-03∆tfn−3 + 1.2935818910074356 e-02∆tfn−2

+ 1.3058278778359050 e-02∆tfn−1 + 2.8138644992434614 e-01∆tfn

+ 5.1306660081829913 e-01Y4 + 3.2832456220115930 e-01∆tF4.

5.3 Sixth-order methods Huang [15] introduced the five-step SSP hy-brid method of order six, called HM56, with good cHM = 0.209 and cHM,eff =0.104.



Y2 = 4.7770452064870433 e-02yn−2 + 5.1822440739673215 e-01yn−1

+ 4.3400514053839745 e-01yn + 3.0925041808997500 e-01∆tfn−1

+ 6.0585420298018144 e-01∆tfn,

Y3 = 1.7881046936624978 e-01yn−2 + 3.1435205819021794 e-01yn−1

+ 4.3625387781624295 e-02∆tfn−2 + 4.3882317945301935 e-01∆tfn−1

+ 5.0683747244353228 e-01Y2 + 7.0752528996969044 e-01∆tF2,

Y4 = 2.9550265031416362 e-02yn−2 + 7.2551231803535421 e-01yn

+ 1.6050124780249204 e-02∆tfn−2 + 3.2606121209759092 e-01∆tfn

+ 2.4493741693322948 e-01Y3 + 3.4192305494818703 e-01∆tF3,

yn+1 = 1.2753583942514621 e-02yn−2 + 2.8933666952256465 e-01yn−1

+ 3.7340911321864906 e-01yn + 1.2858329242989275 e-01∆tfn−1

+ 4.7739129368719369 e-01∆tfn + 1.1871654869920525 e-04Y2

+ 3.2438191676757244 e-01Y4 + 4.5282446977609897 e-01∆tF4.


Y2 = 3.1178857109809961 e-02yn−3 + 4.4689624054579768 e-01yn−1

+ 5.2192490234439237 e-01yn + 1.6598587642875223 e-02∆tfn−3


+ 2.8486592470334560 e-01∆tfn−1 + 4.8017895283881568 e-01∆tfn,

Y3 = 5.1959540622281350 e-02yn−3 + 3.5941507823114022 e-01yn−1

+ 3.0209737556498341 e-02∆tfn−3 + 3.3066741043451991 e-01∆tfn−1

+ 5.8862538114657859 e-01Y2 + 5.4154442116810342 e-01∆tF2,

Y4 = 8.0231772899364701 e-03yn−3 + 1.3131587577573756 e-01yn−1

+ 2.5397806261967282 e-01yn + 1.1796160318280763 e-03∆tfn−3

+ 1.2081262924584257 e-01∆tfn−1 + 6.0668288431465323 e-01Y3

+ 5.5815760234260159 e-01∆tF3

yn+1 = 1.3097096111310189 e-02yn−3 + 7.4923370275830570 e-02yn−2

+ 5.5794585649361489 e-02yn−1 + 4.5776927884707702 e-01yn

+ 6.8930655196950130 e-02∆tfn−2 + 5.1331878572652812 e-02∆tfn−1

+ 4.2115479060534788 e-01∆tfn + 3.9841566911642062 e-01Y4

+ 3.6654855503457545 e-01∆tF4.


Y2 = 4.6120419425091331 e-03yn−4 + 3.8804008423349307 e-01yn−1

+ 6.0734787382399780 e-01yn + 3.0712958274783479 e-03∆tfn−4

+ 1.8453264917414514 e-01∆tfn−1 + 4.8003003326508370 e-01∆tfn,

Y3 = 1.3742255978927375 e-02yn−4 + 1.0047901584430190 e-02yn−3

+ 3.3328095723617213 e-01yn−1 + 1.0861478040859079 e-02∆tfn−4

+ 7.9415681517904582 e-03∆tfn−3 + 2.6341554137894352 e-01∆tfn−1

+ 6.4292888520047009 e-01Y2 + 5.0815222618084077 e-01∆tF2,

Y4 = 2.5677299858163655 e-03yn−4 + 5.9479920528846368 e-03yn−3

+ 1.9228455092468361 e-01yn−1 + 9.6050754043419492 e-02yn

+ 4.7011193190314403 e-03∆tfn−3 + 1.5197609698636338 e-01∆tfn−1

+ 7.0314897299319612 e-01Y3 + 5.5574842597382068 e-01∆tF3,

yn+1 = 2.1050616474468436 e-03yn−4 + 1.5128610436953198 e-03yn−3

+ 7.1375068260431468 e-02yn−2 + 5.0395888964586244 e-01yn

+ 1.1957212141998266 e-03∆tfn−3 + 5.6412770782632737 e-02∆tfn−2


+ 3.9831439770717336 e-01∆tfn + 4.2104811940256387 e-01Y4

+ 3.3278414476113632 e-01∆tF4.


