On leap-frog-Chebyshev schemes

Marlis Hochbruck, Andreas Sturm

CRC Preprint 2018/17, August 2018

KARLSRUHE INSTITUTE OF TECHNOLOGY

KIT – The Research University in the Helmholtz Association www.kit.edu

Participating universities

Funded by

ISSN 2365-662X

2

ON LEAP-FROG-CHEBYSHEV SCHEMES∗

MARLIS HOCHBRUCK† AND ANDREAS STURM†

Abstract. This paper is dedicated to the improvement of the efficiency of the leap-frog methodfor second order differential equations. In numerous situations the strict CFL condition of the leap-frog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomialsnew methods have been constructed that exhibit a much weaker CFL condition than the leap-frogmethod. However, these methods do not even approximately conserve the energy of the exact solutionwhich can result in a bad approximation quality.

In this paper we propose two remedies to this drawback. For linear problems we show byusing energy techniques that damping the Chebyshev polynomial leads to approximations whichapproximately preserve a discrete energy norm over arbitrary long times. Moreover, with a completelydifferent approach based on generating functions, we propose to use special starting values thatconsiderably improve the stability. We show that the new schemes arising from these modificationsare of order two and can be modified to be of order four. These convergence results apply tosemilinear problems. Finally, we discuss the efficient implementation of the new schemes and givegeneralizations to fully nonlinear equations.

Key words. time integration, Hamiltonian systems, wave equation, 2nd order ode, leap-frogmethod, CFL condition, Chebyshev polynomials, stability analysis, error analysis, energy techniques,generating functions

AMS subject classifications. Primary: 65L04, 65L20. Secondary: 65L05, 65L06, 65L70

1. Introduction. In this paper we are concerned with the second order differ-ential equation

(1.1) q(t) = −Lq(t)− g(q(t)

), q(0) = q0, q(0) = q0,

with a symmetric and positive definite matrix L of large norm and a “nice” func-tion g. Such equations are used to model a plurality of phenomena. Among othersHamiltonian problems and (spatially discretized) wave-type problems are describedby (1.1).

The most natural approach to discretize (1.1) is to replace the second order timederivative by a centered second-order difference quotient — the well-known leap-frog (LF) scheme. Thanks to a variety of nice features such as symplecticity, time-reversibility [8] and an easy implementation the LF scheme serves as the standardtime integrator for problems of the type (1.1).

However, its efficiency can be severely limited by the time step size restriction(CFL condition) arising from the large norm of L. This forces a large number ofevaluations of the nonlinear function g. In many situation such an evaluation iscostly which renders the LF method prohibitively expensive.

The same issue arises for first order parabolic problems and explicit Runge–Kutta(RK) methods. In this setting Runge–Kutta–Chebyshev (RKC) methods [10, 16, 17,18] have been found a remedy. First order RKC methods are constructed by using ascaled and shifted Chebyshev polynomial as stability function. This choice maximizesthe stability region and thus alleviates the CFL condition compared to standard RK

∗Submitted to the editors August 2018Funding: We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft

(DFG) through CRC 1173.†Institute for Applied and Numerical Analysis, Karlsruhe Institute of Technology, 76128 Karlsru-

he, Germany, marlis.hochbruck@kit.edu, andreas.sturm@kit.edu

1

2 MARLIS HOCHBRUCK AND ANDREAS STURM

methods. Based on this idea higher order methods and further extensions, as, e.g.,the ROCK familiy [1, 2], have been proposed.

In [6, 12] the authors applied analogous ideas to the linear case (g ≡ 0) of (1.1)and the LF method. In accordance with RKC schemes the resulting methods maybe called leap-frog-Chebyshev (LFC) schemes. Unfortunately, these schemes sufferfrom stability problems. As numerical examples indicate they fail to reproduce thestability behavior of the exact solution. This can result in an energy drift and poorapproximation quality.

The aim of this paper is to improve the LFC methods such that they generateapproximations with the correct stability behavior and a good approximation grade.In fact, we propose the following two remedies to the aforementioned problem:

• Motivated by stabilized RKC methods [10, 16, 17, 18] we construct a dampedversion of the Chebyshev polynomial. Using an energy technique we showthat this modified scheme nearly conserves a discrete energy and leads touniformly bounded approximations over arbitrary long times.

• We replace the standard starting values based on a Taylor expansion of theexact solution by ones involving the Chebyshev polynomial and its derivative.With this modification we show via a generating functions technique thestability of the new scheme.

A main feature of the novel schemes emanating from these modifications is that theirCFL condition is alleviated by a factor p compared to the LF method, if p is thedegree of the Chebyshev polynomial.

Having these methods at hand, they can be combined, e.g., with the LF schemefor g to integrate the semilinear problem (1.1). As we will show in the course of thispaper this multi rate method can be employed with an (approximately) p times largertime step than the LF method. This renders the method considerably more efficientthan the LF scheme since it requires p times less evaluations of the nonlinearity g.

Our paper is organized as follows: In Section 2 we present a general two-steptime integration method for (1.1) which comprises among others the LF and the LFCscheme. Section 3 is dedicated to linear problems. Here, we discuss the stability ofthe general scheme for g ≡ 0. We derive conditions which guarantee the stability ofthe scheme both in the standard and in the energy norm. Moreover, we constructthe special starting value mentioned above and prove the stability of the resultingscheme. In Section 4 we present the error analysis. We show that the general schemeis of order two and can be adapted to converge with fourth order. Then, in Section 5we show that all required assumptions apply for the LFC methods. Subsequently, wediscuss in Section 6 the efficiency and the implementation of the LFC method andalso generalize it to fully nonlinear problems. We conclude our paper in Section 7with numerical examples. In particular, we look at the problems of the LFC methodsin [6, 12] and show that our modifications overcome them.

2. A general class of two-step schemes. The LF scheme for the semilinearproblem (1.1) is given by

qn+1 − 2qn + qn−1 = −τ2Lqn − τ2g(qn),(2.1a)

q0 = q0, q1 =(I− 1

2τ2L)q0 − 1

2τ2g(q0)

+ τ q0,(2.1b)

where τ > 0 is the time step size and qn approximates the exact solution q(tn) attime tn = nτ .

Our aim is to modify the “linear part” of the LF method such that the resultingscheme remains stable for larger time step sizes than the standard LF method (2.1).

ON LEAP-FROG-CHEBYSHEV SCHEMES 3

For this purpose we use a polynomial P satisfying

(2.2) P (0) = 0, P ′(0) = 1,

and propose the scheme

qn+1 − 2qn + qn−1 = −P (τ2L)qn − τ2gn(2.3a)

q0 = q0, q1 =(I− 1

2P (τ2L))q0 − 1

2τ2g0 + τP ′(τ2L)q0,(2.3b)

where gn is a suitable approximation to g(q(tn)). The obvious choice is to use gn =g(qn) but other choices are possible, e.g., filtered versions as in Gautschi-type methods

[8, 9]. For such methods one could set gn = P (τ2L)g(P (τ2L)qn

)with a suitable filter

function P . Then, the scheme could be interpreted as a particular implementation ofGautschi-type methods, where the trigonometric matrix functions are approximatedby special polynomials P and P , respectively.

