Numer. Math.DOI 10.1007/s00211-014-0632-2

NumerischeMathematik

Gauss–Runge–Kutta time discretization of waveequations on evolving surfaces

Dhia Mansour

Received: 12 April 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract This paper is concerned with an error analysis for a full discretization ofthe linear wave equation on a moving surface. The equation is discretized in space bythe evolving surface finite element method. Discretization in time is done by Gauß–Runge–Kutta (GRK) methods, aiming for higher-order accuracy in time and uncon-ditional stability of the fully discrete scheme. The latter is established in the naturaltime-dependent norms by using the algebraic stability and the coercivity propertyof the GRK methods together with the properties of the spatial semi-discretization.Under sufficient regularity conditions, optimal-order error estimates for this class offully discrete methods are shown. Numerical experiments are presented to confirmsome of the theoretical results.

Mathematics Subject Classification (2000) 65M12 · 65M15 · 65M60 · 35L99 ·35R01 · 35R37

1 Introduction

The analytical and numerical study of partial differential equations on fixed and mov-ing surfaces has attracted considerable attention over the last years. These equationsappear in many applications such as fluid dynamics [17], material science [3], imageprocessing [26], and physiology [16]. In particular, the numerical analysis of parabolicdifferential equations on moving surfaces has been studied in several works. We referto [8–11,19,23] and the references cited therein.

This work was supported by DFG, SFB/TR 71 “Geometric Partial Differential Equations”.

D. Mansour (B)Mathematisches Institut, Universität Tübingen,Auf der Morgenstelle 10, 72076 Tübingen, Germanye-mail: mansour@na.uni-tuebingen.de

123

D. Mansour

In this paper, we deal with the numerical solution of the linear wave equation

∂•∂•u(x, t) + ∂•u(x, t) ∇Γ (t) · v(x, t) − ΔΓ (t)u(x, t) = 0 (1.1)

on a compact moving hypersurface Γ (t) ⊂ Rm+1, t ∈ [0, T ], with a given velocity

v(x, t). Here ∂•u denotes the material derivative of u:

∂•u = ∂u

∂t+ v · ∇u.

Based on the fact that the solution of (1.1) makes the action integral

S [u] =T∫

0

⎛⎜⎝1

2

∫

Γ (t)

∣∣∂•u∣∣2 − 1

2

∫

Γ (t)

|∇Γ u|2⎞⎟⎠ dt

stationary under all paths with fixed endpoints, the authors in [22] developed andanalyzed a fully variational discrete method for (1.1), stable under a CFL-condition.In work [22], the variational principle was discretized with functions that are piecewiselinear in space and time. This yielded a discretization of the wave equation in space bythe evolving surface finite element method of Dziuk and Elliott [7], and a discretizationin time by a variational integrator, namely, a version of the second order leapfrog orStörmer–Verlet method. In order to fulfill the CFL-condition, it was necessary tochoose a time step size τ = O(h), where h is the spatial mesh size. Since we aredealing here with moving meshes, such a time step restriction can be inconvenient.For this reason, and in order to obtain higher order accuracy in time, we consider fullyimplicit Gauß–Runge–Kutta (GRK) methods for the time discretization.

Based on the weak form of the wave equation

d

dt

∫

Γ

∂•uϕ +∫

Γ

∇Γ u · ∇Γ ϕ =∫

Γ

∂•u∂•ϕ, (1.2)

where ϕ : ⋃t∈[0,T ] Γ (t) × {t} → R is an arbitrary test function, we consider a finiteelement approximation using piecewise linear finite elements on a triangulated surfaceinterpolating Γ (t) as described in [7]. This leads to the system of ordinary differentialequations (ODEs)

d

dt(M(t)q(t)) + A(t)q(t) = 0, (1.3)

where M(t) and A(t) are the evolving mass and stiffness matrices and q(t) is thenodal vector of the spatially discrete solution. In order to construct the fully discretesolution, the ODE system (1.3) is discretized in time by the s-stage GRK method. Thelowest order method of this class is the implicit midpoint rule, which reads

123

Gauss–Runge–Kutta time discretization of wave equations

qn+1 = qn + τ M−1n+ 1

2

(pn + pn+1

2

)

pn+1 = pn − τ An+ 12

(qn + qn+1

2

),

where τ is the time step size, M−1n+ 1

2= M(tn + 1

2τ)−1, and An+ 12

= A(tn + 12τ).

The main purpose of this paper is to derive optimal-order error estimates for theconcerned fully discrete scheme (piecewise linear finite elements in combination withthe s-stage GRK method). As in [11,22], the keystone is to show stability estimates inthe natural time-dependent norms for the time discretization. This will be achieved byusing the algebraic stability and the coercivity property of the GRK methods togetherwith the properties of the spatial semi-discretization. Our treatment here is inspired bythe B-convergence theory which was originally developed to study the convergence ofimplicit Runge–Kutta methods when applied to stiff systems of ordinary differentialequations (cf. [2,4]). In particular, we prove that the order in time of the fully discretemethod is at least the B-convergence order of the s-stage GRK method. This order isequal to 2 for s = 1, whereas, for s ≥ 2, the B-convergence order is only equal to s.Under additional regularity assumptions which we expect to be satisfied for closedsmooth surfaces, we show that the order is indeed the full classical order of the GRKmethods, i.e., 2s.

The paper is organized as follows: In Sect. 2, we start by recalling the basic notationfor the wave equations on evolving surfaces and describe the spatial discretization usingthe piecewise linear finite elements of Dziuk and Elliott [7]. This leads to the timedependent ODE system (1.3), which we further reformulate as a Hamiltonian system.By taking up an idea of Kraaijevanger [18], stability estimates for the 1-stage GRKmethod (implicit midpoint rule) are established in Sect. 3. In Sect. 4, we recall someproperties of the GRK methods which we use together with the properties of the spatialsemi-discretization to show stability estimates for general GRK methods with s ≥ 2.In Sect. 5, we prove error bounds for a projection of the exact solution onto the finiteelement space on the discretized surface. In Sect. 6, we summarize various resultsestablished in [22] which we later use in Sect. 7 to investigate the convergence orderof the fully discrete scheme. Finally, in Sect. 8, we present some numerical examples.

2 The wave equation on evolving surfaces

2.1 Basic notation

Let Γ (t), t ∈ [0, T ], be a smoothly evolving family of smooth m-dimensional com-pact closed hypersurfaces in R

m+1 without boundary and with unit outward point-ing normal ν. We let v(x(t), t) denote the given velocity of the surface Γ (t), i.e.,x(t) = v(x(t), t) for x(t) ∈ Γ (t). The space–time surface is

GT =⋃

t∈[0,T ]Γ (t) × {t}.

123

D. Mansour

The tangential gradient of a smooth function g : GT → R is given by (omittingthe argument t in Γ (t))

∇Γ g = ∇ g − ∇ g · ν ν,

where g is an extension of g to an open neighborhood of Γ , ∇ g denotes the usual(m+1)-dimensional gradient and a ·b = ∑m+1

j=1 a j b j for vectors a and b in Rm+1. The

tangential gradient only depends on the values of g on the surface Γ and is independentof the choice of the extension.

The Laplace–Beltrami operator on Γ is the tangential divergence of the tangentialgradient

ΔΓ g = ∇Γ · ∇Γ g =m+1∑j=1

(∇Γ ) j (∇Γ ) j g,

and Green’s formula on Γ (∂Γ = ∅) reads

∫

Γ

∇Γ g · ∇Γ ϕ = −∫

Γ

ϕΔΓ g. (2.1)

We let ∂•g denote the material derivative

∂•g = ∂ g

∂t+ v · ∇ g, (2.2)

which only depends on the values of the function g on the space–time surface GT

and is independent of the choice of the extension. For a more detailed discussionconcerning the surface gradients and material derivatives, we refer to [7,12].

2.2 The mathematical model

We consider the initial value problem for the wave equation on evolving surfaces

⎧⎪⎨⎪⎩

∂•∂•u(x, t) + ∂•u(x, t) ∇Γ (t) · v(x, t) − ΔΓ (t)u(x, t) = 0 on GT

u(·, 0) = u0 on Γ (0)

∂•u(·, 0) = u0 on Γ (0)

(2.3)

with given initial data u0 ∈ H2(Γ (0)) and u0 ∈ H1(Γ (0)). For more details con-cerning the derivation of the wave equation, wellposedness and regulartiy results, werefer the reader to [22,24].