Y2 = 3.0226059771207344 e-03yn−5 + 8.7545896410865780 e-03yn−4

+ 7.8435884898216568 e-02yn−2 + 2.9916309843217620 e-01yn−1

+ 6.1062382105139978 e-01yn + 2.2866986950151632 e-03∆tfn−5

+ 6.6231288031580260 e-03∆tfn−4 + 5.9339271144419857 e-02∆tfn−2

+ 2.2632651161273898 e-01∆tfn−1 + 4.6195657168437948 e-01∆tfn,

Y3 = 9.9896934435634438 e-03yn−4 + 1.1801509303913174 e-02yn−3

+ 3.1510376244531735 e-01yn−1 + 3.5392632861649552 e-13yn

+ 7.5575245777678842 e-03∆tfn−4

+ 8.9282215838712414 e-03∆tfn−3 + 2.3838613694016850 e-01∆tfn−1

+ 6.6310503480685223 e-01Y2 + 5.0166029883763630 e-01∆tF2,

Y4 = 7.9297252475052740 e-04yn−5 + 3.8136999490602317 e-03yn−4

+ 4.6610951552942086 e-03yn−3 + 2.4540415971449447 e-01yn−1

+ 2.8851867437259052 e-03∆tfn−4 + 3.5262684880633466 e-03∆tfn−3

+ 1.8565614440588796 e-01∆tfn−1 + 7.4532807265640044 e-01Y3

+ 5.6386467306766652 e-01∆tF3,

yn+1 = 5.5222989416631608 e-04yn−5 + 5.4233441687091179 e-03yn−3

+ 5.8183634765173811 e-02yn−2 + 5.0182884439958442 e-01yn

+ 4.1029343973635205 e-03∆tfn−3 + 4.4017792162080203 e-02∆tfn−2

+ 3.7964967061390170 e-01∆tfn + 4.3401194677236649 e-01Y4

+ 3.2834400507960038 e-01∆tF4.

5.4 Seventh-order methods Huang’s investigation of hybrid methods[15] shows that a seven-step SSP hybrid method of order 7, called HM77,exists with cHM = 0.234 and cHM,eff = 0.117.

We present SSP HB methods of order 7 with 4, 5, 6 and 7 steps.



Y2 = 1.4181171301608114 e-01yn−3 + 2.6746986944809459 e-01yn−2

+ 8.8729483281577440 e-02yn−1 + 5.0198893425424684 e-01yn

+ 4.7424507571486046 e-01∆tfn−2 + 1.5732433938761134 e-01∆tfn−1

+ 8.9006578806300507 e-01∆tfn,

Y3 = 5.6455583476077723 e-02yn−3 + 1.7475976037994695 e-01yn−2

+ 3.7086510533962463 e-01yn−1 + 1.8183631650368465 e-01yn

+ 4.7513634361493660 e-02∆tfn−3 + 6.5757294578528802 e-01∆tfn−1

+ 3.2241006385482263 e-01∆tfn + 2.1608323430066603 e-01Y2

+ 3.8313253759417537 e-01∆tF2,

Y4 = 2.4221142166745917 e-02yn−3 + 4.8697070969724393 e-02yn−2

+ 1.9256679584245262 e-12yn−1 + 7.1071176389436097 e-01yn

+ 8.6343729694796853 e-02∆tfn−2 + 3.4143798899322064 e-12∆tfn−1

+ 3.6397677596282374 e-01∆tfn + 1.3133260011104431 e-02Y2

+ 2.0323676295613866 e-01Y3 + 3.6035473541396956 e-01∆tF3,

yn+1 = 3.3921902937070918 e-02yn−3 + 2.5714116547015001 e-02yn−2

+ 3.9501177301951948 e-02yn−1 + 5.3391509806078963 e-01yn

+ 1.3755767756843626 e-02∆tfn−3 + 4.5593147272786197 e-02∆tfn−2

+ 7.0038688316725384 e-02∆tfn−1 + 3.0756492028491300 e-01∆tfn

+ 8.6160403455941698 e-02Y2 + 2.8078730169723076 e-01Y4

+ 4.9785792855077493 e-01∆tF4.


Y2 = 2.3982101438859836 e-02yn−4 + 8.7678139889331416 e-02yn−3

+ 5.3701393015570177 e-02yn−2 + 3.8841208288909290 e-01yn−1

+ 4.4622628276714571 e-01yn + 9.9887959665517417 e-02∆tfn−3

+ 6.1179703245210756 e-02∆tfn−2 + 4.3486379190199109 e-01∆tfn−1

+ 5.0836654371316070 e-01∆tfn,


Y3 = 1.1796058355794571 e-02yn−4 + 8.3827812174208732 e-02yn−3

+ 4.0125168106181686 e-01yn−1 + 7.7216192941462425 e-02∆tfn−3

+ 4.5712890104892312 e-01∆tfn−1 + 5.0312444840817983 e-01Y2

+ 5.7318819346265659 e-01∆tF2,

Y4 = 1.4755308341935032 e-02yn−4 + 6.1776665970132170 e-02yn−3

+ 3.0637732023705277 e-01yn−1 + 1.5743252991886703 e-01yn

+ 7.0379516792679092 e-02∆tfn−3 + 3.4904259425320944 e-01∆tfn−1

+ 1.7935615671620367 e-01∆tfn + 4.5965817553201299 e-01Y3

+ 5.2366892540627263 e-01∆tF3,

yn+1 = 1.1895330671295582 e-02yn−4 + 7.4076433114412518 e-02yn−2

+ 1.2066305045361436 e-02yn−1 + 5.4016773987737765 e-01yn

+ 6.0405726702878262 e-03∆tfn−4 + 8.4392116124240218 e-02∆tfn−2

+ 1.3746625934402956 e-02∆tfn−1 + 3.9593568274754259 e-01∆tfn

+ 3.6179419129155271 e-01Y4 + 4.1217666835272870 e-01∆tF4.