In this paper we examine the general scheme (2.3a)–(2.3b) with particular atten-tion to the choice

(2.3c) P (z) = Pp(z) = 2− 2

Tp(νp)Tp

(νp −

z

αp

), αp = 2

T ′p(νp)

Tp(νp).

Here, Tp denotes the pth Chebyshev polynomial of first kind, so that Pp is a polynomialof degree p ≥ 1, and νp ≥ 1 is a damping parameter whose choice will be discussedlater.

We note that (2.3c) is motivated by the construction of RKC methods [10, 16, 17,18] and accordingly we name methods from the class (2.3a)–(2.3c) leap-frog-Chebyshevschemes. In particular, we discuss in this paper the method with g ≡ 0 and the multirate case with gn = g(qn).

Let us remark that for p = 1 the LFC method (2.3a)–(2.3c) reduces to the stan-dard LF method, i.e., P (z) = P1(z) = z for any choice of νp ≥ 1.

As already indicated above the general scheme (2.3a)–(2.3b) comprises the LFbut also (for g ≡ 0) the modified equation leap-frog (modified LF) [15] method with

PLF(z) = z, PmodLF(z) = z − 112z

2.(2.4)

For p > 1, νp = 1, g ≡ 0 and standard starting values (2.1b) the method (2.3a),(2.3c) has been constructed in [6, 12]. Unfortunately, as we will show in the nextsection, these methods suffer from stability issues. In particular, they are not suitedto construct a stable multi rate method. We propose to remedy these problems byusing the starting values (2.3b) which also involve Pp and its derivative P ′p. A secondcrucial modification is the use of a damping parameter νp > 1. This approach followsthe idea of damped RKC methods [10, 16, 17, 18]. In the next section, we show thatthese modifications lead to a stable scheme.

3. Linear stability. In this section we consider the associated linear homoge-neous problem to (1.1), i.e.,

(3.1) q(t) = −Lq(t), q(0) = q0, q(0) = q0.

Recall that L ∈ Rd×d is a symmetric, positive definite matrix w.r.t. a given innerproduct

(·, ·), i.e., L satisfies(

Lq, q)

=(q,Lq

),

(Lq,q

)> 0, for all q, q ∈ Rd.

4 MARLIS HOCHBRUCK AND ANDREAS STURM

The solution

q(t) = cos(tL1/2)q0 + t sinc(tL1/2)q0, sinc(ζ) =sin ζ

ζ,

of (3.1) satisfies

(3.2) ‖q(t)‖ ≤ ‖q0‖+ t‖q0‖ and ‖|q(t)|‖ = ‖|q(0)|‖

for all t ≥ 0. Here, we denoted the standard norm by ‖ · ‖2 =(·, ·)

and the energynorms by

(3.3) ‖|q(t)|‖2 = ‖q(t)‖2 + ‖q(t)‖2L, ‖q‖2L =(Lq,q

).

We show stability and error bounds in these norms by two completely differenttechniques, namely energy techniques and generating functions.

3.1. Stability in the energy norm. Before we start to study the stability ofthe recursion (2.3a), we introduce a short notation for differences and means:

[qn+ 12] = qn+1 − qn, qn+ 1

2 = 1

2 (qn+1 + qn),

[[qn]] = qn+1 − 2qn + qn−1, qn = 14

(qn+1 + 2qn + qn−1

).

Using the identity

(3.5) qn = qn − 14 [[qn]],

we can write the recursion (2.3a) in the equivalent form

(3.6)(I− 1

4P)[[qn]] + Pqn = 0, P = P (τ2L).

From this form it is easy to see that this recursion has a preserved quantity.

Lemma 3.1. The iterates (qn)n obtained from the recursion (2.3a) with g ≡ 0satisfy

(3.7a) Mq,n+ 12≡Mq, 12

for all n = 1, 2, . . . ,

where

(3.7b) Mq,n+ 12

=((

I− 14P)[qn+ 1

2], [qn+ 1

2])

+(Pqn+ 1

2, qn+ 1

2).

Proof. The statement follows directly by taking the inner product of (3.6) withqn+1 − qn−1.

Motivated by (3.3) we define the discrete energy norm

(3.8) ‖|qn+ 12|‖2τ =

∥∥∥∥ [qn+ 12]

τ

∥∥∥∥2

+∥∥qn+ 1

2∥∥2

L≈ ‖|q(tn+ 1

2)|‖2.

It is easy to see that if Mq,n+ 12

is equivalent to the energy norm ‖|qn+ 12|‖2τ then the

recursion (2.3a) yields stable approximations for arbitrary starting values in the sensethat there exists a constant Cstb such that

(3.9) ‖|qn+ 12|‖τ ≤ Cstb‖|q 1

2|‖τ , n = 1, 2, . . . .

ON LEAP-FROG-CHEBYSHEV SCHEMES 5

Moreover, we will show in Theorem 4.5 below that (under suitable assumptions)1τ2Mq,n+ 1

2coincides with ‖|qn+ 1

2|‖2τ up to a second order pertubation, i.e., the recur-

sion (2.3a) approximately preserves the discrete energy. This reflects the behavior ofthe exact solution which is also energy conserving, see (3.2).

The aforementioned of equivalence Mq,n+ 12

to ‖|qn+ 12|‖2τ requires the following

assumption.

Assumption 3.2. We assume that there exist constants τCFL,m1,m2 > 0 suchthat for all

(3.10) 0 < τ ≤ τCFL

we have

(3.11) m1

∥∥∥∥ [qn+ 12]

τ

∥∥∥∥2

+m2

∥∥qn+ 12∥∥2

L≤Mq,n+ 1

2

τ2≤∥∥∥∥ [qn+ 1

2]

τ

∥∥∥∥2

+∥∥qn+ 1

2∥∥2

L.

The restriction (3.10) on the time step size is usually called a CFL condition.

Theorem 3.3. Let Assumption 3.2 be satisfied. Then, for all τ ≤ τCFL themethod consisting of the recursion (2.3a) and arbitrary starting values is stable. Infact, for all n = 1, 2, . . . the iterates of this scheme are bounded by (3.9) with Cstb =minm1,m2−1/2.

Proof. This follows directly from (3.7) and (3.11).

In the next lemma we give conditions on P that ensure Assumption 3.2.

Lemma 3.4. Assumption 3.2 is satisfied if

(3.12) m1 ≤ 1− 14P (z) ≤ 1, m2z ≤ P (z) ≤ z, for all z ∈

[0, β2

],

with

(3.13) τ2CFL =

β2

‖L‖.

Proof. This follows by considering Mq,n+ 12

in the orthonormal eigenbasis of thematrix L.

In the following we call the largest interval [0, β2] such that (3.12) holds true thestability interval of the method given by the recursion (2.3a) with arbitrary startingvalues.

Example 3.5. The LF method satisfies the second condition in (3.12) withm2 = 1.However, the first condition requires z ∈ [0, 4), i.e., the stability interval is [0, 4ϑ2]with a ϑ < 1. Then, we obtain m1 = 1 − ϑ2 under the CFL condition (3.13) withβ2 = β2

LF = 4ϑ2.The modified LF scheme fulfills the first condition in (3.12) for all z ∈ [0, 12] with

m1 = 14 . However, the second condition requires that z ∈ [0, 12), i.e., the stability

interval is [0, 12ϑ2] with ϑ < 1. Then, we have m2 = 1−ϑ2 and β2mLF = 12ϑ2 = 3β2

LF.This means the modified LF method allows a

√3 ≈ 1.732 times larger time step size

than the LF method.