123

Gauss–Runge–Kutta time discretization of wave equations

2.3 Weak formulation

Let ϕ : GT → R be a smooth test function. Multiplying the above Eq. (2.3) by ϕ,integrating over Γ , and performing integration by parts, we obtain

0 =∫

Γ

∂•∂•u ϕ + ∂•u ϕ ∇Γ · v + ∇Γ u∇Γ ϕ

=∫

Γ

∂•(∂•uϕ) − ∂•u∂•ϕ + ∂•u ∇Γ · vϕ + ∇Γ u∇Γ ϕ

= d

dt

∫

Γ

∂•uϕ +∫

Γ

∇Γ u · ∇Γ ϕ −∫

Γ

∂•u∂•ϕ,

where we made use of the Leibniz formula on surfaces [7, Lemma 2.2]

d

dt

∫

Γ

g =∫

Γ

∂•g + g∇Γ · v. (2.4)

The weak formulation of (2.3) then reads: Find u ∈ H1(GT ) such that:

• For almost every t ∈ [0, T ],d

dt

∫

Γ

∂•uϕ +∫

Γ

∇Γ u · ∇Γ ϕ =∫

Γ

∂•u∂•ϕ for all ϕ ∈ H1(GT ). (2.5)

• u(·, 0) = u0 and ∂•u(·, 0) = u0.

2.4 Evolving surface finite element method [7]

In order to construct a finite element approximation based on the weak form (2.5) ofthe wave equation, we first approximate the smooth surface Γ (t) by a triangulatedsurface Γh(t). Let the discrete surface

Γh(t) =⋃

E(t)∈Th(t)

E(t)

be a union of m-dimensional simplices E(t) that is assumed to form an admissibletriangulation Th(t). The vertices {ai (t)}N

i=1 of all simplices E(t) are taken to sit onthe surface Γ (t) for all time t ∈ [0, T ] and to move with the given velocity v(ai (t), t).By h, we denote the maximum diameter of the whole triangulation.

The surface gradient on Γh(t) is given by

∇Γh g = ∇g − ∇g · νhνh,

where νh denotes the normal to Γh(t).

123

D. Mansour

For each t ∈ [0, T ], we define the finite element space

Sh(t) = {φh ∈ C0(Γh(t)) : φh |E ∈ P1 for all E ∈ Th(t)},

where P1 denotes the space of polynomials of degree at most 1. The moving nodalbasis functions {χi }N

i=1 of Sh(t) are determined by χi (a j (t), t) = δi j for all j , so, wehave

Sh(t) = span{χ1(·, t), . . . , χN (·, t)}.

The discrete velocity Vh of the discrete surface Γh(t) is the piecewise linear inter-polant of v:

Vh(x, t) =N∑

j=1

v(a j (t), t)χ j (x, t), x ∈ Γh(t). (2.6)

Thereby, we define the discrete material derivative on Γh(t),

∂•hφh = ∂φh

∂t+ Vh · ∇φh . (2.7)

Then, the discrete material derivative of the basis functions satisfies the remarkabletransport property [7, Proposition 5.4]:

∂•hχ j = 0. (2.8)

2.5 The spatial semi-discretization

The spatial semi-discretization of the wave equation reads as follows: Find Uh(·, t) ∈Sh(t) such that

• For all temporally smooth φh with φh(·, t) ∈ Sh(t) and for all t ∈ (0, T ],d

dt

∫

Γh

∂•h Uh φh +

∫

Γh

∇Γh Uh · ∇Γh φh =∫

Γh

∂•h Uh∂•

hφh . (2.9)

• Uh(·, 0) = U 0h and ∂•

hUh(·, 0) = U 0h , where U 0

h and U 0h ∈ Sh(0) are appropriate

approximations of u0 and u0, respectively.

2.6 The Hamiltonian ODE system

By choosing φh = χ j for j = 1, . . . , N in (2.9) and using the transport property (2.8),it follows that (2.9) is equivalent to

123

Gauss–Runge–Kutta time discretization of wave equations

d

dt

∫

Γh

∂•hUh χ j +

∫

Γh

∇Γh Uh · ∇Γh χ j = 0. (2.10)

Since Uh(·, t) ∈ Sh(t), there exists a unique vector q(t) = (q j (t)) ∈ RN such that

Uh(·, t) =N∑

j=1

q j (t)χ j (·, t).

Using the transport property (2.8), it follows that ∂•h Uh(·, t) is also a finite element

function belonging to Sh(t). Writing q j = dq j/dt , we find

∂•h Uh(·, t) =

N∑j=1

q j (t)χ j (·, t).

Inserting these expressions into (2.10), we obtain the equivalent initial value problem:Find q : [0, T ] → R

N such that⎧⎪⎨⎪⎩

ddt (M(t)q(t)) + A(t)q(t) = 0

q(0) = q0 = (U 0h (a j ))

q(0) = q0 = (U 0h (a j )),

(2.11)

where M(t) and A(t) are the evolving mass and stiffness matrices given by

M(t)i j =∫

Γh(t)

χi (·, t)χ j (·, t), A(t)i j =∫

Γh(t)

∇Γh(·,t)χi (·, t) · ∇Γh(t)χ j (·, t).

The mass matrix is symmetric and positive definite. The stiffness matrix is symmetric,and because we consider closed surfaces, only positive semidefinite.

With the conjugate momenta

p(t) = M(t)q(t) ∈ RN ,

the ODE system (2.11) can be written in the variable y(t) = (p(t), q(t))T as Hamil-ton’s equations

{y(t) = J−1 H(t)y(t)

y(0) = y0 = (q0, p0)T = (q0, M(0)q0)

T ,(2.12)

with

J =(

0 IN

−IN 0

), H(t) =

(M(t)−1 0

0 A(t)

)∈ R

2N×2N , (2.13)

and IN is the identity matrix of dimension N .

123

D. Mansour

We use the notation: For a symmetric positive definite or semidefinite matrix G(t) ∈R

N×N , we define the norm or semi-norm, respectively, for w ∈ RN :

|w|2G(t) = 〈w |G(t)|w〉 = wTG(t)w.

The following lemma from [11, Lemma 4.1] and [23, Lemma 2.2] will be the onlyresult that we need from the evolving surface finite element method in order to provestability estimates for the time discretization schemes considered in this paper.

Lemma 2.1 There are constants μ, κ (independent of the mesh-width h) such that

wT(M(s) − M(t))z ≤ μ|s − t | |w|M(t) |z|M(t) (2.14)

wT(M(s)−1 − M(t)−1)z ≤ μ|s − t | |w|M(t)−1 |z|M(t)−1 (2.15)

wT(A(s) − A(t))z ≤ κ|s − t | |w|A(t) |z|A(t) (2.16)

for all w, z ∈ RN and s, t ∈ [0, T ].

2.7 Time dependent energy norm

With the symmetric positive definite matrix

H(t) =(

M(t)−1 00 A(t) + M(t)

)∈ R

2N×2N ,

we define the associate time-dependent energy norm for y = (p, q)T on R2N :

‖y‖2t = 〈q|A(t) + M(t)|q〉 + 〈p|M(t)−1|p〉 = 〈y|H(t)|y〉 = yT H(t)y. (2.17)

Note that for finite element functions Uh(·, t) = ∑Nj=1 q j (t)χ j (·, t) ∈ Sh(t) with

the vector of nodal values q(t) = (q j (t)) ∈ RN and p(t) = M(t)q(t), for y(t) =

(p(t), q(t))T,we have

‖y(t)‖2t = 〈q(t)|M(t)|q(t)〉 + 〈q(t)|A(t)|q(t)〉 + 〈p(t)|M(t)−1|p(t)〉

= ‖Uh(·, t)‖2L2(Γh(t)) + ‖∇Γh(t)Uh(·, t)‖2

L2(Γh(t)) + ‖∂•h Uh(·, t)‖2

L2(Γh(t)).

(2.18)

In the stability analysis we will make use of the following estimates:

• Using the above definitions, the fact that 〈p|(A + M)−1|p〉 ≤ 〈p|M−1|p〉, theCauchy–Schwarz inequality, and Young’s inequality yield

yT H(t)J−1 H(t)y = qT p ≤ 1

2‖y‖2

t . (2.19)

123

Gauss–Runge–Kutta time discretization of wave equations

• Due to Lemma 2.1, we have

yT(H(s) − H(t))z ≤ (μ + κ)|s − t |‖y‖t‖z‖t (2.20)

for all y, z ∈ R2N and s, t ∈ [0, T ].

3 Stability analysis of the implicit midpoint rule

3.1 Method formulation

For the numerical integration of (2.12), for simplicity, we choose an equidistant grid0 = t0 < t1 < · · · tn ≤ T with step size τ . The approximation yn of y(tn) is determinedvia the implicit midpoint rule

Yn+ 12

= yn + τ

2J−1 Hn+ 1

2Yn+ 1

2(3.1a)

yn+1 = yn + τ J−1 Hn+ 12Yn+ 1

2, (3.1b)

with Hn+ 12

= H(tn + 12τ).

3.2 Defects and errors

We consider the perturbed scheme

Yn+ 12

= yn + τ

2J−1 Hn+ 1

2Yn+ 1

2+ Δn+ 1

2(3.2a)

yn+1 = yn + τ J−1 Hn+ 12Yn+ 1

2+ δn+1, (3.2b)

where Δn+ 12

and δn+1 are the defects obtained when inserting the values Yn+ 12

and yn

(e.g., Yn+ 12

= y(tn + 12τ) and yn = y(tn)) into (3.1).