Y2 = 5.6855872260172732 e-03yn−5 + 7.9297284979339799 e-02yn−3

+ 4.3500789258590154 e-02yn−2 + 3.2326556643968146 e-01yn−1

+ 5.4825077209637141 e-01yn + 7.7458278661341431 e-02∆tfn−3

+ 4.2491949847438738 e-02∆tfn−2 + 3.1576862099913094 e-01∆tfn−1

+ 5.3553612954593099 e-01∆tfn,

Y3 = 2.0822724469317017 e-03yn−5 + 3.2215004365335170 e-02yn−4

+ 9.2042336478407524 e-03yn−3 + 4.2684197678291180 e-01yn−1

+ 3.1467896862504119 e-02∆tfn−4 + 8.9907755977306839 e-03∆tfn−3

+ 4.1694296079144122 e-01∆tfn−1 + 5.2965651275698045 e-01Y2

+ 5.1737309506389240 e-01∆tF2,

Y4 = 3.3380614950874847 e-03yn−5 + 5.0440369871285376 e-02yn−3

+ 2.5660915848408855 e-01yn−1 + 1.8461916259929675 e-01yn

+ 4.9270592634906657 e-02∆tfn−3 + 2.5065806111888317 e-01∆tfn−1


+ 1.8033760609289001 e-01∆tfn + 5.0499324755024178 e-01Y3

+ 4.9328180278850281 e-01∆tF3,

yn+1 = 2.9026243385924483 e-03yn−5 + 4.0701406471331039 e-03yn−3

+ 4.9510098281349141 e-02yn−2 + 5.2856101283494494 e-01yn

+ 2.0405771171904080 e-03∆tfn−5 + 3.9757488357719736 e-03∆tfn−3

+ 4.8361895243025151 e-02∆tfn−2 + 3.3011253799304868 e-01∆tfn

+ 8.4004096111150329 e-03Y2 + 4.0655571428686538 e-01Y4

+ 8.2055932772060330 e-03∆tF2 + 3.9712716288833166 e-01∆tF4.


Y2 = 1.4810747800482122 e-03yn−6 + 6.2117213460275887 e-04yn−4

+ 7.4054828988991600 e-02yn−3 + 4.1606073711662151 e-03yn−2

+ 3.7182580251349062 e-01yn−1 + 5.4785651421170045 e-01yn

+ 5.4072108500486532 e-04∆tfn−4 + 6.4463624895834862 e-02∆tfn−3

+ 3.6217467054521736 e-03∆tfn−2 + 3.2366881926613467 e-01∆tfn−1

+ 4.7690093017610774 e-01∆tfn,

Y3 = 7.8285481545311750 e-05yn−6 + 2.8227209326948936 e-02yn−4

+ 4.0839155782542635 e-01yn−1 + 2.4571365010904813 e-02∆tfn−4

+ 3.5549876427636362 e-01∆tfn−1 + 5.6330294736607944 e-01Y2

+ 4.9034682001795038 e-01∆tF2,

Y4 = 7.1704107456275394 e-04yn−6 + 6.1235432988951365 e-03yn−4

+ 2.0048927610024553 e-02yn−3 + 2.7641589466966882 e-01yn−1

+ 5.3304531742563235 e-03∆tfn−4 + 1.7452292668294348 e-02∆tfn−3

+ 2.4061591651073727 e-01∆tfn−1 + 6.9669459334684869 e-01Y3

+ 6.0646225972844570 e-01∆tF3,

yn+1 = 1.4527906634020011 e-03yn−6 + 1.8351723030834131 e-02yn−3

+ 4.7521482658094340 e-02yn−2 + 6.8090256545031591 e-02yn−1

+ 4.2153263269698360 e-01yn + 1.1542564365902303 e-03∆tfn−6

+ 1.5974901377839960 e-02∆tfn−3 + 4.1366742377066279 e-02∆tfn−2


+ 5.9271553481442418 e-02∆tfn−1 + 3.6693787409287626 e-01∆tfn

+ 3.2143791984829299 e-02Y2 + 4.1090732242082512 e-01Y4

+ 2.7980690748119430 e-02∆tF2 + 3.5768869986082236 e-01∆tF4.

5.5 Eighth-order methods We present SSP HB methods of order 8 with5, 6 and 7 steps.


Y2 = 1.1240806184358765 e-01yn−4 + 2.3425521053653969 e-01yn−2

+ 2.2640298225573424 e-01yn−1 + 4.2693374536413853 e-01yn

+ 6.5470343019684962 e-02∆tfn−4 + 4.7749624431538534 e-01∆tfn−2

+ 1.6853188701135904 e-01∆tfn−1 + 8.7024429260701008 e-01∆tfn,

Y3 = 7.6866971450876870 e-02yn−4 + 1.9188166805763143 e-01yn−3

+ 3.8225805090004583 e-01yn−1 + 1.4490725338067179 e-01yn

+ 3.0082537934918074 e-01∆tfn−3 + 7.7917918344710779 e-01∆tfn−1

+ 2.9537302117997444 e-01∆tfn + 2.0408605621077405 e-01Y2

+ 4.1600067351578807 e-01∆tF2,

Y4 = 1.3379640088295723 e-02yn−4 + 1.5102357622019259 e-02yn−3

+ 4.9961391525236702 e-02yn−2 + 6.1124255630347346 e-03yn−1

+ 7.1696847190691049 e-01yn + 8.4163142856899329 e-03∆tfn−4

+ 1.0183925795900226 e-01∆tfn−2 + 1.2459318378965911 e-02∆tfn−1

+ 3.4874152013934234 e-01∆tfn + 1.9847571329450317 e-01Y3

+ 4.0456477987779804 e-01∆tF3,

yn+1 = 5.6758862391967811 e-03yn−4 + 1.6498330025689711 e-01yn−3

+ 1.2555868880225157 e-01yn−2 + 1.7392006935005835 e-01yn−1

+ 2.9860166752988093 e-01yn + 6.8744743680103665 e-02∆tfn−3

+ 2.5593369815305611 e-01∆tfn−2 + 3.5451155914745591 e-01∆tfn−1

+ 6.0865743163304931 e-01∆tfn + 2.3126038782171532 e-01Y4

+ 4.7139172012809527 e-01∆tF4.