3.2. Stability in the standard norm. In this section we prove the stability ofschemes relying on the recursion (2.3a) in the standard norm ‖ · ‖ with the generatingfunctions technique. Our analysis shows that the choice (2.3b) of the starting valueq1 yields a stable scheme even if (3.12) is only satisfied with m1 = m2 = 0.

6 MARLIS HOCHBRUCK AND ANDREAS STURM

Theorem 3.6. Assume P satisfies (3.12) with m1 = m2 = 0. Then, the scheme(2.3a), (2.3b) is stable for τ ≤ τCFL with τCFL given in (3.13). The iterates arebounded by

(3.14) ‖qn‖ ≤ ‖q0‖+ P ′maxtn‖q0‖, P ′max = max

z∈[0,β2]|P ′(z)|.

Proof. Following the generating functions technique we define the formal powerseries

q(ζ) =

∞∑n=0

qnζn.

Multiplying the recursion (2.3a) by ζn+1 and summing over n ≥ 1 we obtain

(3.15) %(ζ)q(ζ) = q0 + ζq1 − ζ(2I−P

)q0, %(ζ) = ζ2I− ζ(2I−P) + I.

The matrix-valued roots ζ± of % are given by

ζ± = I− 12P± i 1

2

√P(4I−P),

where i =√−1 is the imaginary unit. By (3.12), we have ‖ζ±‖ = 1 so that we can

write ζ± = e±iΦ with a matrix Φ whose spectrum is contained in [0, π]. Clearly, this

yields ζ+ = ζ−1− and thus

%(ζ) = (ζI− ζ+)(ζI− ζ−) = (I− ζζ−)(I− ζζ+) = (I− ζe−iΦ)(I− ζeiΦ).

Using the Neumann series and the Cauchy product we have for |ζ| < 1

%(ζ)−1 =

∞∑n=0

e−inΦζnn∑`=0

e2i`Φ =

∞∑n=0

sin((n+ 1)Φ

)sin Φ

ζn.

Here, the second equality follows with the geometric sum identity. Using this in (3.15)we deduce by comparing the coefficients of ζn that

(3.16) qn =sin((n+ 1)Φ

)sin Φ

q0 +sin(nΦ)

sin Φ

(q1 − 2(I− 1

2P)q0),

where we further used q0 = q(0) = q0. For the second starting value q1, we employthe ansatz

(3.17) q1 = a(τ2L)q0 + τb(τ2L)q0,

where a, b : [0,∞) → R are suitable analytic functions satisfying the consistencyconditions a(0) = b(0) = 1 and a′(0) = − 1

2 . Using I− 12P = cos Φ and a trigonometric

identity, we infer

qn = cos(nΦ)q0 +sin(nΦ)

sin Φ

((a(τ2L)− cos Φ

)q0 + τb(τ2L)q0

).

This motivates the choice a(z) = 1 − 12P (z) in (2.3b) since then a(τ2L) = cos Φ.

The choice b(z) = P ′(z) is not so obvious. It is based on the observation thatsin Φ = 1

2

√P(4I−P) becomes singular if Φ has eigenvalues z for which P (z) ∈ 0, 4.

ON LEAP-FROG-CHEBYSHEV SCHEMES 7

However, in the interior of (0, β2), these eigenvalues are stationary points of P , i.e.,P ′(z) = 0. Our choice of b thus removes all singularities in the interior of the stabilityinterval. This yields the simplified expression

qn = cos(nΦ)q0 + τ

sin(nΦ)

sin ΦP′q0.

Now, using that for all Φ ∈ R we have | cos Φ| ≤ 1 and∣∣ sinnΦ

sin Φ

∣∣ ≤ n completes theproof.

In summary, we have seen that the recursion (2.3a) is stable under the CFLcondition τ ≤ τCFL with τCFL given in (3.13), both in the standard and in the en-ergy norm, if the polynomial P satisfies the conditions (3.12) with positive constantsm1,m2. However, for m1 = 0 or m2 = 0 this requires special starting values, e.g.,(2.3b).

4. Error analysis. In the previous section we established the stability of thegeneral scheme (2.3a), (2.3b). The aim of this section is to provide its error analysis.The outcome will be a convergence result in the energy norm ‖|·|‖τ for the linearproblem and a convergence result in the standard norm ‖ · ‖ for the semilinear case.

Let us denote the error of the scheme (2.3a), (2.3b) with

(4.1) en = qn − qn, qn = q(tn),

where q(t) is the exact solution of (1.1). We denote bounds on derivatives of q(t) by

(4.2) B(k)n = max

0≤t≤tn‖q(k)(t)‖, k = 1, 2, . . . .

Our error analysis requires the following additional assumption.

Assumption 4.1. There exist constants m3,m4,m′3 and m′4 such that for all z ∈

[0, β2] it holds∣∣P (z)− z∣∣ ≤ m3z

2,∣∣P (z)− z +m3z

2∣∣ ≤ m4z

3,(4.3a) ∣∣P ′(z)− 1∣∣ ≤ m′3z, ∣∣P ′(z)− 1 +m′3z

∣∣ ≤ m′4z2.(4.3b)

This assumption implies that

|Q(z)| ≤ m4, Q(z) =P (z)− z +m3z

2

z3,(4.4a)

|Q(z)| ≤ m′4, Q(z) =P ′(z)− 1 +m′3z

z2,(4.4b)

for all z ∈[0, β2

].

4.1. Error analysis in the energy norm for linear problems. In this sec-tion we restrict ourselves to linear problems (3.1). Then, we can prove the followingerror recursion.

Lemma 4.2. For q ∈ C6(0, T ) the error en satisfies the recursion

(4.5a) [[en]] + Pen = dn, dn = ∆n + δ(6)n ,

where

∆n =(

112 −m3

)τ4q(4)(tn)− τ6Q(τ2L)q(6)(tn),(4.5b)

δ(k)n = τk−1

∫ tn+1

tn

κ(k−1)n,+ (t)q(k)(t) dt− τk−1

∫ tn

tn−1

κ(k−1)n,− (t)q(k)(t) dt,(4.5c)

8 MARLIS HOCHBRUCK AND ANDREAS STURM

with κ(`)n,±(t) = (tn±1 − t)`/(`!τ `). The defect is bounded by

(4.6) ‖dn‖ ≤M3B(4)n τ4 +

(m4 + 1

60

)B

(6)n+1τ

6, M3 =∣∣ 1

12 −m3

∣∣.Proof. We insert the exact solution qn into the scheme (2.3a)

[[qn]] + Pqn = dn,

which yields (4.5a) with a defect dn. In order to determine dn we use Taylor expansionand (3.1) to obtain

[[qn]] = τ2q(tn) + 112τ

4q(4)(tn) + δ(6)n = −τ2Lqn + 1

12τ4q(4)(tn) + δ(6)

n .

This implies

dn =(P− τ2L

)qn + 1

12τ4q(4)(tn) + δ(6)

n ,

which shows (4.5) by definition (4.4a) of Q. The bound follows directly from Assump-tion 4.1.

From the error recursion (4.5) we can derive an error bound in the energy norm.

