Abstract We present a new class of one-step, multi-value Exponential Integrator (EI)methods referred to as Exponential Almost Runge-Kutta (EARK) methods whichinvolve the derivatives of a nonlinear function of the solution. In order to approx-imate such derivatives to a sufficient accuracy, the EARK methods will be imple-mented within the broader framework of Exponential Almost General Linear Meth-ods (EAGLMs) to accommodate past values of this nonlinear function and becomingmultistep in nature as a consequence. Established EI methods, such as ExponentialTime Differencing (ETD) methods, Exponential Runge-Kutta (ERK) methods andExponential General Linear Methods (EGLMs) become special cases of EAGLMs.We present order conditions which facilitate the construction of two- and three-stageEARK methods and, when cast in an EAGLM format, we perform a stability analysisto enable a comparison with existing EI methods. We conclude with some numericalexperiments which confirm the convergence order and also demonstrate the compu-tational efficiency of these new methods.
Keywords Exponential integrators · Exponential almost Runge-Kutta · EARK ·Exponential almost general linear methods · EAGLM · Semilinear · Stiff PDEs
Over the last decade, we have observed significant interest in the construction ofExponential Integrator (EI) methods for semilinear initial-value problems (see [5, 11,16, 23]) of the form
y′ = Ly + N(t, y), y(0) = y0, (1.1)
where y ∈ R → Rd , L ∈ Rd × Rd and N(:) ∈ Rd represent the linear and nonlin-ear terms of the equation respectively. Spatial semi-discretisation of parabolic partialdifferential equations leads to systems of this type where d represents the number ofspatial grid points and matrix L will typically be large and sparse.
Exponential integrators are characterised both by their ability to exactly eval-uate the contribution of the linear part of the equation (i.e. if the nonlinear partis zero, then the numerical method simplifies to an evaluation of the exponentialfunction of L) and also by stability properties that are comparable to those of typ-ical implicit methods. We present a class of explicit exponential integrator meth-ods which when re-formulated into Exponential Almost General Linear Methods(EAGLMs) can be viewed as an extension of the explicit Exponential General Lin-ear Methods (EGLMs) [27] which combine the Exponential Runge-Kutta (ERK)methods [11, 15, 17] and Exponential Time Differencing (ETD) methods [5], pro-viding high-order methods with good stability properties for problems describedby (1.1).
We begin in Sect. 2 with a representation of the exact solution of (1.1) usingexponential-like functions referred to as ϕ-functions. Exponential Time Differencing(ETD) methods and Exponential Runge-Kutta (ERK) methods are reviewed brieflyin Sect. 3, where we also discuss how these methods can be combined to produce thefamily of Exponential General Linear Methods (EGLMs). For a more comprehensivereview of exponential integrators and their history, we refer the reader to Minchevand Wright [23].
In Sect. 4, we present a new class of methods, which we call Exponential Al-most Runge-Kutta (EARK) methods, which were motivated by the work on AlmostRunge-Kutta (ARK) methods by Butcher [8] and, in Sect. 5, we show how theycan be combined with ETD methods to produce a class of explicit Exponential Al-most General Linear Methods (EAGLMs). In Sect. 6, we study the stability prop-erties of the new EARK family of methods and establish their correspondence toEGLMs.
In Sect. 7, we consider the implementation costs of EARK methods when com-pared with the more established families of methods as this is an important consider-ation in the design of efficient algorithms. We conclude in Sect. 8 by presenting theresults of some numerical experiments applied to semi-discretised parabolic partialdifferential equations which support the convergence properties of these methods andalso demonstrate their computational performance (in terns of both accuracy and ef-ficiency) when compared with competitive exponential integrator methods over fixedintegration step sizes.
Exponential almost Runge-Kutta methods for semilinear problems 569
2 Exponential integrators and ϕ-functions
The exact solution of (1.1) is given by the variation of constants formula.