Y2 = 4.6472690717261362 e-03yn−5 + 1.7617740666490314 e-02yn−2

+ 5.1087509618159122 e-01yn−1 + 4.6685989408019241 e-01yn

+ 4.0074586314538660 e-03∆tfn−5 + 2.4400693250375298 e-02∆tfn−2

+ 2.2519192095834503 e-01∆tfn−1 + 6.4660419755303700 e-01∆tfn,

Y3 = 6.3437465306319159 e-03yn−5 + 3.8280022711921196 e-03yn−4

+ 1.3378083962303952 e-01yn−3 + 4.0276427099509055 e-01yn−1

+ 1.0731673311116895 e-01yn + 5.3018097467378913 e-03∆tfn−4

+ 1.1632225850184115 e-01∆tfn−3 + 5.5783131417384202 e-01∆tfn−1

+ 1.4863442111273212 e-01∆tfn + 3.4596640746887680 e-01Y2

+ 4.7916587849650372 e-01∆tF2,

Y4 = 2.4354830266242293 e-02yn−5 + 1.0614148803253295 e-01yn−3

+ 3.0848403901567312 e-01yn−1 + 2.1975355520489917 e-01yn

+ 1.2996464181697261 e-02∆tfn−5 + 1.4700669851193610 e-01∆tfn−3

+ 4.2725253772042082 e-01∆tfn−1 + 3.0436020104627681 e-01∆tfn

+ 3.4126608748065268 e-01Y3 + 4.7265590264986124 e-01∆tF3,

yn+1 = 7.0409864973861782 e-03yn−5 + 3.1910209804637511 e-02yn−4

+ 1.1534180942145611 e-01yn−2 + 7.1301418909879871 e-02yn−1

+ 4.6441610170111464 e-01yn + 3.5260849749088656 e-02∆tfn−4

+ 1.5974920756946681 e-01∆tfn−2 + 9.8752960670244611 e-02∆tfn−1

+ 4.9054264305477624 e-01∆tfn + 3.0998947366552543 e-01Y4

+ 4.2933757517186882 e-01∆tF4.


Y2 = 1.1033483152094540 e-02yn−6 + 2.1618741753910483 e-02yn−4

+ 7.3164773446182038 e-02yn−3 + 8.9377315222374923 e-02yn−2

+ 3.3052209858727233 e-01yn−1 + 4.7428358783816565 e-01yn


+ 7.9271467501655368 e-03∆tfn−6 + 2.5357039226603571 e-02∆tfn−4

+ 8.5816374116448504 e-02∆tfn−3 + 1.0483237710410760 e-01∆tfn−2

+ 3.8767574517240444 e-01∆tfn−1 + 5.5629636906003022 e-01∆tfn,

Y3 = 4.1673572573951061 e-03yn−6 + 4.4113926716232821 e-02yn−4

+ 4.0105444119401153 e-02yn−3 + 4.2687049006597683 e-01yn−1

+ 5.1742075598859100 e-02∆tfn−4 + 4.7040449037793017 e-02∆tfn−3

+ 5.0068463208894975 e-01∆tfn−1 + 4.8474278184099362 e-01Y2

+ 5.6856415946279171 e-01∆tF2,

Y4 = 6.7610940286408298 e-03yn−6 + 1.1290295300529569 e-02yn−4

+ 6.5520108757853551 e-02yn−3 + 2.9888049430192454 e-01yn−1

+ 1.8498138261618360 e-01yn + 4.7303133989144205 e-03∆tfn−6

+ 1.3242605146698148 e-02∆tfn−4 + 7.6849799438662283 e-02∆tfn−3

+ 3.5056269714262467 e-01∆tfn−1 + 2.1696823194354331 e-01∆tfn

+ 4.3256662499486692 e-01Y3 + 5.0736573862494427 e-01∆tF3,

yn+1 = 2.5881604396226902 e-03yn−6 + 7.6357016234845469 e-03yn−5

+ 1.6304328340768028 e-02yn−4 + 1.2567327447111440 e-01yn−2

+ 7.7112740468168656 e-02yn−1 + 4.3369750806348006 e-01yn

+ 8.9560617261325842 e-03∆tfn−5 + 1.9123661219807291 e-02∆tfn−4

+ 1.4740460785250881 e-01∆tfn−2 + 9.0447020792438215 e-02∆tfn−1

+ 5.0648288653910711 e-01∆tfn + 3.3698828659336139 e-01Y4

+ 3.9526006181688789 e-01∆tF4.

6 Numerical results We numerically compare our new methods withthe following SSP methods:

• HM of Huang [15],

• RK104 of Ketcheson [18],

• RK54 of Spiteri and Ruuth [40],

• RK105 of Ruuth [29],

• GL24, GL34 and GL44 of Constantinescu and Sandu [3, D.3, D.9, D.17].

and the non-SSP classic 4-stage RK44 of order 4.


6.1 Percentage efficiency gain

Definition 2. The percentage efficiency gain (PEG) of the SSP coefficientsCm2,eff of method 2 over Cm1,eff of method 1 is

(29) PEG(Cm2,eff, Cm1,eff) =Cm2,eff − Cm1,eff

Cm1,eff.

Table 1 lists cHB, cHB,eff , C, and Ceff of HBkp and known methods of orderp in columns 3, 4, 6, and 7, respectively. Column 8 lists PEG(cHB,eff , Ceff) ofHB methods over other known methods. It is seen that the PEG(cHB,eff , Ceff)is not negligible. For HBkp and HMkp, it increases as both k and p increasesimultaneously, but decreases as k increases and p remains constant. In otherwords, the PEG(cHB,eff , Ceff) of the new HB methods over [27] Huang’s hybridmethods is generally larger when k is smaller.