Theorem 4.3. Let q ∈ C6(0, T ) be the solution of (3.1) and let Assumptions 3.2and 4.1 be satisfied. Then, for all τ ≤ τCFL and tn+1 ≤ T we have

(4.7)1

Cstb‖|en+ 1

2|‖τ ≤ C2τ

2 + C3τ3 + C4τ

4,

where

C2 = M ′3B(3)0 +

tn√m1

M3B(4)n , C3 = 1

2M3B(4)0 , M ′3 =

∣∣ 16 −m

′3

∣∣,and where Cstb is defined in Theorem 3.3. The constant C4 only depends on tn, the

bounds B(5)n , B

(6)n+1, and the constants in Assumption 4.1.

Proof. This first part of the proof is inspired by [4, Lemma 2.6] and [11, Sec-tion 2.4].

Using (3.5), we write (4.5a) in the equivalent form

(4.8)(I− 1

4P)[[en]] + Pen = dn.

Taking the inner product of (4.8) with en+1 − en−1 we obtain

Me,n+ 12−Me,n− 1

2=(dn, en+1 − en−1

)≤ ‖dn‖

(‖[qn+ 1

2]‖+ ‖[qn− 1

2]‖)

≤ 1√m1‖dn‖

(√Me,n+ 1

2+√Me,n− 1

2

),

where Me,n+ 12

was defined in (3.7b) and where we used (3.11) for the last estimate.This is equivalent to √

Me,n+ 12−√Me,n− 1

2≤ 1√m1‖dn‖.

ON LEAP-FROG-CHEBYSHEV SCHEMES 9

Summing this inequality and again applying (3.11) yields

(4.9)1

Cstb‖|en+ 1

2|‖τ ≤

1

τ‖e1‖+

1√m1

1

τ

n∑`=1

‖d`‖.

Here, we further used that that by q0 = q0 we have e0 = 0 and thus Me, 12= ‖e1‖2.

It remains to bound ‖e1‖. A Taylor expansion of q(τ) shows

e1 = 12

(P− τ2L + 1

12τ4L2

)q(0) + τ

(I− 1

6τ2L−P′

)q(0) + τ4

∫ τ

0

κ(4)0,+(t)q(5)(t) dt.

By (4.4), we find

e1 = 12

(112 −m3

)τ4q(4)(0) +

(16 −m

′3

)τ3q(3)(0)

− 12Q(τ2L)τ6q(6)(0)− Q(τ2L)τ5q(5)(0) + τ4

∫ τ

0

κ(4)0,+(t)q(5)(t) dt.

Using the bounds in (4.4) we obtain

‖e1‖ ≤M ′3B(3)0 τ3 + 1

2M3B(4)0 τ4 +m′4B

(5)0 τ5 + 1

24B(5)1 τ5 + 1

2m4B(6)0 τ6.

Inserting this bound and (4.6) into (4.9) completes the proof.

Theorem 4.3 shows that in general the scheme (2.3a), (2.3b) with a polynomial Psatisfying (3.12) and (4.3) is of order two and how the error constants depend on theassumptions on P . In particular, we see that if the polynomial P satisfies (4.3) with

(4.10) m3 =1

12and m′3 =

1

6

we have error constants M3 = M ′3 = 0. Then, the resulting scheme satisfies the bound(4.7) with constants C2 = C3 = 0 and thus is a fourth order scheme. Moreover, for0 < m3 <

16 and 0 < m′3 <

13 , the error constants M3 and M ′3 are smaller than the

ones of the LF scheme, where M3 = 112 and M ′3 = 1

6 . This results in smaller errors,as will be confirmed in our numerical examples in Section 7.

The second order error bound can also be proved under the weaker regularityconditions q ∈ C4(0, T ).

Corollary 4.4. Let the assumptions of Theorem 4.3 be fulfilled for q ∈ C4(0, T ).Then, for all τ ≤ τCFL and tn+1 ≤ T we have

(4.11)1

Cstb‖|en+ 1

2|‖τ ≤ C2τ

2 + C3τ3,

where

C2 = m′3B(3)0 +

tn√m1

(m3 + 1

3

)B

(4)n+1

and C3 only depends on m3 and on the bounds B(3)1 and B

(4)0 .

Proof. The statement follows as in the proof of Theorem 4.3 with two minorchanges. We write the defect in (4.5) as

dn =(P− τ2L

)qn + δ(4)

n ,

10 MARLIS HOCHBRUCK AND ANDREAS STURM

and the error e1 as

e1 = 12 (P− τ2L)q0 + τ(I−P′)q0 + τ2

∫ τ

0

κ(2)0,+q(3)(t) dt.

By (4.3) this yields ‖dn‖ ≤(m3 + 1

3

)B

(4)n+1τ

4 and ‖e1‖ ≤ m′3B(3)0 τ3 + 1

2m3B(4)0 τ4 +

12B

(3)1 τ3. Inserting these bounds in (4.9) proves the error bound.

Let us end this section on the linear error analysis by proving that the recursion(2.3a) nearly preserves the discrete energy by showing that it is order two close to thepreserved quantity Mq, 12

=Mq,n+ 12.

Theorem 4.5. We consider the method (2.3a), (2.3b) with a polynomial P sat-isfying (3.12) and (4.3a). Then, we have∣∣∣ ‖|qn+ 1

2|‖τ −

Mq,n+ 12

τ2

∣∣∣ ≤ Cτ2.

Proof. We have

τ2‖|qn+ 12|‖2τ −Mq,n+ 1

2= 1

4

(P[qn+ 1

2], [qn+ 1

2])−((P− τ2L)qn+ 1

2, qn+ 1

2).

Using (3.12) and (4.3a) we can bound this by

0 ≤ ‖|qn+ 12|‖2τ −

Mq,n+ 12

τ2≤ 1

4τ2

∥∥∥∥ [qn+ 12]

τ

∥∥∥∥2

+m3τ2∥∥Lqn+ 1

2∥∥2,

which proves the result.

4.2. Error analysis in the standard norm for semilinear problems. Inthis section we prove an error bound for the scheme (2.3a), (2.3b) for semilinearproblems (1.1). We assume that g : Rd → Rd is a Lipschitz-continuous function with

(4.12) ‖g(q)− g(q)‖ ≤ Lg‖q− q‖, for all q, q ∈ Rd.

Theorem 4.6. Let q ∈ C4(0, T ) and let Assumptions 3.2 and 4.1 be satisfied.Then, for all τ ≤ τCFL and tn ≤ T we have

(4.13) ‖en‖ ≤ (C1tn + Cdt2n)eLgt

2nτ2.

The constants C1, Cd are independent of L, n, and τ .

Proof. Analogously to Lemma 4.2 and Corollary 4.4 the error en satisfies therecursion

(4.14) en+1 − (2I−P)en + en−1 = dn + rn, rn = −τ2(g(qn)− (qn)),

with

(4.15) e0 = 0, ‖e1‖ ≤ C1τ3, ‖dn‖ ≤ Cdτ4, ‖rn‖ ≤ τ2Lg‖en‖.

By the same procedure as for the proof of Theorem 3.6 we have

en =sin(nΦ)

sin Φe1 +

n−1∑`=1

sin((n− `)Φ

)sin Φ

(d` + r`

).