y(tn + h) = ehLyn +∫ h
0e(h−τ)LN
(tn + τ, y(tn + τ)
)dτ (2.1)
where h = tn+1 − tn is the integration step-size. If we substitute the Taylor expansionfor N(tn + τ, y(tn + τ)) about the point tn
∞∑m=0
τmN(m)(y(tn))
m! (2.2)
into (2.1), and define
ϕj (hL) = 1
hj
∫ h
0e(h−τ)L τ j−1
(j − 1)! dτ, (2.3)
we can then represent the exact solution for (1.1) by the expansion
y(tn+1) = ehLy(tn) +∞∑i=1
hiϕi(hL)N(i−1)(tn, y(tn)
)
where tn+1 = tn + h (see [23, Lemma 5.1]).A key element, and the main computational difficulty, in the implementation of
exponential integrators is the evaluation of the matrix exponential and exponential-like functions (2.3) referred to as ϕ-functions. An additional complication lies in theobservation that, if the matrix exponential or ϕ-function is computed explicitly, theresultant matrix will typically not retain any of the sparse properties of the origi-nal matrix [24], making the direct approach unsuitable for very large matrices dueto excessive storage requirements. As a consequence, a number of alternative ap-proaches have been developed which work with the application of the ϕ-functions ona vector, ϕj (A)×v, without generating ϕj (A) explicitly (see [1], for example). Algo-rithms which adopt this approach include Krylov subspace methods [12, 13, 25], realLeja points methods [2, 3, 9], contour integration methods based on rational approx-imations [18, 19, 28] and, more recently, contour approximation techniques whichemploy the Carathéodory-Fejér method [29]. We will conduct our numerical experi-ments in Sect. 8 using two of these approaches, namely the PHIPM implementationof a Krylov subspace-based approach by Niesen and Wright [25], and an implemen-tation utilising the real Leja points method by Caliari and Ostermann [9], which werefer to as ReLPM.
3 Multi-step and multi-stage methods
3.1 Exponential time differencing methods
Some of the earliest EI schemes to appear were members of the multistep ExponentialTime Differencing (ETD) family of methods [23]. Following the approach of Sect. 2,
570 J. Carroll, E. O’Callaghan
ETDs methods can be constructed by approximating N(tn + τ, y(tn + τ)) in (2.1) bythe Newton interpolation polynomial (2.2) up to the required order and then solvingthe resulting integral exactly. The simplest case is to take one term of the series andapproximate N(tn + τ, y(tn + τ)) by the constant value Nn = N(tn, yn), leading tothe one-step exponential time differencing Euler method:
yn+1 = ehLyn + hϕ1Nn, (3.1)
which is referred to as ETD1, or the ETD Euler method, as it reduces to the classicalEuler method when L = 0. Taking the first two terms of the series, i.e. approximatingN(tn + τ, y(tn + τ)) by a linear polynomial Nn + τ
h(Nn − Nn−1) results in the two-
step method:
yn+1 = ehLyn + h[ϕ1 + ϕ2]Nn − hϕ2Nn−1 (3.2)
which will be referred to as ETD2.
3.2 Exponential Runge-Kutta methods
Cox and Matthews [11] developed a 1-step Runge-Kutta-type extension to ETDmethods which they referred to as ETD Runge-Kutta Methods. These schemes arenow considered as belonging to the family of Exponential Runge-Kutta (ERK) meth-ods. The application of an ERK method to the general problem (1.1) results in thefollowing equations:
yn+1 = ehLyn + h
s∑i=1
bi(hL)Ki.
K1 = Nn = N(tn, yn)
Ki = N
(tn + cih, ecihLyn + h
i−1∑j=1
aij (cihL)Kj
), i = 2, . . . , s
where s is the number of stages.The order of a classical Runge-Kutta method can be established by comparing a
Taylor expansion of the exact solution y(tn + h) with its numerical approximationyn+1. This approach is more complicated for ERK methods arising from couplingconditions between the nonlinear and linear parts of the problem, despite the fact thatthe linear part has been solved exactly [21].
Definition 3.1 [20, Definition 1] An exponential method has stiff order p if the localerror has order p + 1 with respect to hn+1 when the method is applied to an abstractsemi-linear ODE (1.1) in which N(t, y(t)) is a sufficiently smooth function of t .
When deriving order conditions for constructing schemes of higher order, it issometimes not possible to satisfy all the conditions unless we weaken some of them.We will therefore make a distinction between schemes of strong and weak stiff order.
Exponential almost Runge-Kutta methods for semilinear problems 571
Definition 3.2 An exponential method is said to be of strong order p if it satisfiesthe p stiff order conditions for the general semi-linear ODE (1.1).
Definition 3.3 An exponential method is said to be of weak order p if it has strongorder p − 1 and satisfies the p stiff order conditions only in the weakened case whereL = 0.