Table 1 also shows that the new methods are generally competitive withknown methods. As a first example, HB54 has a larger cHB,eff than everyk-step method of order 4 on hand including the seven-step SSP hybrid HM74with large Ceff = 0.469. As a second example, these new methods are compet-itive with fifth-order hybrid methods given by Huang [15] based on the samenumber of steps, and with fifth-order RK methods with ten stages given byRuuth [29]. This ten-stage method has Ceff = 3.395/10 ≈ 0.339, which is lessthan cHB,eff = 0.390 of HB55.

6.2 Validating the order preservation property To illustrate theboundary/source order reduction phenomenon we consider a classic test prob-lem with a nonlinear source described in [34]:

(30)∂

∂tu(x, t) = −

∂

∂xu(x, t) + b(x, t),

over 0 ≤ x ≤ 1 and 0 ≤ t ≤ 1 with initial condition u(x, 0) = 1 + x, boundaryconditions u(0, t) = 1/(1 + t) and source term b(x, t) = (t − x)/(1 + t)2. Theexact solution, u(x, t) = (1 + x)/(1 + t), is linear in space, allowing the useof first-order upwind space discretization without introducing discretizationerrors. For the time integration, the SSP 5-stage RK method of order 4and the classic RK method of order 4 are used. All considered explicit RKmethods have stage order equal to one. Sanz-Serna et al. [34] show thatexplicit RK methods with p ≥ 3 suffer from order reduction on problems withnonhomogeneous boundary conditions or nonzero source terms such as (30).

For the test problem (30), we distinguish two cases, one that illustrates theorder reduction phenomenon and, for validation purposes, one that does not.Specifically, if the spatial and temporal grids are refined simultaneously, onenotices that low stage-order methods suffer from order reduction [34]. If the


p Meth. cHB cHB,eff Meth. C Ceff PEG4 HB24 1.593 0.398 GL24 1.59 0.398 0 %

HB34 1.843 0.461 HM34 0.494 0.247 87 %” GL34 1.84 0.461 0 %

HB44 1.932 0.483 HM44 0.682 0.341 42 %” GL44 1.93 0.483 0 %

HB54 1.979 0.494 HM54 0.793 0.396 25 %” HM74 0.938 0.469 5 %” RK104 6.000 0.600 -18 %” RK54 1.508 0.302 64 %

5 HB45 1.537 0.384 HM45 0.371 0.185 108 %HB55 1.562 0.390 HM55 0.525 0.262 49 %

” RK105 3.395 0.339 15 %HB65 1.569 0.392 HM65 0.657 0.328 20 %

6 HB56 1.265 0.316 HM56 0.209 0.104 202 %HB66 1.321 0.330 HM66 0.362 0.181 82 %

7 HB77 1.148 0.287 HM77 0.234 0.117 145 %

TABLE 1: PEG(cHB,eff , Ceff) for k-step HBkp, HMkp, LMkp and GLkpand k-stage RKkp, all of order p. Comparison is row-wise.

space grid is kept fixed, that is, the ODE problem is fixed, then the (classic)order of consistency is preserved.

In the cases of RK44 and RK54, Table 2 shows the discretization errorversus the time step without order reduction (when ∆x = 1/10) and withorder reduction (when ∆x = 1/20). In the former case, the order of the RKmethods is preserved (if ∆x is maintained fixed at ∆x = 1/10), whereas, inthe later case, the order clearly drops for all RK methods. A special bound-ary/source treatment can be used to alleviate this problem, but with greateffort and limited success [1, 33, 34]. This discussion also applies to implicitRK methods with low stage orders such as DIRK [26].

The HBkp methods listed in Table 2 maintain well their consistency orderscompared to RK44 and RK54.

6.3 Comparing SSP HB with other methods on Burgers’ equa-tion with a unit-step initial condition As a first comparison of ournew methods with RK methods, following Huang [15], we consider Burgers’equation in Problem 1.

Problem 1. Burgers’ equation with a unit-step initial condition:


HB45 HB46 HB55

∆t\∆x 1/10 1/20 1/10 1/20 1/10 1/20

1/20 2.04e-8 2.33e-8 1.46e-9 1.90e-9 2.08e-8 2.25e-81/40 1.29e-9 1.60e-9 4.85e-11 6.36e-11 1.32e-9 1.62e-91/80 5.13e-11 7.41e-11 1.01e-12 1.53e-12 5.26e-11 7.49e-11

HB56 HB66 HB57

∆t\∆x 1/10 1/20 1/10 1/20 1/10 1/20

1/20 2.19e-9 2.81e-9 3.07e-9 3.86e-9 2.41e-10 5.98e-101/40 6.85e-11 8.98e-11 9.80e-11 1.25e-10 4.44e-12 5.78e-121/80 1.54e-12 2.13e-12 2.11e-12 2.97e-12 5.21e-14 7.66e-14

RK44 RK54

∆t\∆x 1/10 1/20 1/10 1/20

1/20 2.62e-6 1.63e-5 1.14e-6 5.89e-61/40 1.27e-7 6.55e-7 6.08e-8 2.89e-71/80 6.91e-9 3.24e-8 3.47e-9 1.56e-8

TABLE 2: L∞-error at t = 1 for the listed methods applied to problem(30).

(31)∂

∂tu(x, t) +

∂

∂x

»

1

2u(x, t)2

–

= 0, u(x, 0) =

(

1, −1 ≤ x < 0,

0, 0 < x ≤ 1.

and boundary condition u(−1, t) = 1 for t ≥ 0.