By subtracting (3.2) from (3.1) and setting En+ 12

= Yn+ 12

− Yn+ 12

and en = yn − yn ,we get the error equations

En+ 12

= en + 1

2τ En+ 1

2− Δn+ 1

2(3.3a)

en+1 = en + τ En+ 12

− δn+1, (3.3b)

where

En+ 12

= J−1 Hn+ 12

En+ 12. (3.3c)

123

D. Mansour

3.3 Stability

We are now ready to state and prove the main result of this section.

Lemma 3.1 There exists τ0 > 0 (depending only on μ and κ of Lemma 2.1 ( suchthat for τ ≤ τ0, the error for the implicit midpoint rule is bounded , for tn = nτ ≤ T ,by

‖en‖tn ≤ C

(‖e0‖t0 +

∥∥∥Δ 12

∥∥∥t0

+∥∥∥δn − Δn− 1

2

∥∥∥tn

)

+ Cn−1∑j=1

∥∥∥δ j + Δ j+ 12

− Δ j− 12

∥∥∥t j

.

The constant C is independent of h, τ and n subject to the stated conditions.

Proof The first step is to modify the errors en and the defects δn+1 in such a way thatwe obtain new error equations of the form (3.3), where the first equation contains nodefects. For this purpose, we follow the idea of Kraaijevanger in his study of the B-convergence of the implicit midpoint rule [18] and define, for given n, the new errors{ek}n

k=0 and the new defects {δk}nk=0 in the following way:

ek = ek − Δk+ 12

k = 0, . . . , n − 1

en = en

δ0 = Δ 12

δk = δk + Δk+ 12

− Δk− 12

k = 1, . . . , n − 1

δn = δn − Δn− 12.

This gives the desired error equations for k = 1, . . . , n − 1:

Ek+ 12

= ek + 1

2τ En+ 1

2(3.4a)

ek+1 = ek + τ En+ 12

− δk+1. (3.4b)

Now, we start from (3.4b) by taking the squared norm of (ek+1 + δk+1) at (tk+ 12

=tk + 1

2τ ) and estimate the terms in

∥∥ek+1 + δk+1∥∥2

tk+ 1

2

= ‖ek‖2tk+ 1

2

+ 2τ⟨ek |Hk+ 1

2|Ek+ 1

2

⟩+ τ 2

∥∥Ek∥∥2

tk+ 1

2

.

Expressing ek by (3.4a), using (3.3c) and relation (2.19), it follows that

∥∥ek+1 + δk+1∥∥2

tk+ 1

2

≤ ‖ek‖2tk+ 1

2

+ τ

∥∥∥Ek+ 12

∥∥∥2

tk+ 1

2

. (3.5)

123

Gauss–Runge–Kutta time discretization of wave equations

In order to bound the norm of Ek+ 12, we first multiply the Eq. (3.4a) by ET

k+ 12

Hk+ 12

and obtain

∥∥∥Ek+ 12

∥∥∥2

tk+ 1

2

=⟨ek

∣∣∣Hk+ 12

∣∣∣ Ek+ 12

⟩+ 1

2τ⟨Ek+ 1

2

∣∣∣Hk+ 12

∣∣∣ Ek+ 12

⟩.

Then, using the relation (2.19), the Cauchy–Schwarz inequality and Young’s inequal-ity, for sufficiently small τ , yield

∥∥∥Ek+ 12

∥∥∥2

tk+ 1

2

≤ C ‖ek‖2tk+ 1

2

. (3.6)

Next, we write Hk+1 = Hk+ 12

+ (Hk+1 − Hk+ 12) and use the condition (2.20) to

estimate

∥∥ek+1 + δk+1∥∥2

tk+1= ⟨

ek+1 + δk+1∣∣Hk+1

∣∣ ek+1 + δk+1⟩

=⟨ek+1 + δk+1

∣∣∣Hk+ 12

∣∣∣ ek+1 + δk+1

⟩

+⟨ek+1 + δk+1

∣∣∣Hk+1 − Hk+ 12

∣∣∣ ek+1 + δk+1

⟩

≤ (1 + Cτ)∥∥ek+1 + δk+1

∥∥2tk+ 1

2

. (3.7)

In the same way, we find

‖ek‖2tk+ 1

2

≤ (1 + Cτ)‖ek‖2tk . (3.8)

Combining (3.5)–(3.8) yields

‖ek+1‖tk+1≤ (1 + Cτ) ‖ek‖tk + ∥∥δk+1

∥∥tk+1

.

Summing over k and knowing that e0 = e0 − δ0 gives

∥∥etn

∥∥n ≤ Cτ

n−1∑j=0

∥∥e j∥∥

t j+

n∑j=0

∥∥δ j∥∥

t j+ ‖e0‖t0 .

Thus, using a discrete Gronwall inequality and the fact that en = en yield the statedresult.

4 Gauß–Runge–Kutta methods

In this section, we study the stability of the GRK methods. At this point, we mentionthe connection to the earlier work [22], where the wave equation is discretized by a

123

D. Mansour

fully discrete variational integrator of second order. We note that the GRK methods canalso be interpreted as variational integrators (cf. [13, Section VI.6] and [25] for moredetails). In the following, we will see that we have a class of fully discrete variationalschemes with an arbitrarily high order in time. Let us now start by giving a brief reviewof the GRK methods.

4.1 Method formulation and properties

For a given step size τ > 0, the s-stage GRK method applied to the Hamiltoniansystem (2.12) reads

Yni = yn + τ

s∑j=1

ai j Yn j , i = 1, . . . , s, (4.1a)

yn+1 = yn + τ

s∑i=1

bi Yni , (4.1b)

where the internal stages satisfy

Yni = J−1 Hni Yni i = 1, . . . , s,

with Hni = H(tn + ciτ).The method is uniquely defined via the so-called Butcher-tableau (cf. [15, Section

IV.5]):

Here, the {bi }si=1 are the weights of the s-stage Gauß-quadrature and the {ci }s

i=1are the nodes of this quadrature transformed to the interval [0, 1]. The coefficients ofthe matrix Oι are determined from the conditions

s∑j=1

ai j ck−1j = ck

i

ki, k = 1, . . . , s.

Note that the implicit midpoint rule is the 1-stage GRK method with

Now, we summarize the various properties of the method (cf. [15, Section IV.14]),which are crucial in order to show stability estimates.

123

Gauss–Runge–Kutta time discretization of wave equations

The GRK method is algebraically stable, i.e.,

bi ≥ 0, i = 1, . . . , s, (4.2a)

bi ai j + b j a ji − bi b j = 0 i, j = 1, . . . , s. (4.2b)

The matrix Oι is invertible and we denote its inverse by Oι−1 = [wi j ]. Further, 0 <

ci < 1(i = 1, . . . , s), and by defining

α:= mini=1,...,s

1

2ci (1 − ci ), B = diag(b1, b2, . . . , bs), C = diag(c1, c2, . . . , cs),

D = diag(d1, d2, . . . , ds):=B(C −1 − I ),

we have the coercivity condition

wTDOι−1w ≥ αwTDw for all w ∈ Rs, (4.3)

with α > 0 and di > 0.The s-stage GRK method is of stage order s and classical order 2s.

4.2 Defects and errors

Let yn, Yni and ˙Y ni be reference values that we want to compare with yn, Yni andYni , respectively (e.g., Yni = y(tn + ciτ), ˙Y ni = y(tn + ciτ) and yn = y(tn)). Wheninserted into (4.1), they yield defects δn+1 and Δni in

Yni = yn + τ

s∑j=1

ai j˙Y nj + Δni , i = 1, . . . , s, (4.4a)

yn+1 = yn + τ

s∑i=1

bi˙Y ni + δn+1, (4.4b)

where the new internal stages Yni satisfy

˙Y ni = J−1 Hni Yni , i = 1, . . . , s.