Hochbruck and Ostermann [15] developed an approach based on trees for deriv-ing stiff order conditions using the elementary differentials of F(u) = Lu + N(u)
which arise from a Taylor expansion of the exact solution. Such an approachfacilitated the construction of conditions for arbitrary orders in a manner simi-lar to that for classical RK methods. An example of a strongly 2nd order, two-stage family of schemes is ERK22 [15, Scheme 5.3] parametrised by the free vari-able c2:
0 I
c2 c2ϕ1,2 ec2hL
ϕ1 − 1c2
ϕ21c2
ϕ2
(3.3)
In the construction of higher order methods, the number of conditions to be satis-fied grows rapidly. This makes it increasingly difficult to achieve higher orders, andthis difficultly makes the notion of weak order, as defined in Definition 3.3, very im-portant. For example, there are no strongly 3rd order 3-stage ERKs, only weakly 3rdorder methods.
For comparison purposes in numerical experiments to follow in Sect. 8, we willuse Krogstad’s 4-stage scheme [17]:
012
12ϕ1,2
12
12ϕ1,3 − ϕ2,3 ϕ2,3
1 ϕ1,4 − 2ϕ2,4 0 2ϕ2,4
ϕ1 − 3ϕ2 + 4ϕ3 2ϕ2 − 4ϕ3 2ϕ2 − 4ϕ3 −ϕ2 + 4ϕ3
(3.4)
We refer to this scheme as ERK34. It is a strongly 3rd order, weakly 4th order schemeand was consistently the best performing 4-stage scheme among those considered byHochbruck and Ostermann [15].
3.3 Exponential general linear methods
General Linear Methods (GLMs) are a class of multi-step, multi-stage explicit meth-ods introduced by Butcher [6]. They are a generalisation of linear multi-step methodswith the multi-stage nature of Runge-Kutta methods. The advantage which they of-fer is that they allow one to easily construct higher-order schemes while retaining aninherent Runge-Kutta stability [7].
572 J. Carroll, E. O’Callaghan
Ostermann, Thalhammer and Wright [27] introduced a class of explicit Expo-nential General Linear Methods (EGLMs) based on the Adams-Bashforth schemes.EGLMs are an extension of GLMs into the exponential framework and con-tain, as special cases, the ETD and ERK method families. For given start valuesy0, y1, . . . , yq−1, the internal stages Ki are defined by
Ki = N
(ehLyn + h
i−1∑j=1
aij (hL)Kj + h
q∑j=1
uij (hL)Nn−j
)
for 1 < i ≤ s, and the numerical approximation yn+1 at time tn+1 is given by
yn+1 = ehLyn + h
s∑i=1
bi(hL)Ki + h
q∑j=1
vi(hL)Nn−j
Following the ERK notation, we identify EGLMs through the use of the sub-scripts p, s and q , where p is the order of the method, s represents the num-ber of stages while q is the number of steps. Ostermann, Thalhammer and Wright[27] concluded that EGLMs, like their classical counterparts, combine the ease ofconstruction of high-order methods with the superior stability properties of ERKmethods. They derived order conditions for 2-stage schemes [27, (2.3) and (2.7)]and considered the case where c2 = 1. This choice for c2 reduces the numberof distinct ϕ’s thereby improving the computational efficiency of the schemes.These schemes are referred to as EGLM322 [27, Table 4.1] and EGLM423 [27, Ta-ble 4.2].