We discretize the spatial derivative by the difference quotient

(32)1

∆x

»

1

2(uj(t))

2 −1

2(uj−1(t))

2

–

with space stepsize ∆x = 1/75, where uj(t) is an approximation to u(xj , t)with xj = j∆x, j = . . . ,−2,−1, 0, 1, 2, . . .. This leads to the semi-discretesystem

d

dtuj(t) = −

1

∆x

»

1

2(uj(t))

2 −1

2(uj−1(t))

2

–

,

to which a time discretization can be applied.

We consider the total variation norm of the numerical solution at tfinal =1.8. For this purpose, we let numeff be the largest effective CFL number

defined as

(33) numeff = max∆t

∆t

∆x

1

l

ff

,


such that the TV error in the numerical solution satisfies the inequality

(34) |TV(u(x, tfinal)) − TV(u(x, 0))| ≤ 5.0 e-02,

and we let max ∆tnum = l∆x numeff be the maximum numerical step size.Here l is the number of function evaluations per step of the method on hand.We note that inequality (34) is used because tfinal is small.

Finally, we let max∆ttheor of SSP HBs for Problem 1 be taken as

(35) max∆ttheor = cHB ∆tFE,

where cHB is the SSP coefficient of SSP HB methods.

In Table 3, we list the maximum effective CFL number, numeff, and theratio max∆tnum/ max ∆ttheor of SSP HB for Problem 1 and Problem 2 below.It is seen that:

• Generally, the k-step HB methods of orders 4 to 7 have higher numeff thank-step hybrid methods, for the same k,

• An increase in the step number k improves the numeff similar to a simul-taneous increase of the number of stages and orders.

• HB24, HB34 and HB44 behave almost like GL24, GL34 and GL44, respec-tively, since their coefficients are almost identical.

Definition 3. The percentage efficiency gain of numeff for method 2 overmethod 1 is

(36) PEG(numeff) =numeff (method 2) − numeff (method 1)

numeff (method 1).

The numeff of HB methods and hybrid methods applied to Problem 1are listed in columns 3 and 5 of Table 4, respectively. Column 6 lists thePEG(numeff) of HB methods over hybrid and other known methods. ForProblem 1, it is seen that

• In Tables 4 and 3, numeff of HB54, HB45, HB66 and HB77 compare favor-ably with numeff of most of the other methods of the same order.

• PEG(numeff) decreases as the step number k increases. On the other hand,a simultaneous increase of k and p improves PEG(numeff). In other words,the PEG(numeff) of the new methods over Huang’s hybrid methods is gen-erally larger when k is smaller.


Problem 1 Problem 2

Method numeff Rnum/theor numeff Rnum/theor

HB24 0.516 1.111 0.540 1.141HB34 0.572 1.065 0.594 1.085HB44 0.610 1.083 0.634 1.105HB54 0.636 1.102 0.664 1.130HM34 0.302 1.049 0.310 1.056HM44 0.408 1.026 0.406 1.002HM54 0.486 1.051 0.470 0.998RK44 0.414 2.130 0.348 1.758RK54 0.496 1.410 0.442 1.234RK104 0.662 0.946 0.618 0.867GL24 0.516 1.111 0.540 1.141GL34 0.572 1.065 0.594 1.085GL44 0.610 1.083 0.634 1.105

HB25 0.385 1.547 0.369 1.455HB35 0.496 1.246 0.504 1.242HB45 0.538 1.201 0.538 1.179HB55 0.516 1.133 0.532 1.147HM45 0.342 1.581 0.294 1.334HM55 0.430 1.405 0.382 1.225

HB36 0.325 1.557 0.325 1.528HB46 0.378 1.194 0.394 1.221HB56 0.396 1.074 0.412 1.097HB66 0.422 1.096 0.430 1.096HM56 0.292 2.396 0.252 2.030HM66 0.258 1.222 0.274 1.274

HB47 0.316 1.922 0.316 1.886HB57 0.368 1.439 0.384 1.474HB67 0.408 1.368 0.424 1.395HB77 0.410 1.225 0.424 1.244HM77 0.208 1.525 0.232 1.669

HB58 0.244 1.708 0.260 1.787HB68 0.294 1.397 0.310 1.446HB78 0.310 1.248 0.326 1.288

FE 1.166 1.000 1.188 1.000RK33 0.499 1.281 0.419 1.058RK53 0.754 1.220 0.582 0.924

TABLE 3: numeff and ratio Rnum/theor = max ∆tnum/ max ∆ttheor ofSSP HB for Problems 1 and 2.


p Meth. numeff Meth. numeff PEG(numeff)

4 HB24 0.517 GL24 0.517 0 %HB34 0.572 HM34 0.302 89 %

” GL34 0.572 0 %HB44 0.610 HM44 0.408 50 %

” GL44 0.610 0 %HB54 0.636 HM54 0.486 31 %HB54 0.636 RK104 0.662 -4 %

” RK54 0.496 28 %” RK44 0.414 54 %

5 HB25 0.385HB35 0.496HB45 0.538 HM45 0.342 57 %HB55 0.516 HM55 0.430 20 %

6 HB36 0.325HB46 0.378HB56 0.396 HM56 0.292 36 %HB66 0.422 HM66 0.258 64 %

7 HB47 0.316HB57 0.368HB67 0.408HB77 0.410 HM77 0.208 97 %

8 HB58 0.244HB68 0.294HB78 0.310

TABLE 4: Percentage efficiency gain of numeff for HBkp over HMkpand RKsp applied to Problem 1.