ON LEAP-FROG-CHEBYSHEV SCHEMES 11

Since∣∣ sinnΦ

sin Φ

∣∣ ≤ n for all Φ ∈ R and using the bounds (4.15) we obtain

(4.16) ‖en‖ ≤ (C1tn + Cdt2n)τ2 + Lgtnτ

n−1∑`=0

‖e`‖.

The statement now follows from a discrete Gronwall lemma.

Corollary 4.7. For linear problems (g ≡ 0), we have ‖en‖ ≤ (C1tn + Cdt2n)τ2,

so that the error only grows quadratically in tn.

5. Error and stability analysis of LFC methods. In this section we showthat the LFC method (2.3a)–(2.3c) satisfies the assumptions of Sections 3 and 4.

Theorem 5.1. Let p > 1 and νp > 1. The polynomial Pp defined in (2.3c)satisfies Assumptions 3.2 and 4.1 with

m1 =1

2

(1− 1

Tp(νp)

), m2 = 4

m1

β2p

, β2 = β2p = αp(νp + 1),(5.1a)

m3 =T ′′p (νp)

α2pTp(νp)

, m4 =T ′′′p (νp)

3α3pTp(νp)

,(5.1b)

m′3 = 2m3, m′4 = 3m4.(5.1c)

Remark 5.2. (i) For p = 2, . . . , 5 the following choices of νp fulfill (4.10):

ν2 =

√6

2≈ 1.224745, ν3 ≈ 1.029086, ν4 ≈ 1.008261, ν5 ≈ 1.003233,

and thus give a fourth order scheme. In the case of p = 2 and ν2 =√

62 we retrieve

the modified LF method, see (2.4).(ii) We have the following limits,

limνp→1

αp = 2p2, limνp→∞

αp = 0, limνp→1

β2p = 4p2, lim

νp→∞β2p = 2p,

limνp→1

m1 = 0, limνp→∞

m1 =1

2, lim

νp→1m2 = 0, lim

νp→∞m2 =

1

p,

and

limνp→1

m3 =p2 − 1

12p2, lim

νp→∞m3 =

p− 1

4p,

limνp→1

m4 =(p2 − 1)(p2 − 4)

360p4, lim

νp→∞m4 =

(p− 1)(p− 2)

24p2,

see also Figure 5.1. This means the stability constants m1, m2 improve and theCFL condition degrades with larger νp. Moreover, we see that the error constant M3

defined in (4.6) depends on the size of νp.

Proof. Throughout this proof we change between the coordinates

x = νp −z

αp∈ [−1, νp] and z = αp(νp − x) ∈

[0, β2

p

].

(a) We have to prove that the inequalities (3.12) hold true with constants (5.1a):(i) First inequality: It is well-known that for νp ≥ 1 we have

−1 ≤ Tp(x) ≤ Tp(νp), for x ∈ [−1, νp],

12 MARLIS HOCHBRUCK AND ANDREAS STURM

1 1.05 1.1

50

100

0νp

β2p

1 1.05 1.1

0.25

0.5

0νp

m1

1 1.05 1.1

0.05

0νp

m2

1 1.05

0.025

0.05

0νp

M3

Fig. 5.1: Dependence of β2p , m1, m2 and M3 = | 1

12 −m3| on νp for p = 3, 4, 5 (dotted,dashed, solid).

see also Figures 5.2a and 5.2b. This is equivalent to

1

2

(1− 1

Tp(νp)

)≤ 1− 1

4Pp(x) ≤ 1, for x ∈ [−1, νp],

which is the desired bound with m1 given in (5.1a).(ii) Second inequality: For the lower bound note that Tp is bounded by the line `Tthrough (−1, 1) and (νp, Tp(νp)) (the blue line in Figures 5.2a and 5.2b), i.e., forx ∈ [−1, νp], we have

Tp(x) ≤ `T (x) = Tp(νp) +1− Tp(νp)

1 + νp(νp − x)

= Tp(νp) +1− Tp(νp)

β2p

α(νp − x).

From this we obtain

Pp(z) ≥2

β2p

(1− 1

Tp(νp)

)z = `P (z), for z ∈ [0, β2

p ],

which is the claimed bound (see also the blue line in Figures 5.2c and 5.2d).For the upper bound we use that

T ′p(x) ≤ T ′p(1), for x ∈ [−1, 1],

see, e.g., [7, Thm 2.1] or the original work [13]. Because T ′p is monotonically increasingon [1,∞) we deduce that

T ′p(x) ≤ T ′p(νp), for x ∈ [−1, νp].

Integrating from x to νp gives

Tp(νp)− Tp(x) ≤ T ′p(νp)(νp − x),

and from this we obtain

Pp(z) ≤T ′p(νp)

Tp(νp)(νp − x) = αp(νp − x) = z.

(b) We have to show the inequalities (4.3a) with constants (5.1b): Markov broth-ers’ inequality, see, e.g., [7, Thm 2.2] or the original work [14], states that

(5.2) T (n)p (x) ≤ T (n)

p (νp) for x ∈ [−1, νp], n ∈ N.

ON LEAP-FROG-CHEBYSHEV SCHEMES 13

Using n = 2 in this inequality and integrating it twice from x to νp we obtain

Tp(νp)− Tp(x) ≥ T ′p(νp)(νp − x)−T ′′p (νp)

2(νp − x)2.

Choosing n = 3 and integrating three times we get

Tp(νp)− Tp(x) ≤ T ′p(νp)(νp − x)−T ′′p (νp)

2(νp − x)2 +

T ′′′p (νp)

6(νp − x)3.

From these two inequalities we conclude

Pp(z) ≥ z −m3z2, Pp(z) ≤ z −m3z

2 +m4z3,

and together with the second bound of (3.12)

0 ≥ Pp(z)− z ≥ −m3z2, 0 ≤ Pp(z)− z +m3z

2 ≤ m4z3.

(c) It remains to show (4.3b) with constants (5.1c): We have

P ′p(z) =2

αpTp(νp)T ′p(x) =

1

T ′p(νp)T ′p(x).

By (5.2) with n = 1 it holds P ′p(z)− 1 ≤ 0. Integrating (5.2) with n = 2 once from xto νp we obtain

T ′p(νp)− T ′p(x) ≤ T ′′p (νp)(νp − x).

Moreover, by integrating (5.2) with n = 3 twice we get

T ′p(νp)− T ′p(x) ≥ T ′′p (νp)(νp − x)−T ′′′p (νp)

2(νp − x)2.

So, we can conlcude

2m3z ≤ P ′p(z)− 1 ≤ 0, 0 ≤ P ′p(z)− 1 + 2m3z ≤ 3m4z2.

This finishes the proof.

In the next corollary we show the convergence of the LFC scheme.

Corollary 5.3. For p > 1 and νp > 1, we consider the LFC scheme (2.3a)–(2.3c).

(a) If the exact solution of (3.1) satisfies q ∈ C4(0, T ), then the error is boundedby (4.11).

(b) If the exact solution of (3.1) satisfies q ∈ C6(0, T ), then the error is boundedby (4.7).

(c) Let g fulfill (4.12). If the exact solution of (1.1) satisfies q ∈ C4(0, T ), thenthe error is bounded by (4.13).

The constants are given in Theorem 5.1.