They are particular cases of the more general families parametrised by c2 [26]namely the strongly 3rd order family EGLM322c2 family of methods:
c2 a2,1 ec2hL u2,1
b1 b2 ehL v1
a2,1 = c2ϕ1,2 + c22ϕ2,2
b1 = ϕ1 + c2 − 1
c2ϕ2 + −2
c2ϕ3 b2 = 1
c22 + c2
ϕ2 + 2
c22 + c2
ϕ3
u2,1 = −c22ϕ2,2 v1 = −c2
c2 + 1ϕ2 − 2
c2 + 1ϕ3
(3.5)
and the strongly 4th order EGLM423c2 family of methods:
c2 a21 ec2hL u21 u22
b1 b2 ehL v1 v2
Exponential almost Runge-Kutta methods for semilinear problems 573
a2,1 = c2ϕ1,2 + 3c22
2ϕ2,2 + c3
2ϕ3,2
u2,1 = −2c22ϕ2,2 − 2c3
2ϕ3,2 u2,2 = c22
2ϕ2,2 + c3
2ϕ3,2 (3.6)
b1 = ϕ1 +3c2−2
2 ϕ2 + c2 − 3ϕ3 − 3ϕ4
c2b2 = 2ϕ2 + 6ϕ3 + 6ϕ4
c32 + 3c2
2 + 2c2
v1 = −2c2ϕ2 − 2c2 − 4ϕ3 + 6ϕ4
c2 + 1v2 =
c22 ϕ2 + c2 − 1ϕ3 − 3ϕ4
c2 + 2
4 Exponential almost Runge-Kutta methods
Almost Runge-Kutta (ARK) methods were introduced by Butcher in 1997 [8]. Theyare a special case of GLMs in that they retain the multi-stage nature of RK methodsbut allow for the passing of more than one value from step to step. In this way, theyare multi-value rather than multi-step schemes. For ARK methods, three values formthe inputs and outputs at each step, namely the approximation to the solution, togetherwith approximations to its first and second derivatives.
However, in the design of an Exponential Almost Runge-Kutta (EARK) method,the input and output values passed from step-to-step are the function evaluations ofthe approximate solution, Nn+1, together with the derivatives N
(i)n+1 of increasing
order. Using a similar notation to EGLMs, we will use the subscripts p, s, q andr to denote a p-order, q = 1-step s-stage, r-value method. The general form of anEARKpsqr tableau is:
c2 a21 w21 · · · w2r
......
. . ....
...
cs−1 as−1,1 · · · as−1,s−2) ws−1,1 · · · ws−1,r
1 b1 · · · bs−1 0 z1 · · · zr
b1 · · · bs−1 0 z1 · · · zr
β11 · · · β1,s−1 β1s δ11 · · · δ1r
......
......
...
βr1 · · · βr,s−1 βrs δr1 · · · δrr
(4.1)
The wij and zi entries represent the ϕ-functions operating on the incoming deriva-
tives approximations, N(i)n , while the β and δ entries are scalars used in the approxi-
mation of the outgoing derivative approximations.As an illustration, we will consider a 3-stage EARK method. The repetition in the
final row of the tableau’s upper half and the first row of its lower half means thatan s-stage EARK method has more in common with an (s − 1)-stage EGLM. Thesimplest form of an EARK method is a 3-stage, 1-value method whose tableau has
574 J. Carroll, E. O’Callaghan
the form:
c2 a21 w21
1 b1 b2 z1
b1 b2 0 z1
β21 β22 β23 δ21
representing the equations
K1 = Nn = N(tn, yn)
Y2 = ec2hLyn + h(a21K1 + w21hN ′
n
)K2 = N(tn + c2h,Y2)
Y3 = ehLyn + h(b1K1 + b2K2 + z1hN ′
n
)yn+1 = Y3
Nn+1 = N(tn + h,Y3)
hN ′n+1 = β21K1 + β22K2 + β23Nn+1 + δ21hN ′
n
(4.2)
We can see from (4.2), that the value of Y3 can be reused for the output values yn+1
and Nn+1, as such, the final extra row on the upper tableau does not have any detri-mental effect on performance.
To produce the vector of outgoing approximations, v = (N ′n+1, . . . ,N
(r)n+1), we
construct a matrix M = (β δ), where β and δ, from the lower section of (4.1), arer × s and r × r matrices respectively, such that,
M
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Nn
Nn+c2...
Nn+cs
Nn+1hN ′
n...
hP N(r)n
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
=⎛⎜⎝
hN ′n+1...
hP N(r)n+1
⎞⎟⎠
This matrix is constructed by solving a set of linear equations generated from Taylorexpansions of the elements of the incoming vector of approximations, together withthe Nn+ci
and Nn+1 values produced by the internal stages. Specifically, mj , the j throw of the matrix M , is the solution to the linear system
XmTj = ej+1
Exponential almost Runge-Kutta methods for semilinear problems 575
where ei is the standard basis vector and the (r + s) × (r + s) matrix X = [W, Z]where,
We summarise the convergence properties of EARK methods using the followingresults.
Theorem 4.1 EARK methods of the form (4.1) will be exact in the case Nn = N , aconstant, if the following conditions are met
i−1∑j=1
aij = ciϕ1,i (4.3a)
∑i
bi = ϕ1 (4.3b)
for i = 1, . . . , s.