For the errors, we introduce the notations

en = yn − yn (4.5a)

Eni = Yni − Yni (4.5b)

Eni = Yni − ˙Y ni , (4.5c)

123

D. Mansour

and subtract (4.4) from (4.1) to get the error equations

Eni = en + τ

s∑j=1

ai j En j − Δni , i = 1, . . . , s, (4.6a)

en+1 = en + τ

s∑i=1

bi Eni − δn+1, (4.6b)

where

Eni = J−1 Hni Eni , i = 1, . . . , s. (4.7)

4.3 Error equations in compact form

We rewrite the Runge–Kutta scheme (4.6) in a more compact form. The s × s and the2N × 2N−identity matrices will be denoted by Is and I2N , respectively. The vectorwith all components equal to one in R

s is denoted by 1. Then, we put:

OιOιOι = Oι ⊗ I2N , bbbT = bT ⊗ I2N , 111 = 1 ⊗ I2N ,

where ⊗ denotes the Kronecker product of two matrices. Further, with the vectors

ΔΔΔn = (Δn1, . . . , Δns)T, EEEn = (En1, . . . , Ens)

T, EEEn = (En1, . . . , Ens)T,

and the block diagonal matrices

JJJ−1 = Is ⊗ J−1, HHHn = diag (Hn1, Hn2, . . . , Hns) ,

the Runge–Kutta relations (4.6) and (4.7) can be written as

EEEn = 111en + τOιOιOιEEEn − ΔΔΔn, (4.8a)

en+1 = en + τbbbTEEEn − δn+1, (4.8b)

with EEEn satisfying the relation

EEEn = JJJ−1HHHn EEEn . (4.9)

For a vector EEE = (E1, E2, . . . , Es)T ∈ R

2N ·s (Ei ∈ R2N ), we define the norm

‖|EEE‖|2t = ⟨EEE∣∣Is ⊗ H(t)

∣∣ EEE⟩ = EEET (Is ⊗ H(t)

)EEE =

s∑i=1

‖Ei‖2t .

123

Gauss–Runge–Kutta time discretization of wave equations

4.4 Stability estimate

The following stability lemma will play a key role in estimating the total error.

Lemma 4.1 There exists τ0 > 0 (depending only on μ and κ of Lemma 2.1) suchthat for τ ≤ τ0, the error for the s-stage GRK method (with s ≥ 2) is bounded, fortn = nτ ≤ T , by

‖en‖tn ≤ C

⎛⎝‖e0‖t0 +

n−1∑j=0

‖|ΔΔΔ j‖|t j+

n∑j=1

‖δ j‖t j

⎞⎠ .

The constant C is independent of h, τ and n subject to the stated conditions.

Proof We prove this lemma in three steps:(a) Local error: Here, we analyze the error after one step, starting with en = 0. Thus,the error equation (4.8) simply reads

EEEn = τOιOιOιEEEn − ΔΔΔn (4.10a)

en+1 = τbbbTEEEn − δn+1. (4.10b)

We multiply the equation (4.10a) by EEETn HHHn(DOι−1 ⊗ I2N ) and obtain

EEETn HHHn

(DOι−1 ⊗ I2N

)EEEn = τEEET

n HHHn (D ⊗ I2N ) EEEn

+ EEETn HHHn

(DOι−1 ⊗ I2N

)ΔΔΔn . (4.11)

We handle each term separately, starting by bounding the term on the left-hand sidefrom below, and then bounding the terms on the right-hand side from above.

We write HHHn = (Is ⊗ Hn0

)+ (HHHn − (

Is ⊗ Hn0))

, where Hn0 = H(tn), and get

EEETn HHHn

(DOι−1 ⊗ I2N

)EEEn = EEET

n

(Is ⊗ Hn0

) (DOι−1 ⊗ I2N

)EEEn

+EEETn

(HHHn − (

Is ⊗ Hn0)) (

DOι−1 ⊗ I2N

)EEEn .

(4.12)

Since Hn0 is symmetric and positive definite, we define EEEn = (Is ⊗ Hn0

)1/2EEEn .

Then, using the coercivity condition (4.3) and the fact that di > 0, for the first termon the right-hand side of (4.12), we get

EEETn

(Is ⊗ Hn0

) (DOι−1 ⊗ I2N

)EEEn = EEE

Tn

(DOι−1 ⊗ I2N

)EEEn

≥ αEEETn

(D ⊗ Hn0

)EEEn

≥ c‖|EEEn‖|2tn (4.13)

with a constant c > 0.

123

D. Mansour

The last term on the right-hand side of (4.12) is estimated using condition (2.20) andYoung’s inequality as follows

EEETn

(HHHn − (

Is ⊗ Hn0)) (

DOι−1 ⊗ I2N)

EEEn =s∑

i, j=1

(DOι−1)

i j

⟨Eni

∣∣Hni − Hn0∣∣ Enj

⟩

≤ Cτ

s∑i, j

‖Eni‖tn

∥∥Enj∥∥

tn

≤ Cτ

s∑i, j

(‖Eni‖2tn + ‖Enj‖2

tn

)

≤ Cτ‖|EEEn‖|2tn . (4.14)

Therefore, we deduce by (4.12), (4.13) and (4.14), for sufficiently small τ , that

‖|EEEn‖|2tn ≤ C EEETn HHHn(DOι−1 ⊗ I2N )EEEn . (4.15)

For the first term on the right-hand side of (4.11), we use the relations (4.9), (2.19),and (2.20) to get

τEEETn HHHn(D ⊗ I2N )EEEn = τEEET

n HHHn(D ⊗ I2N )JJJ−1HHHnEEEn

= τ

s∑i=1

di Eni Hni J−1 Hni Eni

≤ Cτ

s∑i=1

‖Eni‖2tni

≤ Cτ‖|EEEn‖|2tn . (4.16)

As above, the right-hand side of (4.11) is estimated using the Cauchy–Schwarz inequal-ity and condition (2.20)

EEETn HHHn(DOι−1 ⊗ I2N )ΔΔΔn ≤ (1 + Cτ) ‖|EEEn‖|tn · ‖|ΔΔΔn‖|tn . (4.17)

Combining (4.11), (4.15), (4.16) and (4.17), for sufficiently small τ , yields

‖|EEEn‖|tn ≤ C ‖|ΔΔΔn‖|tn . (4.18)

Now, we go back to (4.10b) and rewrite with the help of (4.10a)

en+1 = (bTOι−1 ⊗ I2N )(EEEn + ΔΔΔn) − δn+1,

123

Gauss–Runge–Kutta time discretization of wave equations

then, using the Cauchy–Schwarz inequality, Young’s inequality, (4.18) and condition(2.20), it follows that

‖en+1 + δn+1‖tn+1≤ C

(‖|EEEn‖|tn+1+ ‖|ΔΔΔn‖|tn+1

)≤ C ‖|ΔΔΔn‖|tn .

For the local error, we thus find

‖en+1‖tn+1 ≤ C ‖|ΔΔΔn‖|tn + ‖δn+1‖tn+1 . (4.19)

(b) Error propagation: Here, we analyze the error after one step of the GRK methodbetween two numerical solutions starting from different start values. Instead of (4.6),we thus have the following error equations

Eni = en + τ

s∑j=1

ai j En j , i = 1, . . . , s, (4.20a)

en+1 = en + τ

s∑i=1

bi Eni , (4.20b)

with

Eni = J−1 Hni Eni , i = 1, . . . , s. (4.21)

We start from (4.20b) by taking the squared energy norm at tn+1 and then express en

by (4.20a) to find

‖en+1‖2tn+1

= ‖en‖2tn+1

+ 2τ

s∑j=1

b j⟨Enj |Hn+1|En j

⟩

− τ 2s∑

i=1

s∑j=1

bi ai j + b j a ji − bi b j⟨Eni |Hn+1|En j

⟩. (4.22)

The first term on the right-hand side of (4.22) is estimated using condition (2.20)

‖en‖2tn+1

= ⟨en|Hn|en

⟩+ ⟨en|Hn+1 − Hn|en

⟩≤ (1 + Cτ)‖en‖2

tn . (4.23)

For the second term, we use the relations (2.20), (4.21) and (2.19) to estimate

⟨Enj |Hn+1|En j

⟩ = ⟨Enj |Hn+1 − Hnj |En j

⟩+ ⟨Enj |Hnj |En j

⟩

≤ C τ‖Enj‖tn+c j τ‖En j‖tn+c j τ + 1

2‖Enj‖2

tn+c j τ

≤ C τ‖Enj‖tn ‖En j‖tn + C‖Enj‖2tn . (4.24)

123

D. Mansour

As in the first step by the local error (4.18), with Δnj = en , we estimate

‖Enj‖tn ≤ C ‖en‖tn . (4.25)

On the other hand, rewriting (4.20a) as

En j = τ−1s∑

i=1

wi j (Eni − en),

and using (4.25), it follows that

‖En j‖tn ≤ C τ−1‖en‖tn .

Therefore, by (4.24) and (4.25), we find

〈Enj |Hn+1|En j 〉 ≤ C ‖en‖2tn . (4.26)

Thanks to the algebraic stability of the method (4.2), the last term on the right-handside of (4.22) vanishes. Thus, by (4.22), (4.23) and (4.26), we obtain

‖en+1‖tn+1 ≤ (1 + Cτ)‖en‖tn . (4.27)

(c) Error accumulation: A standard application of Lady Windermere’s fan (see [14,15]) completes the proof.

5 Error bounds for a projection to the finite element space

In this section, we analyze the error between the fully discrete numerical solution U nh

and some projection of the exact solution u(·, t) of the wave equation to the finiteelement space Sh(t) at time t = tn .