Proof. Theorem 5.1 shows that the assumptions of Theorem 4.3 and Corollary 4.4are satisfied.

Let us end this section by remarking that it is possible to slightly increase thestability bound β2

p given in (5.1a). However, this degrades either m1 or m2 dependingon whether the polynomial degree p is odd or even. The next lemma gives the details.

14 MARLIS HOCHBRUCK AND ANDREAS STURM

1 νp−1•••

•1

•−1

•Tp(νp)

x

(a) T4(x) and `T (x).

1 νp−1•••

•1

•−1

•Tp(νp)

x

(b) T5(x) and `T (x).

0 β2p

•

•4

•0

z

(c) P4(z) and `P (z).

0 β2p

•

•4

•0

z

(d) P5(z) and `P (z).

Fig. 5.2: Illustration of the Chebyshev polynomial Tp(x) and the line `T (x) (top) andof the LFC polynomial Pp(z) and the line `P (z) (bottom) for p = 4, 5.

Lemma 5.4. Let p > 1, νp > 1 and θ ∈(

1νp, 1). Moreover, let

(5.3) m1 =1

2

(1− Tp(θνp)

Tp(νp)

), m2 = 4

m1

β2p

, β2p = αpνp(1 + θ).

Then, Pp satisfies Assumption 3.2 with β2p, m1 and m2 if p is even, and β2

p, m1

and m2 if p is odd, respectively. The bounds from Assumption 4.1 stay true with theconstants (5.1b), (5.1c) for z ∈ [0, β2

p ].

Remark 5.5. (i) For θ = 1νp

we obtain m1 = m1, m2 = m2 and β2p = β2

p .

(ii) For θ = 1 we get m1 = m2 = 0.

Proof. We show how the proof of Theorem 5.1 has to be adapted. For θ ∈[

1νp, 1)

we define νp = θνp ∈ [1, νp), so that we have the coordinates

x = νp −z

αp∈ [−νp, νp] and z = αp(νp − x) ∈

[0, β2

p

].

Modifications of part (a)(i): For p even the considerations for the lower boundhold true for x ∈ [−νp, νp] while for p odd we have

−Tp(νp) ≤ Tp(x) ≤ Tp(νp), for x ∈ [−νp, νp].

This yields (3.12) with m1.Modifications of part (a)(ii): For p odd all considerations for the lower and the

upper bound hold true for x ∈ [−νp, νp]. For p even, Tp is bounded by the line through (−νp, Tp(νp)) and (νp, Tp(νp)), whence

Tp(x) ≤ T (x) = Tp(νp) +Tp(νp)− Tp(νp)

β2p

α(νp − x), for x ∈ [−νp, νp],

ON LEAP-FROG-CHEBYSHEV SCHEMES 15

and thus

Pp(z) ≥2

β2p

(1− Tp(νp)

Tp(νp)

)z, for z ∈ [0, β2

p ].

This is the desired bound. The proof for the upper bound holds true for x ∈ [−νp, νp].Parts (b) and (c) remain unchanged since (5.2) stays true for x ∈ [−νp, νp].

6. Efficiency, implementation, and generalizations of the LFC method.In this section we discuss the efficiency and the implementation of the LFC method(2.3a)–(2.3c) for semilinear differential equations and we generalize it to fully nonlinearproblems.

6.1. Semilinear LFC method. In Algorithm 6.1 we present an efficient im-plementation of the nth time step of the LFC method (2.3a)–(2.3c) to integrate thesemilinear problem (1.1).

Algorithm 6.1 Leap-frog-Chebyshev scheme for (1.1).

1: P0 = 0, P1 = 2αpνp

τ2Lqn2: for k = 2, . . . , p do

3: Pk = 2νpTk−1(νp)Tk(νp) Pk−1 + 2

αp

Tk−1(νp)Tk(νp) τ2L

(2qn − Pk−1

)− Tk−2(νp)

Tk(νp) Pk−2

4: end for5: qn+1 = 2qn − qn−1 − Pp − τ2g(qn)

The parameters αp and T0(νp), . . . , Tp(νp) have to be precomputed only once bymeans of the Chebyshev recursions. Hence, each time step requires p matrix-vectormultiplications with L and one evaluation of g. As we show below this makes thealgorithm attractive in applications where on the one hand the evaluation of g isexpensive compared to a matrix-vector multiplication by L but on the other hand thetime step is restricted by a CFL condition dominated by L.

We compare the CFL conditions of the standard LF scheme and the generalrecursion (2.3a) for the special case of a linear function g, i.e., g(q) = Gq, where Gis a symmetric and positive semidefinite matrix with ‖G‖ ‖L‖.

Lemma 6.1. Let g(q) = Gq and assume that the CFL conditions

(6.1) τ2‖L‖ ≤ β2, τ2‖G‖ ≤ 4ϑ2, ϑ2 ∈ (0,m1),

are satisfied. Then, the recursion (2.3a) with P satisfying Assumption 3.2 is stablewith bound

(6.2) minm1 − ϑ2,m2‖|qn+ 12|‖2τ ≤

∥∥∥∥ [q 12]

τ2

∥∥∥∥2

+ ‖q 12‖2L+G.

Remark 6.2. Note that the CFL conditions (6.1) can only be satisfied if m1 > 0.For LFC methods this requires a sufficiently large damping parameter νp > 1 to allowfor a reasonable ϑ.

Proof. For the scheme (2.3) we have

Mq,n+ 12

=((

I− 14P)[qn+ 1

2], [qn+ 1

2])

+(Pqn+ 1

2, qn+ 1

2)

− 14

(τ2G[qn+ 1

2], [qn+ 1

2])

+(τ2Gqn+ 1

2, qn+ 1

2).

16 MARLIS HOCHBRUCK AND ANDREAS STURM

The first two terms can be treated with Assumption 3.2 using the first CFL conditionin (6.1) and for the last two terms we can use the second CFL condition in (6.1) andthe fact that G is positive semidefinite to obtain

(m1 − ϑ2)

∥∥∥∥ [qn+ 12]

τ2

∥∥∥∥2

+m2‖qn+ 12‖2L ≤

Mq,n+ 12

τ2≤∥∥∥∥ [qn+ 1

2]

τ2

∥∥∥∥2

+ ‖qn+ 12‖2L+G.

The assertion follows similar to the proof of Theorem 3.3.

Let ‖L‖ = r‖G‖ with a factor r 1. For p2 . r the first CFL condition in (6.1)limits the time step size, whereas for p2 & r the second CFL condition applies. Thismeans that a larger polynomial degree p of P in (2.3a) improves the CFL conditionuntil p ≈

√r. A further increase of the polynomial degree does not alleviate the CFL

condition anymore.So, let p2 . r. Then, the CFL condition of the recursion (2.3a) and of the LF

method are

τ2 .4p2

r‖G‖and τ2 .

4

(r + 1)‖G‖,

respectively. The fraction is r+1r p2 ∼ p2 since we assume r 1. Thus, the recursion

(2.3a) allows an (approximately) p times larger time step than the LF method.In summary, we conclude that N time steps of the LFC method (via Algo-

rithm 6.1) cost

pN matrix vector multiplications with L +N evaluations of g.

Due to its stricter CFL condition the LF method has to perform pN time steps withcosts

pN matrix vector multiplications with L + pN evaluations of g.