Proof See [26, Theorem 13]. �
Theorem 4.2 EARK methods of the form (4.1) where s = 2 or 3, can achieve orderp if they satisfy (4.3a), (4.3b), and
j−1∑k=2
ajk
cik
i! + wki = ci+1k ϕi+1,k
576 J. Carroll, E. O’Callaghan
for i = 1, . . . , p − 3, j = 1, . . . , s,
s∑j=2
bj
(j−1∑k=2
ajk
cik
i! + wji − ci+1j ϕi+1,j
)= 0
for i = p − 2, and
s∑j=2
cij
i! bj + zi = ϕi+1
for i = 1, . . . , p − 1.
Proof See [26, Theorem 13]. �
If we pass the two values hN ′ and h2N ′′ from step to step we can construct the 4thorder family of schemes knows as EARK4312c2 , which is parametrised by the freevariable c2:
c2 c2ϕ1,2 c22ϕ2,2 c3
2ϕ3,2
1 ϕ1 − 6c3
2ϕ4
6c3
2ϕ4 0 ϕ2 − 6
c22ϕ4 ϕ3 − 3
c2ϕ4
ϕ1 − 6c3
2ϕ4
6c3
2ϕ4 0 ϕ2 − 6
c22ϕ4 ϕ3 − 3
c2ϕ4
− 2 c22−3 c2
c22−2 c2+1
− 1c2
3−2 c22+c2
2 c2+1c2
− c2c2−1 0
2 c22−6
c22−2 c2+1
− 4c2
3−2 c22+c2
2 c2+4c2
− 2 c2+2c2−1 0
(4.4)
Numerical experiments have shown that a value of c2 = 0.75 in EARK4312c2 isthe optimal choice in terms of accuracy against stepsize performance. The choice ofc2 = 1 would allow for a more efficient implementation. However, this would result inthe stage approximation at t = tn+c2 and the method approximation at t = tn+1 beingevaluated at the same point in time. In (4.4), this results in division-by-zero withinthe 2nd and 3rd lower tableau rows. To construct an EARK scheme with c2 = 1, anew lower tableau section must be derived, namely one which does not involve K2,i.e. a scheme where β12 = β22 = 0:
ϕ1 − 6c3
2ϕ4
6c3
2ϕ4 0 ϕ2 − 6
c22ϕ4 ϕ3 − 3
c2ϕ4
−2 0 2 −1 0
−2 0 2 −2 0
(4.5)
Note: (4.5) shows only the lower section of the tableau—the upper section is identicalto that of (4.4).
To study the accuracy performance of both (4.4) and (4.5), we conduct a fixedstepsize numerical experiment for the Brusselator System [14], described in greaterdetail in Sect. 8.1. For comparison purposes, we also include EGLM423c2 (3.6) withc2 = 1, as it’s the most computationally efficient 4th order EGLM.
Exponential almost Runge-Kutta methods for semilinear problems 577
Fig. 1 Fixed step integration of Brusselator System
From Fig. 1(a), we can see that the choice of c2 = 1 allows for a significantly fasterimplementation of (4.4). However, Fig. 1(b) shows that the modified lower tableau(4.5) has caused a serious impact on accuracy. This is because the approximationsto hN ′ and h2N ′′ are calculated by (4.5) to a lower order than that provided for inthe lower tableau section of (4.4) and, as a consequence, compromise the scheme’saccuracy.
If we wish to persevere with the more computationally-efficient choice of c2 = 1,we will need additional data to produce sufficiently high-order approximations. Anobvious source for such additional data are previous values of N . To this end, weintroduce a broader framework of multi-stage, multi-step and multi-value methods,providing the flexibility required to construct EARK methods which exhibit highaccuracy without compromising implementation efficiency.
5 Combining EARK methods with EGLMs
5.1 Exponential almost general linear methods
To begin constructing schemes of a multi-stage, multi-step and multi-value nature, itis necessary to use an expanded tableau to represent all method families consideredso far. This representation gives rise to a broader family of methods, which we referto as Exponential Almost General Linear Methods (EAGLMs):
for 1 ≤ i ≤ s. Such methods combine the multi-stage, multi-step nature of EGLMswith the multi-value format of EARK methods. We summarise the convergence prop-erties of EAGLMs using the following results.