5.1 The fully discrete solution

Let yn = (pn, qn)T be generated by the s-stage GRK method (4.1) (keeping in mindthat the 1-stage GRK is the implicit midpoint rule). Then, from the vectors qn = (qn

j )

and qn = (qnj ):=M(tn)−1 pn , we obtain the fully discrete numerical solution and its

numerical material derivative

U nh =

N∑j=1

qnj χ j (·, tn), ∂•

h U nh =

N∑j=1

qnj χ j (·, tn), (5.1)

which are finite element functions defined on the surface Γh(tn).

123

Gauss–Runge–Kutta time discretization of wave equations

5.2 Projection to Sh(t)

Let Ph : H1(Γ (t)) → Sh(t) ⊂ H1(Γh(t)) be an arbitrary projection of the exactsolution of the wave equation to the finite element space Sh(t). We set

Phu(·, t) =N∑

j=1

q j (t)χ j (·, t), ∂•h (Phu)(·, t) =

N∑j=1

˙q j (t)χ j (·, t).

We define the finite element residual Rh(·, t) = ∑Nj=1 r j (t)χ j (·, t) ∈ Sh(t) by

∫

Γh

Rhφh = d

dt

∫

Γh

∂•h (Phu) φh +

∫

Γh

∇Γh (Phu) · ∇Γh φh −∫

Γh

∂•h (Phu) ∂•

hφh, (5.2)

where φh is a temporally smooth function with φh(·, t) ∈ Sh(t). The equivalent matrixversion for the vector r(t) = (r j (t)) ∈ R

N is

d

dt(M(t) ˙q(t)) + A(t)q(t) = M(t)r(t), (5.3)

where q(t) = (q j (t)) ∈ RN . We reformulate (5.3) as

˙p(t) = −A(t)q(t) + M(t)r(t)˙q(t) = M(t)−1 p(t).

Further, we set y(t) = ( p(t), q(t))T and Λ(t) = (M(t)r(t), 0)T to get

˙y(t) = J−1 H(t)y(t) + Λ(t). (5.4)

5.3 Error bounds for the implicit midpoint rule

We use the stability Lemma 3.1 together with the norm identity (2.18) to prove thefollowing error estimates for the projection Phu.

Theorem 5.1 Let U nh and ∂•

hU nh be determined by the 1-stage GRK method (5.1)

(implicit midpoint rule). Under sufficient regularity conditions on the exact solutionu of the wave equation (2.3), the errors En

h = U nh − Phu(·, tn) and ∂•

h Enh = ∂•

hU nh −

∂•h (Phu)(·, tn) are bounded, for sufficiently small h ≤ h0 and for tn = nτ ≤ T , by

‖Enh‖L2(Γh(tn)) + ‖∇Γh En

h‖L2(Γh(tn)) + ‖∂•h En

h‖L2(Γh(tn))

≤ C(‖E0

h‖L2(Γh(t0)) + ‖∇Γh E0h‖L2(Γh(t0)) + ‖∂•

h E0h‖L2(Γh(t0))

)

+Cβhτ 2 + Cτ

n−1∑k=0

‖Rh(·, tk + 1

2τ)‖L2(Γh(tk+ 1

2 τ)).

123

D. Mansour

Here, C is independent of h (but depends on T ), and

βh =T∫

0

(‖∇Γh ∂

(2)h (Phu)(·, t)‖L2(Γh(t)) + ‖∇Γh ∂

(3)h (Phu)(·, t)‖L2(Γh(t))

+4∑

�=1

‖∂(�)h (Phu)(·, t)‖L2(Γh(·,t))

)dt,

where the superscript (�) denotes the �-th discrete material derivative.

Proof Considering the errors

en = yn − y(tn)

En+ 12

= Yn+ 12

− y(

tn+ 12

),

the defects appearing in the error equations for the implicit midpoint rule (3.3)satisfy

Δk− 12

= y(

tk− 12

)− y(tk−1) − 1

2τ J−1 Hk− 1

2y(

tk− 12

)

δk = y(tk) − y(tk−1) − τ J−1 Hk− 12

y(

tk− 12

).

By (5.4) and Taylor expansion, we obtain

Δ 12

= 1

2τΛ

(t 1

2

)+

t 12∫

0

(t 1

2− t

2

)¨y(t) dt (5.5a)

δk + Δk+ 12

− Δk− 12

= 1

2τΛ

(tk− 1

2

)+ 1

2τΛ

(tk+ 1

2

)

+τ 2

tk+ 1

2∫

tk− 1

2

ϑ

(t − tk− 1

2

τ

)...y (t) dt (5.5b)

δn − Δn− 12

= 1

2τΛ

(tn− 1

2

)+

tn∫

tn− 1

2

(tn− 1

2− t

2

)¨y(t) dt, (5.5c)

with bounded Peano kernel ϑ .By Lemma 2.1 and the norm identity (2.18), we first note that

‖Λ(t)‖2σ =〈M(t)r(t)|M(σ )−1|M(t)r(t)〉≤2|r(t)|2M(t) =2‖Rh(·, t)‖2

L2(Γh(t)), (5.6)

provided that μ|t − σ | ≤ 1.

123

Gauss–Runge–Kutta time discretization of wave equations

We shall make use of Lemma 9.2 in [11] which shows that for Zh(·, t) =∑mj=1 z j (t)χ j (·, t) with z(t) = (z j (t)):

|(Mz)(k)(t)|2M(t)−1 ≤ Ck∑

�=0

‖∂(�)h Zh(·, t)‖2

L2(Γh(t)), (5.7)

where f (k)(t) denotes the k-th time derivative of f (t) if k ≥ 1 and f (0)(t) = f (t).The system (5.4), Lemma 2.1, (5.7) and the norm identity (2.18) yield

‖y(k)(t)‖2σ = | p(k)(t)|2M(σ )−1 + |q(k)(t)|2(A+M)(σ )

≤ 2|(M ˙q)(k)(t)|2M(t)−1 + 2|q(k)(t)|2(A+M)(t)

≤Ck+1∑�=1

(‖∂(�)

h (Phu)(·, t)‖2L2(Γh(t))

)+2‖∇Γh ∂

(k)h (Phu)(·, t)‖2

L2(Γh(t)) (5.8)

The identities (5.5) together with the bounds (5.6) and (5.8) yield

‖Δ 12‖t0 + ‖δn − Δn− 1

2‖tn + C

n−1∑j=1

‖δ j + Δ j+ 12

− Δ j− 12‖t j

≤ Cτ 2βh + Cτ

n−1∑k=0

‖Rh

(·, tk+ 1

2

)‖

L2

(Γh

(tk+ 1

2

)).

Inserting this bound into Lemma 3.1 and using the norm identity (2.18) completes theproof.

5.4 Error bounds for the Gauß–Runge–Kutta methods

For the general GRK method, we use the stability Lemma 4.1 together with the normidentity (2.18) to prove the following results similar to Theorem 5.1 for the implicitmidpoint rule.

Theorem 5.2 For s ≥ 2, let U nh and ∂•

hU nh be determined by the s-stage GRK method

(5.1). Under sufficient regularity conditions on the exact solution u of the wave equa-tion (2.3), the errors En

h = U nh − Phu(·, tn) and ∂•

h Enh = ∂•

hU nh − ∂•

h (Phu)(·, tn) arebounded for sufficiently small h ≤ h0 and for tn = nτ ≤ T by

‖Enh‖L2(Γh(tn)) + ‖∇Γh En

h‖L2(Γh(tn)) + ‖∂•h En

h‖L2(Γh(tn))

≤ C(‖E0

h‖L2(Γh(t0)) + ‖∇Γh E0h‖L2(Γh(t0)) + ‖∂•

h E0h‖L2(Γh(t0))

)

+Cβh,sτs + Cτ

n−1∑k=0

s∑i=1

‖Rh(·, tk + ciτ)‖L2(Γh(tk+ci τ)).

123

D. Mansour

Here, C is independent of h (but depends on T ), and

βh,s =T∫

0

(‖∇Γh ∂

(s+1)h (Phu)(·, t)‖L2(Γh(t)) +

s+2∑�=1

‖∂(�)h (Phu)(·, t)‖L2(Γh(t))

)dt.