We see that the effort on the “linear part” are equal for the LFC and the LF method,but the evaluations of the nonlinearity g can be (considerably) reduced by using theLFC method.

6.2. Nonlinear LFC method. In this section we derive the LFC method forthe nonlinear problem

(6.3) q(t) = −f(q), q(0) = q0, q(0) = q0.

This requires a recursion for Pp inspired by RKC methods [16] which we provide inthe next lemma.

Lemma 6.3. The polynomial

Pk,p(z) = 2− 2

Tk(νp)Tk

(νp −

z

αp

)

ON LEAP-FROG-CHEBYSHEV SCHEMES 17

satisfies the recursion

P0,p(z) = 0,

P1,p(z) =2

αpνpz,

Tk(νp)Pk,p(z) = 2νpTk−1(νp)Pk−1,p(z)

+2

αpTk−1(νp)z

(2− Pk−1,p(z)

)− Tk−2(νp)Pk−2,p(z),

for k = 2, . . . , p.

Proof. The result follows easily from the recursion of Chebyshev polynomials.

Lemma 6.3 and Pp(z) = Pp,p(z) imply the following algorithm to implement one time-step of the LFC scheme for nonlinear ordinary differential equations (6.3). One timestep requires p evaluations of f as the main cost.

Algorithm 6.2 Leap-frog-Chebyshev scheme for (6.3).

1: P0 = 0, P1 = 2αpνp

τ2f(qn)

2: for k = 2, . . . , p do

3: Pk = 2νpTk−1(νp)Tk(νp) Pk−1 + 2

αp

Tk−1(νp)Tk(νp)

(2τ2f(qn)− τ2f(Pk−1)

)− Tk−2(νp)

Tk(νp) Pk−2

4: end for5: qn+1 = 2qn − qn−1 − Pp

7. Numerical examples. In our last section we illustrate our theoretical find-ings on LFC schemes by numerical examples. It turns out that already the mostsimple examples show the lack of stability for general starting values or for the un-damped case νp = 1. All implementations have been performed in Python. The codeswill be made available by the authors on request.

7.1. Harmonic oscillator. We consider the harmonic oscillator

(7.1) q(t) = −ω2q(t), q(0) = q0, q(0) = q0,

where ω > 0 is a fixed frequency. Recall that the solution q satisfies (3.2) and inparticular it preserves the energy.

Now, we examine the LFC method (2.3a), (2.3c) with the standard starting values(2.1b) obtained from Taylor expansion and the new ones we proposed in (2.3b).

In Figure 7.1 we present the results for ω2 = 4, q0 = 2 and q0 = 1. We used thefifth order polynomial P5 in (2.3c) without damping (ν = 1) and employ a range oftime steps τ so that the product 0 ≤ τ2ω2 ≤ β2

5 = 100. In Figure 7.1a we depict the(discrete) energy norm of the approximations qn obtained with starting value (2.3b).For this choice the energy norm stays bounded independent of the simulation time. Incontrary, for the standard choice (2.1b) illustrated in Figure 7.1b we observe resonanceeffects appearing at z = τ2ω2 where Pp(z) = 4 or Pp(z) = 0. This fits perfectly toour analysis because these values force m1 = 0 or m2 = 0 in (3.12), respectively, andthus prevent a stability result as obtained in Theorem 3.3.

18 MARLIS HOCHBRUCK AND ANDREAS STURM

0 20 40 60 80 1000

10

20

30

τ2ω2

maxn‖|q

n+

1 2|‖ τ

(a) Starting values (2.3b).

0 20 40 60 80 1000

1,000

2,000

3,000

τ2ω2

(b) Starting values (2.1b).

0 20 40 60 80 1000

1

2

3

4

0 β2p

z

(c) Polynomial P5(z).

Fig. 7.1: Time integration of the harmonic oscillator with the LFC recursion (2.3a),(2.3c) and different starting values q1. We ran the simultion for N = 5, 10, 15 and 20time steps.

7.2. Wave equation. Next, we consider the homogeneous wave equation withhomogeneous Dirichlet boundary conditions in the unit square Ω = (0, 1)2,

(7.2a)

q(t, x, y) = ∆q(t, x, y)− dq(t, x, y), (x, y) ∈ Ω, t ∈ [0, T ],

q(t, x, y) = 0, (x, y) ∈ ∂Ω, t ∈ [0, T ],

q(0, x, y) = q0(x, y), q(0, x, y) = q0(x, y), (x, y) ∈ Ω,

with a parameter d ≥ 0. As initial data we choose

(7.2b) q0(x, y) = sin(πx) sin(πy), q0(x, y) =√

2π2 + d sin(πx) sin(πy).

Then, the solution of (7.2) is given by

(7.3) q(t, x, y) = sin(πx) sin(πy)(cos(t

√2π2 + d ) + sin(t

√2π2 + d )

).

We discretize (7.2) with a symmetric interior penalty discontinuous Galerkinmethod [3], [5, Chapter 4] using piecewise polynomials of degree three on an un-structured mesh with 312 triangles with smallest and largest diameter 0.0301 and0.0744, respectively. This results in the following system if odes

(7.4) Mq(t) = −Aq(t)− dMq(t) q(0) = q0, q(0) = q0.

The block diagonal mass matrix M and the stiffness matrix A are symmetric (w.r.t.the standard Euclidean inner-product) and positive definite. The boundary conditionin (7.2a) is enforced through A.

Because the mass matrix is block-diagonal it can be inverted at low costs. Thus,(7.2) can be written in the form (1.1) with L = −M−1A and g(q) = dq. Note thatL is symmetric w.r.t. the inner-product

(q, q

)= qTMq.

In the following, we integrate (7.4) with the LFC method (2.3a)–(2.3c) until thefinal time T = 4.2 and consider the error

(7.5) eh,n = qh(tn)− qn

ON LEAP-FROG-CHEBYSHEV SCHEMES 19

0.020.0050.00125

10−3

10−2

10−1

τ

maxn‖|e

h,n

+1/2|‖ τ

(a) p = 3.

0.020.0050.00125

τ

(b) p = 4.

0.020.0050.00125

τ

(c) p = 5.

0.020.0050.00125

10−3

10−2

10−1

τ

maxn‖|e

h,n

+1/2|‖ τ

(d) p = 3.

0.020.0050.00125

τ

(e) p = 4.

0.020.0050.00125

τ

(f) p = 5.

Fig. 7.2: Error of the LFC method (2.3a), (2.3c) plotted over time steps τ for startingvalues (2.3b) (top) and starting values (7.6) (bottom). The polynomial degree p is3, 4 and 5 (left to right). For νp we used the following choices: νp = 1, νp = 1.0001,νp = 1.001, νp = 1.01, νp = 1.1, νp from Remark 5.2 (4th order scheme). The solidblack line stems from the LF method. The black dashed lines represent 25τ2 and8000τ4. The dotted lines correspond to integer multiples of the maximum stable timestep size τLF of the LF method, i.e., mτLF, with m = 1, . . . , 5.

between the L2(Ω)-orthogonal projection qh(t) of the exact solution onto the discon-tinuous Galerkin space and the LFC iterate qn. We distinguish the cases d = 0 andd > 0.