6.4 Comparing SSP HBs and other methods on Burgers’ equa-tion with a square-wave initial condition As a second comparison,we consider Burgers’ equation with a square-wave initial value in Problem 2,which is one of Laney’s five test problems [23].

Problem 2. Burgers’ equation with a square wave initial condition:

(37)∂

∂tu(x, t) +

∂

∂x

»

1

2u(x, t)2

–

= 0, u(x, 0) =

(

1, |x| ≤ 13,

0, 13

< |x| ≤ 1.

and boundary condition u(−1, t) = 0 for t ≥ 0.

We discretize the spatial derivative of Problem 2 by the difference quotient(32) and compute the total variation of the numerical solution as a functionof the effective CFL number, ∆t/(`∆x), at tfinal = 0.6.


The numeff of HB methods and hybrid methods applied to Problem 2are listed in columns 3 and 5 of Table 5, respectively. Column 6 lists thePEG(numeff) of HB methods over hybrid methods and other known methods.

It is seen that

• PEG(numeff) decreases as k increases but a simultaneous increase of k and pimproves PEG(numeff). In other words, Problem 2 confirms again that thePEG(numeff) of the new methods over Huang’s hybrid methods is generallylarger when k is smaller.

• In Table 5 and in the last two columns of Table 3 for Problem 2, numeff ofHB54 compares favorably with numeff of other methods of the same order,including RK104,

• numeff of HB45, HB66 and HB77 compare favorably with numeff of othermethods of the same order.

p Meth. numeff Meth. numeff PEG(numeff)

4 HB24 0.540 GL24 0.540 0 %HB34 0.594 HM34 0.310 92 %

” GL34 0.594 0 %HB44 0.634 HM44 0.406 56 %

” GL44 0.634 0 %HB54 0.664 HM54 0.470 41 %

” RK104 0.618 7 %” RK54 0.442 50 %” RK44 0.348 91 %

5 HB25 0.369HB35 0.504HB45 0.538 HM45 0.294 83 %HB55 0.532 HM55 0.382 39 %

6 HB36 0.325HB46 0.394HB56 0.412 HM56 0.252 63 %

HB66 0.430 HM66 0.274 57 %

7 HB47 0.316HB57 0.384HB67 0.424HB77 0.424 HM77 0.232 83 %

8 HB58 0.260HB68 0.310HB78 0.326

TABLE 5: Percentage efficiency gain of numeff of HBkp over HMkp andother known methods for Problem 2.


We observe that, as with RK methods [15], the ratio max ∆tnum/max ∆ttheor

of SSP HB for Problems 1 and 2 are greater than 1. The theoretical strongstability bounds of SSP HB methods are then verified in the numerical compar-ison of maximum time steps for Problem 1 and confirmed again in Problem 2.

7 Conclusion A collection of new SSP explicit 4-stage k-step Hermite-Birkhoff methods, HBkp, of orders p = 4, . . . , 8 with nonnegative coefficientsare constructed as k-step analogues of fourth-order Runge-Kutta methods,incorporating function evaluations at three off-step points. In particular, nocounterparts of HB25, HB36 and HB47 have been found among hybrid andgeneral linear multistep methods in the literature. The new methods tend tohave larger effective SSP coefficients than hybrid methods [15] with the samenumber of steps and other frequently used methods. Most of the proposedgeneral linear methods can attain high-stage orders, a property that allevi-ates the order reduction phenomenon encountered in the classic explicit RKschemes due to nonhomogeneous boundary/source terms (see [3]). Similar to[15], finding more efficient generalized SSP methods appears to be promisingin the light of the present work.

REFERENCES

1. M. Carpenter, D. Gottlieb, S. Abarbanel and W.-S. Don, The theoretical accu-racy of Runge-Kutta time discretizations for the initial boundary value problem:A study of the boundary error, SIAM J. Sci. Comput. 16 (1995), 1241–1252.

2. B. Cockburn and C. W. Shu, TVB Runge-Kutta local projection discontinuousGalerkin finite element method for conservation laws II: General framework,Math. Comp. 52 (1989), 411–435.

3. E. M. Constantinescu and A. Sandu, Optimal explicit strong-stability-preservinggeneral linear methods: Complete results, Tech. Report ANL/MCS-TM-304,Argonne National Laboratory, Mathematics and Computer Science DivisionTechnical Memorandum, Jan. 2009.

4. L. Ferracina and M. N. Spijker, Stepsize restrictions for the total variationdiminishing property in general Runge-Kutta methods, SIAM J. Numer. Anal.42 (2004), 1073–1093.

5. L. Ferracina, M. N. Spijker, An extension and analysis of the Shu-Osher rep-resentation of Runge-Kutta method, Math. Comp. 74 (2005), 201–219.

6. S. Gottlieb, On high order strong stability preserving Runge-Kutta and multistep time discretizations, J. Sci. Comput. 25 (2005), 105–128.

7. S. Gottlieb, L. J. Gottlieb, Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coefficient operators, J.Sci. Comput. 18 (2003), 83–110.

8. S. Gottlieb, D. I. Ketcheson and C. W. Shu, High order strong stability preserv-ing time discretization, J. Sci. Comput. 38(3) (2009), 251–289, DOI: 10.1007/s10915-008-9239-z.


9. S. Gottlieb, D. I. Ketcheson and C. W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, World Scientific, Singapore, 2011.

10. S. Gottlieb and C. W. Shu, Total variation diminishing Runge-Kutta schemes,Math. Comp. 67 (1998), 73–85.

11. S. Gottlieb, C. W. Shu and E. Tadmor, Strong stability-preserving high-ordertime discretization methods, SIAM Rev. 43 (2001), 89–112.