Proof Considering the errors

en = yn − y(tn)

Eni = Yni − y(tn + ciτ),

we rewrite the Runge–Kutta relation (4.8), with ΛΛΛn = (Λn1, . . . , Λns)T, as

EEEn = 111en + τOιOιOιJJJ−1HHHn EEEn − (τOιOιOιΛΛΛn + ΔΔΔn) (5.9a)

en+1 = en + τbbbTJJJ−1HHHn EEEn − (τbbbTΛΛΛn + δn+1). (5.9b)

Due to the stability Lemma 4.1, we get

‖en‖tn ≤ C

⎛⎝‖e0‖t0 + τ

n−1∑j=0

‖|ΛΛΛ j‖|t j+

n−1∑j=0

‖|ΔΔΔ j‖|t j+

n∑j=1

‖δ j‖t j

⎞⎠ . (5.10)

In view of (5.6), we have

τ

n−1∑k=0

‖|ΛΛΛk‖|tk ≤ Cτ

n−1∑k=0

s∑i=1

‖Rh(·, tk + ciτ)‖L2(Γh(tk+ci τ)). (5.11)

By using Taylor series expansion, we find that the defects δn+1 and Δni appearingin the error equation (5.9) satisfy

δn+1 = τ s

tn+1∫

tn

ϑ

(t − tn

τ

)y(s+1)(t) dt

Δni = τ s

tn+1∫

tn

ϑi

(t − tn

τ

)y(s+1)(t) dt,

with bounded Peano kernels ϑ and ϑi . By (5.8), we thus have

‖δn+1‖tn+1 ≤ τ s√

2C

tn+1∫

tn

s+2∑�=1

(‖∂(�)h (Phu)(·, t)‖L2(Γh(t))) dt

123

Gauss–Runge–Kutta time discretization of wave equations

+ τ s√

2C

tn+1∫

tn

‖∇Γh ∂(s+1)h (Phu)(·, t)‖L2(Γh(t)) dt, (5.12a)

‖Δni‖tn ≤ τ s√

2C

tn+1∫

tn

s+2∑�=1

(‖∂(�)h (Phu)(·, t)‖L2(Γh(t))) dt

+τ s√

2C

tn+1∫

tn

‖∇Γh ∂(s+1)h (Phu)(·, t)‖L2(Γh(t)) dt. (5.12b)

Inserting the bounds (5.11) and (5.12) into (5.10) and using the norm identity (2.18)completes the proof.

For the 1-stage GRK method, Theorem 5.1 shows that the order in time of thefully discrete scheme is equal to the classical order O(τ 2). However, Theorem 5.2 forgeneral s-stage GRK method shows only order O(τ s). In order to obtain the classicalorder O(τ 2s), stronger regularity conditions are needed.

For s ≥ 2, we assume that:

‖J−1 H (k j −1)(t) . . . J−1 H (k1−1)(t)y(l)(t)‖t ≤ γ (5.13a)

‖J−1 H (s)(σ )J−1 H (k j −1)(t) . . . J−1 H (k1−1)(t)y(l)(t)‖t ≤ γ, (5.13b)

for all 0 ≤ ki ≤ s − 1 and l ≥ s + 1 with k1 + · · · + k j + l ≤ 2s + 1 and |σ − t | ≤ τ .For ki = 0, the matrix H (−1)(t) is meant to be the identity matrix.

Thereby, we get the following convergence result of full order O(τ 2s).

Theorem 5.3 Under suitable regularity conditions on the exact solution u of the waveequation (2.3) such that conditions (5.13) are satisfied.

The errors Enh = U n

h − Phu(·, tn) and ∂•h En

h = ∂•h U n

h −∂•h (Phu)(·, tn) are bounded,

for sufficiently small h ≤ h0 and for tn = nτ ≤ T , by

‖Enh‖L2(Γh(tn)) + ‖∇Γh En

h ‖L2(Γh(tn)) + ‖∂•h En

h‖L2(Γh(tn))

≤ C(‖E0h‖L2(Γh(t0)) + ‖∇Γh E0

h‖L2(Γh(t0)) + ‖∂•h E0

h‖L2(Γh(t0)))

+ C0τ2s + Cτ

n−1∑k=0

s∑i=1

‖Rh(tk + ciτ)‖L2(Γh(tk+ci τ)).

Here, C0 is independent of h (but depends on T and γ )

Proof The main idea of the proof is to modify the defects appearing in (5.9) so they areof order 2s + 1. To do so, we follow Lubich and Ostermann in their proof of Theorem1 in [21] and first split the matrix Hni and the defects Δni as follows

123

D. Mansour

Hni = Tni + Kni =s−1∑k=0

(ciτ)k

k! H (k)(tn) +tn+ci τ∫

tn

(tn + ciτ − t)s−1

(s − 1)! H (s)(t) dt

Δni = Dni + Rni =2s∑

l=s+1

τ lξ(l)i y(l)(tn) + τ 2s

tn+1∫

tn

ϑi

(t − tn

τ

)y(2s+1)(t) dt

with bounded Peano kernels ϑi and ξ(l)i = 1

l!(

l∑s

j=1 ai j cl−1j − cl

i

).

We introduce the vectors DnDnDn = (Dni )si=1 and RnRnRn = (Rni )

si=1, the block diagonal

matrices

TnTnTn = diag(Tn1, Tn2, . . . , Tns) and KnKnKn = diag(Kn1, Kn2, . . . , Kns),

and the new internal stages

EEEn = EEEn + DDDn, with DDDn =s−1∑k=0

(τOιOιOιJJJ−1TnTnTn)k DDDn .

Next, we rewrite the error equations (5.9) as

en+1 = en + τbbbTJJJ−1HHHn EEEn − (τbbbTΛΛΛn + δδδ′n+1)

EEEn = 111en + τOιOιOιJJJ−1HHHn EEEn − (τOιOιOιΛΛΛn + ΔΔΔ′n),

where the modified defects satisfy

δ′n+1 = δn+1 + τbbbTJJJ−1KnKnKn DDDn + τbbbTJJJ−1TnTnTn DDDn (5.14)

ΔΔΔ′n = RRRn + τOιOιOιJJJ−1KnKnKn DDDn + (τOιOιOιJJJ−1TnTnTn)s DnDnDn .

By the stability Lemma 4.1, we have

‖en‖tn ≤ C

⎛⎝‖e0‖t0 + τ

n−1∑j=0

‖|ΛΛΛ j‖|t j+

n−1∑j=0

‖|ΔΔΔ′j‖|t j

+n∑

j=1

‖δ′j‖t j

⎞⎠ . (5.15)

By the regularity conditions (5.13), for j = (0, . . . , n − 1), we observe that

‖|ΔΔΔ′j‖|t j

≤ C0τ2s+1. (5.16)

Now, we come to the last part of the proof, where we show that δ′j is also of order

O(τ 2s+1). By the regularity condition (5.13), the first and second term of (5.14) clearly

123

Gauss–Runge–Kutta time discretization of wave equations

are of order O(τ 2s+1). Therefore, our problem reduces to show that τbbbTJJJ−1TnTnTn DDDn isof order O(τ 2s+1). We start by introducing the following notation:

C = diag(c1, c2, . . . , cs) and ξ (l) = (ξ(l)i )s

i=1.

Then, we see that τbbbTJJJ−1TnTnTn DDDn consists of a linear combination of expressions of theform

bTOιC k j −1 . . . OιC k1−1ξ (l) · J−1 H (k j −1)(tn) . . . J−1 H (k1−1)(tn)y(l)(tn) · τ |k|+l+1

(5.17)

where |k| = ∑ ji=1 ki , ki ∈ {1, . . . , s} and j ≤ s.

By the order conditions of the Runge–Kutta method (see [13, p. 56]), we have

bTOιC k j −1 . . . OιC k1−1ξ (l) = 0 for |k| + l + 1 ≤ 2s,

therefore, all the expressions of (5.17) vanish for |k| + l + 1 ≤ 2s. Thus, by theregularity conditions (5.13), for j = (1, 2, . . . , n), we get

‖δ′j‖t j ≤ C0τ

2s+1 (5.18)

Inserting the bounds (5.16), (5.18) and (5.11) into (5.15) and using the norm identity(2.18) completes the proof.

6 Ritz map and residual bound

Keeping in mind the estimates above, the problem of estimating the total error reducesto finding an appropriate projection Phu such that the Residual Rh appearing in (5.2)is of optimal order O(h2). This will be achieved by taking Phu to be the modified Ritzprojection introduced in [22]. In the following, we summarize some results from [22],where we define the modified Ritz projection, give some estimates for it, and finallyget to the optimal-order bound of the residual.

We denote by d(x, t), x ∈ Rm+1, t ∈ [0, T ] the signed distance function to the

smooth closed surface Γ (t) and let N (t) be a neighborhood of Γ (t) such that forevery x ∈ N (t) and t ∈ [0, T ] there exists a unique p(x, t) ∈ Γ (t) which is thenormal projection of x onto Γ (t), i.e.,

x − p(x, t) = d(x, t)ν(p(x, t), t). (6.1)

We assume Γh(t) ⊂ N (t). Thus, for each triangle E(t) in Γh(t), there is a uniquecurved triangle e(t) = p(E(t), t) ⊂ Γ (t), and this induces an exact triangulation ofΓ (t) with curved edges. Furthermore, we assume that Γh(t) consists of triangles E(t)in Th(t) with inner radius bounded below by σh ≥ ch for some c > 0.