7.2.1. Wave equation with d = 0. We are in the situation of Section 4.1 andin particular of Theorem 4.3. To show the validity of these elaborations we plot inFigure 7.2a–Figure 7.2c the error (7.5) of the LFC method for polynomial degreesp = 3, 4, 5 and different choices of the damping parameter νp.

We observe that the LFC method allows us to choose an approximately p timeslarger time step compared to the LF method (see the dashed lines which mark integermultiples of the maximum stable time step of the LF method). If we use more dampingthe maximum stable time step gets smaller since β2

p is a monotonically decreasingfunction of the damping parameter νp, see also Figure 5.1. Moreover, one can clearly

20 MARLIS HOCHBRUCK AND ANDREAS STURM

0.020.0050.00125

10−3

10−2

10−1

100

τ

maxn‖|e

h,n

+1/2|‖ τ

(a) d = 10.

0.020.0050.00125

τ

(b) d = 25.

0.020.0050.00125

τ

(c) d = 50.

Fig. 7.3: Error of the LFC method (2.3a)–(2.3c) plotted over time steps τ for dif-ferent values of d. We used polynomial degree p = 5 and damping parametersνp = 1, νp = 1.00001, νp = 1.00005, νp = 1.0001, νp = 1.00025, νp ≈ 1.003233 (4thorder scheme). The solid black line stems from the LF method and the black dashedline represent 25τ2. The dotted lines correspond to integer multiples of the maximumstable time step size τLF of the LF method, i.e., mτLF, with m = 1, . . . , 5.

see the effects of the value of νp on the error constant. In particular, we observethat a choice of νp near the value which gives a fourth order scheme (see (4.10) andRemark 5.2) yields a remarkably better error constant compared to the LF methodand consequently clearly smaller errors.

We can confirm the second order convergence rate of the general LFC method.However, the fourth order achieved via (4.10) is not visible in this example sincethe time discretization error is so small that it is already dominated by the spacediscretization error.

As comparison we give in Figure 7.2d–Figure 7.2f the error of the LFC recursion(2.3a), (2.3c) supplemented with the standard fifth order starting starting value

(7.6) q1 = q0 + τ q0 − 12τ

2Lq0 − 16τ

3Lq0 + 124τ

4L2q0.

We clearly see larger errors compared to the LFC method (2.3a)–(2.3c). In particular,the undamped case νp = 1 suffers from stability problems. However, with enoughdamping this can be controlled and we even can confirm the fourth order convergencerate achieved by the choice (4.10) of νp.

7.2.2. Wave equation with d > 0. Last, we consider the case d > 0 as a modelproblem for the semilinear equation (1.1) to show the effects of the LFC method(2.3a)–(2.3c) discussed in Section 6.1. In Figure 7.3 we plotted the error (7.5) ford = 10, 25, 50, polynomial degree p = 5 and different values of νp. As stated inLemma 6.1 and Remark 6.2 we observe that without enough damping the LFC methodcannot achieve a p times larger time than the LF scheme. The larger d is the moredamping we have to use. However, if νp is sufficiently large we observe an almost ptimes larger maximum stable time step size and second order convergence.

Acknowlegdments. The authors are grateful to Constantin Carle for helpfuldiscussions and for running many numerical tests which inspired us to elaborate the

ON LEAP-FROG-CHEBYSHEV SCHEMES 21

error and stability analysis also in the standard norm. This led us to the suggestionof new starting values.

We also thank Sebastien Imperiale for interesting conversations on using Cheby-shev polynomials in local time stepping methods.

REFERENCES

[1] A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput.,23 (2002), pp. 2041–2054, https://doi.org/10.1137/S1064827500379549.

[2] A. Abdulle and A. A. Medovikov, Second order Chebyshev methods based on orthogonalpolynomials, Numer. Math., 90 (2001), pp. 1–18, https://doi.org/10.1007/s002110100292.

[3] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAMJ. Numer. Anal., 19 (1982), pp. 742–760, https://doi.org/10.1137/0719052.

[4] J. Chabassier and S. Imperiale, Space/time convergence analysis of a class of conservativeschemes for linear wave equations, Comptes Rendus Mathematique, 355 (2017), pp. 282–289, https://doi.org/10.1016/j.crma.2016.12.009.

[5] D. A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods, vol. 69of Mathematiques & Applications (Berlin) [Mathematics & Applications], Springer, Hei-delberg, 2012, https://doi.org/10.1007/978-3-642-22980-0.

[6] J. C. Gilbert and P. Joly, Higher order time stepping for second order hyperbolic prob-lems and optimal CFL conditions, in Partial differential equations, vol. 16 of Com-put. Methods Appl. Sci., Springer, Dordrecht, 2008, pp. 67–93, https://doi.org/10.1007/978-1-4020-8758-5 4.

[7] N. K. Govil and R. N. Mohapatra, Markov and Bernstein type inequalities for polynomials,J. Inequal. Appl., 3 (1999), pp. 349–387, https://doi.org/10.1155/S1025583499000247.

[8] E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, vol. 31 of SpringerSeries in Computational Mathematics, Springer-Verlag, Berlin, second ed., 2006, https://doi.org/10.1007/3-540-30666-8. Structure-preserving algorithms for ordinary differentialequations.

[9] M. Hochbruck and C. Lubich, A Gautschi-type method for oscillatory second-order dif-ferential equations, Numer. Math., 83 (1999), pp. 403–426, https://doi.org/10.1007/s002110050456.

[10] W. H. Hundsdorfer and J. G. Verwer, Numerical solution of time dependent advectiondiffusion reaction equations, Springer series in computational mathematics ; 33, Springer,Berlin, corr. 2. print. ed., 2007, https://doi.org/10.1007/978-3-662-09017-6.

[11] P. Joly, Variational methods for time-dependent wave propagation problems, vol. 31of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2003, https://doi.org/10.1007/978-3-642-55483-4 6.

[12] P. Joly and J. Rodrıguez, Optimized higher order time discretization of second order hyper-bolic problems: construction and numerical study, J. Comput. Appl. Math., 234 (2010),pp. 1953–1961, https://doi.org/10.1016/j.cam.2009.08.046.

[13] A. A. Markov, On a problem of D. I. Mendeleev, Zapiski Imp. Akad. Nauk, 62 (1889), pp. 1–24.

[14] V. A. Markov, Uber Polynome, die in einem gegebenen Intervalle moglichst wenig von nullabweichen, Math. Annalen 77, (1916), pp. 213–258.

[15] G. R. Shubin and J. B. Bell, A modified equation approach to constructing fourth-ordermethods for acoustic wave propagation, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 135–151, https://doi.org/10.1137/0908026.

[16] P. J. van der Houwen and B. P. Sommeijer, On the internal stability of explicit, m-stageRunge–Kutta methods for large m-values, Z. Angew. Math. Mech., 60 (1980), pp. 479–485,https://doi.org/10.1002/zamm.19800601005.

[17] J. G. Verwer, A note on a Runge-Kutta-Chebyshev method, Z. Angew. Math. Mech., 62(1982), pp. 561–563, https://doi.org/10.1002/zamm.19820621008.

[18] J. G. Verwer, W. H. Hundsdorfer, and B. P. Sommeijer, Convergence properties of theRunge-Kutta-Chebyshev method, Numer. Math., 57 (1990), pp. 157–178, https://doi.org/10.1007/BF01386405.