12. A. Harten, High resolution schemes for hyperbolic conservation laws, J. Com-put. Phys. 49 (1983), 357–393.

13. I. Higueras, On strong stability preserving methods, J. Sci. Comput. 21 (2004),193–223.

14. I. Higueras, Representations of Runge-Kutta methods and strong stability pre-serving methods, SIAM J. Numer. Anal. 43 (2005), 924–948.

15. C. Huang, Strong stability preserving hybrid methods, Appl. Numer. Math. 59

(2009), 891–904.

16. W. Hundsdorfer, S. J. Ruuth and R. J. Spiteri, Monotonicity preserving linearmultistep methods, SIAM J. Numer. Anal. 41 (2003), 605–623.

17. W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advec-tion-Diffusion-Reaction Equations, Springer Series in Computational Mathe-matics, 33, Springer, Berlin, 2003.

18. D. I. Ketcheson, Highly efficient strong stability preserving Runge-Kutta meth-ods with low-storage implementations, SIAM J. Sci. Comput. 30(4), (2008),2113–2136.

19. D. I. Ketcheson, Computation of optimal monotonicity preserving general lin-ear methods, Math. Comp. 78 (2009), 1497–1513.

20. D. I. Ketcheson, C. B. Macdonald and S. Gottlieb, Optimal implicit strongstability preserving Runge-Kutta methods, Appl. Numer. Math. 59 (2009), 373–392.

21. J. F. B. M. Kraaijevanger, Absolute monotonicity of polynomials occurringin the numerical solution of initial value problems, Numer. Math. 48 (1986),303–322.

22. J. F. B. M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT 31

(1991), 482–528.

23. C. Laney, Computational Gasdynamics, Cambridge University Press, Cam-bridge, UK, 1998.

24. H.W.J. Lenferink, Contractivity preserving explicit linear multistep methods,Numer. Math. 55 (1989), 213–223.

25. H.W.J. Lenferink, Contractivity-preserving implicit linear multistep methods,Math. Comp. 56 (1991), 177–199.

26. C. Macdonald, S. Gottlieb and S. Ruuth, A numerical study of diagonallysplit Runge-Kutta methods for PDEs with discontinuities, J. Sci. Comput. 36

(2008), 89–112.

27. T. Nguyen-Ba, E. Kengne and R. Vaillancourt, One-step 4-stage Hermite-Birkhoff-Taylor ODE solver of order 12, Can. Appl. Math. Q. 16(1) (2008),77–94.

28. S. Osher and S. Chakravarthy, High resolution schemes and the entropy con-dition, SIAM J. Numer. Anal. 21 (1984), 955–984.

29. S. J. Ruuth, Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comp. 75 (2006), 183–207.

30. S. J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with gen-eral monotonicity and boundedness properties, J. Comput. Phys. 209 (2005),226–248.


31. S. J. Ruuth and R. J. Spiteri, Two barriers on strong-stability-preserving timediscretization methods, J. Sci. Comput. 17 (2002), 211–220.

32. S. J. Ruuth and R. J. Spiteri, High-order strong-stability-preserving Runge-Kutta methods with down-biased spatial discretizations, SIAM J. Numer. Anal.42 (2004), 974–996.

33. J. Sanz-Serna and J. Verwer, Stability and convergence at the PDE/stiff ODEinterface, Appl. Numer. Math. 5 (1989), 117–132.

34. J. Sanz-Serna, J. Verwer and W. Hundsdorfer, Convergence and order reductionof Runge-Kutta schemes applied to evolutionary problems in partial differentialequations, Numer. Math. 50 (1987), 405–418.

35. C. W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci.Statist. Comput. 9 (1988), 1073–1084.

36. C. W. Shu, A survey of strong stability preserving high order time discretiza-tion, in: D. Estep, S. Tavener (Eds.), Collected Lectures on the Preservationof Stability under Discretization, SIAM, 2002.

37. C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatoryshock-capturing schemes, J. Comput. Phys. 77 (1988), 439–471.

38. M. N. Spijker, Contractivity in the numerical solution of initial value problems,Numer. Math. 42 (1983), 271–290.

39. M. N. Spijker, Stepsize conditions for general monotonicity in numerical initialvalue problems, SIAM J. Numer. Anal. 45 (2007), 1226–1245.

40. R. J. Spiteri and S. J. Ruuth, A new class of optimal high-order strong-stability-preserving time-stepping schemes, SIAM J. Numer. Anal. 40 (2002), 469–491.

41. R. J. Spiteri and S. J. Ruuth, Nonlinear evolution using optimal fourth-orderstrong-stability-preserving Runge-Kutta methods, J. Math. Comput. Simul. 62

(2003), 125–135.

42. P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conser-vation laws, SIAM J. Numer. Anal. 21 (1984), 995–1011.

43. E. Tadmor, Approximate solutions of nonlinear conservation laws, in: A.Quarteroni (Ed.), Advanced Numerical Approximation of Nonlinear HyperbolicEquations, Lectures Notes from CIME Course, Cetraro, Italy, 1997, LecturesNotes in Math., 1697, Springer-Verlag, New York, 1998, pp. 1–150.

44. J. A. van de Griend and J. F. B. M. Kraaijevanger, Absolute monotonicity ofrational functions occurring in the numerical solution of initial value problems,Numer. Math. 49 (1986), 413–424.

Department of Mathematics and Statistics, University of Ottawa,

Ottawa, Ontario, Canada K1N 6N5.

E-mail address: [email protected]




Documents

STRONG-STABILITY-PRESERVING 4-STAGE HERMITE-BIRKHOFF TIME … · a higher-order time-discretization (RK, multistep or HB) method that maintains strong stability for the same norm,