123

D. Mansour

For any continuous function ηh : Γh → R, we define its lift ηlh : Γ → R by

ηlh(p, t) = ηh(x, t), p ∈ Γ (t),

where x ∈ Γh(t) is such that p = p(x, t). Then, we have the lifted finite elementspace

Slh(t) = {ϕh = φl

h : φh ∈ Sh(t)}.

For every material point X (t) on Γh(t), there is a unique y(t) = p(X (t), t) ∈ Γ (t)such that

vh(y, t):=y(t) = ∂p

∂t(X (t), t) + Vh(X (t), t) · ∇ p(X (t), t).

We define the bilinear forms for w, ϕ ∈ H1(Γ ) by

m(w, ϕ) =∫

Γ

wϕ a(w, ϕ) =∫

Γ

∇Γ w · ∇Γ ϕ,

and for Wh, φh ∈ Sh we write

mh(Wh, φh) =∫

Γh

Whφh ah(Wh, φh) =∑

E∈Th

∫

E

∇Γh Wh · ∇Γh φh .

It turns out to be convenient in the error analysis to use a modified Ritz projectionPh : H1(Γ (t)) −→ Sh(t) defined in the following way.

Definition 6.1 For given z ∈ H1(Γ (t)) and ∂•z ∈ L2(Γ (t)), there is a unique Phz ∈Sh(t) such that for all φh ∈ Sh(t) we have, with the corresponding lift ϕh = φl

h ,

ah(Phz, φh) + mh(Phz, φh) = a(z, ϕh) + m(z, ϕh)

+ m(∂•z, (v(·, t) − vh(·, t)) · ∇Γ (t)ϕh). (6.2)

We define Phz ∈ Slh(t) as the lift of Phz, i.e., Phz = (Phz)l .

We shall make use of the following result for the error in the Ritz map given in [22,Theorem 8.2].

Lemma 6.1 The error in the Ritz map satisfies the bounds, for 0 ≤ t ≤ T and h ≤ h0with sufficiently small h0,

‖z − Phz‖L2(Γ (t)) + h‖∇Γ (t)(z − Phz)‖L2(Γ (t))

≤ Ch2(‖z‖H2(Γ (t)) + ‖∂•z‖L2(Γ (t))). (6.3)

123

Gauss–Runge–Kutta time discretization of wave equations

In general, ∂•hPhz �= Ph∂•

h z, but we have the following result [22, Theorem 8.3].

Lemma 6.2 The error in the material derivatives of the Ritz map satisfies the bounds,for � ≥ 1, 0 ≤ t ≤ T and h ≤ h0 with sufficiently small h0,

‖∂(�)h (z − Phz)‖L2(Γ (t)) + h‖∇Γ

(∂

(�)h (z − Phz)

)‖L2(Γ (t))

≤ C� h2�∑

i=0

(‖∂(i)z‖H2(Γ (t)) + ‖∂(i+1)z‖L2(Γ (t))

). (6.4)

An optimal-order bound of the residual Rh(·, t) ∈ Sh(t) of (5.2) is obtained if we takethe mapping Ph to be the Ritz map Ph defined in (6.2).

Lemma 6.3 Assume that the solution u of the wave equation is sufficiently smooth.Then, there exist C > 0 and h0 > 0 such that for h ≤ h0 and 0 ≤ t ≤ T , the residualof the Ritz map is bounded by

‖Rh(·, t)‖L2(Γh(t)) ≤ Ch2. (6.5)

7 Error bound for the full discretization

In this section, we compare the lifts of the fully discrete numerical solution unh :=(U n

h )l

and its numerical material derivative ∂•h un

h :=(∂•hU n

h )l with the exact solution u(·, tn)

of the wave equation (2.3) and its material derivative ∂•u(·, tn), respectively.Let yn = (pn, qn)T be generated by the s-stage GRK method (4.1). As in (5.1),

we obtain the lifts of the fully discrete numerical solution and its numerical materialderivative

unh :=(U n

h )l =N∑

j=1

qnj χ

lj (·, tn), ∂•

h unh :=(∂•

hU nh )l =

N∑j=1

qnj χ

lj (·, tn),

which are lifted finite element functions defined on the surface Γ (tn). Then, our mainresult reads as follows:

Theorem 7.1 Let u be a sufficiently smooth solution of the wave equation (2.3) andassume that the discrete initial data satisfy

‖u0h − (Phu)(·, 0)‖L2(Γ (0)) + ‖∇Γ (0)u

0h − ∇Γ (0)(Phu)(·, 0)‖L2(Γ (0))

+‖∂•h u0

h − ∂•h (Phu)(·, 0)‖L2(Γ (0)) ≤ C0h2.

Then, there exist h0 > 0 and τ0 > 0 such that for h ≤ h0 and τ ≤ τ0, the followingerror bound holds for 0 ≤ tn = nτ ≤ T :

‖unh − u(., tn)‖L2(Γ (tn)) + h‖∇Γ (tn)u

nh − ∇Γ (tn)u(., tn)‖L2(Γ (tn))

+ ‖∂•h un

h − ∂•u(., tn)‖L2(Γ (tn)) ≤ C(h2 + τ s).

123

D. Mansour

For the implicit midpoint rule (s = 1), the bound holds with τ 2 instead of τ s . Forgeneral s, assuming that the regularity conditions (5.13) are satisfied, we obtain τ 2s

instead of τ s . The constant C is independent of h, τ , and n subject to the statedconditions.

Proof The total error is divided into two parts such as

unh − u(·, tn) = (un

h − Phu(·, tn)) + (Phu(·, tn) − u(·, tn)). (7.1)

Taking into account that the L2 and H1 norms on the discretized and original surfaceare equivalent [7, Lemma 5.2] and the fact that ‖∂•u −∂•

h u‖L2(Γ ) ≤ Ch2 [8, Corollary5.7], in order to estimate the first part of (7.1), we just need to combine the abovetheorems and lemmas. For example, for the implicit midpoint rule (s = 1), usingTheorem 5.1 together with Lemma 6.3 (residual bound) and Lemmas 6.1 and 6.2 (forestimating βh), we find that the first part is of order O(τ 2 + h2). The second part of(7.1) is already taken care of in Lemmas 6.1 and 6.2.

Remark 7.1 It is already known in the case of plane domains, that Runge–Kutta timediscretization of partial differential equations suffers from order reduction phenomena,which depends on the boundary conditions (cf. [20,21,27]). In particular, condition(5.13) fails to hold uniformly in the mesh size on surfaces with boundary. However,for smooth solutions of equations on smooth closed surfaces, it can be expected thatthe regularity condition (5.13) holds true. A detailed examination of this regularitycondition is outside the scope of this paper.

8 Numerical experiments

In this section, we present three numerical experiments to illustrate some of our con-vergence results. The above fully discrete methods are implemented by using theDUNE-FEM module, which is based on the Distributed and Unified Numerics Envi-ronment (DUNE), see [1,5,6] for details. The implementation of the evolving surfacefinite element method is described in [7].

Example 1

We consider the numerical example [22, Example 1]. Let Γ (t) be an ellipsoid withtime dependent axis

Γ (t) ={

x ∈ R3 : x2

1

1 + 0.25 sin(π · t)+ x2

2 + x23 − 1 = 0

}.

We choose u(x, t) = sin(√

6t)x1x2 to be the exact solution of the wave equation andcompute the right-hand side f (x, t) from

f = ∂•∂•u + ∂•u ∇Γ · v − ΔΓ u, on Γ (t). (8.1)

123

Gauss–Runge–Kutta time discretization of wave equations

Table 1 Errors and observed orders of convergence for the first experiment (implicit midpoint rule)

Level Dof L∞( L2 ) EOC L∞( H1 ) EOC L∞( L2 )• EOC

0 318 5.23 × 10−3 – 1.74 × 10−1 – 2.18 × 10−2 –

1 1,266 1.43 × 10−3 1.870 8.87 × 10−2 0.978 5.47 × 10−3 1.995

2 5,058 3.68 × 10−4 1.957 4.45 × 10−2 0.992 1.37 × 10−3 1.993

3 20,226 9.30 × 10−5 1.986 2.23 × 10−2 0.997 3.44 × 10−4 1.997

4 80,898 2.33 × 10−5 1.996 1.11 × 10−2 0.999 8.61 × 10−5 1.999

5 323,586 5.83 × 10−6 1.998 5.58 × 10−3 0.999 2.15 × 10−5 1.999

6 1,294,338 1.45 × 10−6 1.999 2.79 × 10−3 0.999 5.38 × 10−6 1.999

First experiment: We repeat the experiment conducted in [22, Example 1] with theimplicit midpoint rule. Let {T i

h (t)}6i=0 be a sequence of meshes on the surface Γ (t)

which is obtained through uniform refinement, i.e., hi ≈ 12 hi−1. We choose a time

step size τ0 = 0.125 in order to obtain at least the same accuracy as when using theLeapfrog method (cf. [22, Example 1]). We set τi = 1

2τi−1 to get the time step sizesequence {τi }6

i=0. For each mesh T ih together with the corresponding time step size τi ,

we solve the wave equation using the piecewise linear finite elements in combinationwith the implicit midpoint rule as described above. Then, we compute the error betweenthe lifted numerical solution and the exact solution for 0 ≤ t ≤ 1 in the followingnorms:

L∞(L2) : max0≤n≤N

‖unh − u(·, tn)‖L2(Γ (tn)),

L∞(H1) : max0≤n≤N

‖∇Γ (tn)unh − ∇Γ (tn)u(·, tn)‖L2(Γ (tn)),

L∞( L2 )• : max0≤n≤N

‖∂•h un

h − ∂•u(·, tn)‖L2(Γ (tn)).

Assuming Eri = C(hi + τi )E OC for the error, the experimental order of convergence

(EOC) is determined by

E OC = log Eri−1Eri

log 2, i = 1, . . . k.

In Table 1, we list the errors and the corresponding EOCs. As theoretically expectedfrom Theorem 7.1, we observe E OC ≈ 2 for the L∞(L2) and L∞( L2 )• norms,whereas E OC ≈ 1 for the L∞(H1) norm.

Second experiment: In this experiment, we examine the convergence of the GRKtime discretization with s-stages. We observed, when applying the 2-stage GRKmethod to the resulting ODE system after the space discretization by the evolvingsurface finite element method, the total error is dominated by the spatial error. Forthis reason, we shall compare the fully discrete solution with the exact solution ofthe ODE system. Since this solution is not available, we compute reference solutions

123

D. Mansour

10−3

10−2

10−1

100

10−1

2

10−1

0

10−8

10−6

10−4

10−2

100

Lea

pfr

og

Mid

po

int

2−G

RK

3−G

RK

Sp

atia

l err

or

10−3

10−2

10−1

100

10−1

2

10−1

0

10−8

10−6

10−4

10−2

100

Lea

pfr

og

Mid

po

int

2−G

RK

3−G

RK

Sp

atia

l err

or

10−3

10−2

10−1

100

10−1

2

10−1

0

10−8

10−6

10−4

10−2

100

Lea

pfr

og

Mid

po

int

2−G

RK

3−G

RK

Sp

atia

l err

or

Fig

.1E

rror

svs

.tim

est

epsi

zefo

rfo

urdi

ffer

entt

ime

disc

retiz

atio

nsc

hem

esas

wel

las

the

spat

iale

rror

attim

et=

1

123

Gauss–Runge–Kutta time discretization of wave equations

Fig. 2 Snapshots of the discretesolution of the wave equationusing the Leapfrog method (left)and the implicit midpoint rule(right). The exact solution isconstant: u = 10

qre f and pre f via the 3-stage GRK method with a small time step size τre f = 10−4.Next, we construct the time step size sequence {τi }7

i=0 by setting τi = 12τi−1 with

τ0 = 0.5. For each time step size τi , we solve the ODE system on the time interval0 ≤ t ≤ 1 to obtain the numerical solutions qτi and pτi using 4 different schemes,namely, the Leapfrog method, the implicit midpoint rule, the 2-stage and the 3- stageGRK methods. In Fig. 1, we plot the errors at time t = 1 versus the time step size inthe following norms

Error (M) :(〈qre f − qτi |M(t)|qre f − qτi 〉)1/2,

Error (A) :(〈qre f − qτi |A(t)|qre f − qτi 〉)1/2,

Error (M−1) :(〈pre f − pτi |M(t)−1|pre f − pτi 〉)1/2.

Additionally, we plot the spatial error which dominates the total error. For all schemes,we clearly observe that the experimental convergence rates in time match perfect withthe theoretical ones.

Example 2

In this example, we show how the time step size restriction can be inconvenient. Letus consider the start values u0 = 10 and u0 = 0. Then, independently of the choiceof the moving surface Γ (t), the exact solution of the wave equation (2.3) remains

123

D. Mansour

constant for all time (i.e., u(x, t) = 10). In Fig. 2, we show snapshots of the discretesolution at times t = 0, 1, 1.5, 2, 2.5 (from top to bottom). Since the exact solution isconstant, the torus is not supposed to change its color. On the left-hand side of Fig. 2,the discrete solution is obtained by using the Leapfrog method. We start with a smallstep size τ = 1

128 in order to fulfill the CFL-condition at time t = 0. However, dueto the movement of the mesh (i.e., h might decrease), there is no guarantee for theCFL-condition to remain fulfilled. E.g., see the last two snapshots on the left-hand sideof Fig. 2. One could, of course, use the smallest occurring h. However, in the presentexample, this does not give an accurate solution in a reasonable computing time. Onthe contrary, the implicit midpoint rule integrates this problem without difficulty. Bychoosing the time step size τ = 1

32 , it takes only a few seconds to integrate until t = 2.5and to obtain a good approximation of the exact solution as we clearly recognize fromthe right-hand side of Fig. 2.

Acknowledgments The author would like to thank Christian Lubich for introducing him to the topic ofthis paper, for the fruitful discussions and insightful suggestions.

References

1. Bastian, P., Blatt, M., Dedner, A., Engwer, C., Fahlke, J., Gräser, C., Klöfkorn, R., Nolte, M., Ohlberger,M., Sander, O. (2013) http://www.dune-project.org

2. Burrage, K., Hundsdorfer, W.H., Verwer, J.G.: A study of B-convergence of Runge–Kutta methods.Computing 36(1), 17–34 (1986)

3. Deckelnick, K., Elliott, C., Styles, V.: Numerical diffusion-induced grain boundary motion. InterfacesFree Bound. 3, 393–414 (2001)

4. Dekker, K., Kraaijevanger, J.F.B.M., Spijker, M.N.: The order of B-convergence of the GaussianRunge–Kutta method. Computing 36(1), 35–41 (1986)

5. Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptivescientific computing: abstraction principles and the DUNE-FEM module. Computing 90, 165–196(2010)

6. Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M. (2013) http://dune.mathematik.uni-freiburg.de7. Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27, 262–292

(2007)8. Dziuk, G., Elliott, C.M.: L2-estimates for the evolving surface finite element method. Math. Comp.

82, 1–24 (2013)9. Dziuk, G., Elliott, C.M.: A fully discrete evolving surface finite element method., vol. 50, No. 5,

pp. 2677–2694. ISSN 1095–7170 (2012)10. Dziuk, G., Elliott, C.M.: SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)11. Dziuk, G., Lubich, C., Mansour, D.: Runge–Kutta time discretization of parabolic differential equations

on evolving surfaces. IMA J. Numer. Anal. 32, 394–416 (2012)12. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer,

Berlin (1983)13. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. In: Structure-Preserving Algo-

rithms for Ordinary Differential Equations. 2nd ed. Springer, Berlin (2006)14. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems,

2nd edn. Springer, Berlin (1993)15. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and differential-algebraic

problems, 2nd edn. Springer, Berlin (1996)16. Halpern, D., Jenson, O.E., Grotberg, J.B.: A theoretical study of surfactant and liquid delivery into the

lung. J. Appl. Physiol. 85, 333–352 (1998)17. James, A., Lowengrub, J.: A surfactant-conserving volume-of-fluid method for interfacial flows with

insoluble surfactant. J. Comp. Phys. 201, 685–722 (2004)

123

Gauss–Runge–Kutta time discretization of wave equations

18. Kraaijevanger, J.F.B.M.: B-convergence of the implicit midpoint rule and the trapezoidal rule. BITNumer. Math. 25(4), 652–666 (1985)

19. Lenz, M., Nemadjieu, S.F., Rumpf, M.: A convergent finite volume scheme for diffusion on evolvingsurfaces. SIAM J. Numer. Anal. 49(1), 15–37 (2011)

20. Lubich, C., Ostermann, A.: Runge–Kutta methods for parabolic equations and convolution quadrature.Math. Comp. 60, 105–131 (1993)

21. Lubich, C., Ostermann, A.: Interior estimates for time discretizations of parabolic equations. Appl.Numer. Math. 18, 241–251 (1995)

22. Lubich, C., Mansour, D.: Variational discretization of linear wave equations on evolving surfaces.Math. Comp. (to appear)

23. Lubich, C., Mansour, D., Venkataraman, Ch.: Backward difference time discretization of parabolicdifferential equations on evolving surfaces. IMA J. Numer. Anal., (2013). doi:10.1093/imanum/drs044

24. Mansour, D.: Dissertation (PhD thesis), Univ. Tübingen (2013)25. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514

(2001)26. Memoli, F., Sapiro, G., Thompson, P.: Implicit brain imaging. Human Brain Mapp. 23, 179–188 (2004)27. Sanz-Serna, J.M., Verwer, J.G., Hundsdorfer, W.H.: Convergence and order reduction of Runge-Kutta

schemes applied to evolutionary problems in partial differential equations. Numer. Math. 50, 405–418(1987)

123