Theorem 5.1 EAGLMs of the form (5.1) will be exact in the case Nn = N , a constant,if the following consistency conditions are met
i−1∑j=1
aij +q∑
j=1
uij = ciϕ1,i (5.3a)
for i = 1, . . . , s, and
s∑i
bi +q∑
i=1
vi = ϕ1 (5.3b)
Proof See [26, Theorem 3]. �
Theorem 5.2 EAGLMs of the form (5.2), where s = 2 or 3, that is 2- or 3-stagemethods, that meet the consistency conditions (5.3a), (5.3b), can achieve order p if
j−1∑k=2
ajk
cik
i! +q∑
k=1
(−k)i
i! ukj + wki = ci+1k ϕi+1,k (5.4a)
for i = 1, . . . , p − 3, j = 1, . . . , s,
s∑j=2
bj
(j−1∑k=2
ajk
cik
i! +q∑
k=1
(−k)i
i! ujk + wji − ci+1j ϕi+1,j
)= 0 (5.4b)
where i = p − 2, and
Exponential almost Runge-Kutta methods for semilinear problems 579
s∑j=2
cij
i! bj +q∑
j=1
(−j)i
i! vj + zi = ϕi+1 (5.4c)
for i = 1, . . . , p − 1.
Proof See [26, Theorem 4]. �
In what follows, we will concentrate on 2-stage schemes:
c2 a21 ec2hL u21 u22 w21 w22
b1 b2 ehL v1 v2 z1 z2
K1 = Nn = N(tn, yn)
K2 = N(tn + c2h, ec2hLyn + h
[a21K1 + u21Nn−1 + u22Nn−2 + w21hN ′
n
+ w22h2N ′′
n
])for which we can simplify conditions (5.4a), (5.4b), (5.4c) to
j−1∑k=2
ajk
cik
i! +q∑
k=1
(−k)i
i! ukj + wki = ci+1k ϕk,i+1
for 1 < i ≤ 2, 1 ≤ j ≤ i, and
ci2
i! b2 +q∑
j=1
(−j)i
i! vj + zi = ϕi+1
for 1 ≤ i ≤ 2.We consider a particular form of EAGLMs whose structure is much closer to
EARK methods than to EGLMs. For such schemes, the uij or vi tableau entries arezero and therefore the previous timesteps are not involved in ϕ-function operations.The lower section of the tableau has non-zero γij entries and, as such, uses infor-mation from the previous timesteps in the approximation of the derivatives. We haveadopted the convention of referring to such schemes as EARKpsqr methods. Thisemphasises the EARK structure of the upper tableau while retaining the q subscriptto indicate the number of steps.
Within this restricted format, we present the strongly 3rd Order family of methods:EARK3221c2
c2 c2ϕ1,2 0 c22ϕ2,2
1 ϕ1 − 2ϕ3
c22
2ϕ3
c22
0 ϕ2 − 2ϕ3c2
ϕ1 − 2ϕ3
c22
2ϕ3
c22
0 0 ϕ2 − 2ϕ3c2
−2 0 32
12 0
(5.5)
580 J. Carroll, E. O’Callaghan
EARK4232c2 , the strongly 4th Order family of methods:
c2 c2ϕ1,2 0 0 c22ϕ2,2 c3
2ϕ3,2
1 ϕ1 − 6c3
2ϕ4
6c3
2ϕ4 0 0 ϕ2 − 6
c22ϕ4 ϕ3 − 3
c2ϕ4
ϕ1 − 6c3
2ϕ4
6c3
2ϕ4 0 0 0 ϕ2 − 6
c22ϕ4 ϕ3 − 3
c2ϕ4
−3 0 116
32 − 1
3 0 0−5 0 2 4 −4 0 0
(5.6)
We will also restrict our attention to the choice c2 = 1 as this allows for the mostefficient implementation.
5.2 Equivalence between EGLMs and EARK methods
We will illustrate the equivalence between EGLMs and EARK methods, by compar-ing EGLM322c2 (3.5) with EARK3221c2 (5.5). The EGLM322c2 (3.5) equations are
K2 = N(tn + c2h, ec2hLyn + h
[(c2ϕ21 + c2
2ϕ22)Nn − c2
2ϕ22Nn−1])
yn+1 = ehLyn + h
[(ϕ1 + c2 − 1
c2ϕ2 + −2
c2ϕ3
)Nn +
(1
c22 + c2
ϕ2 + 2
c22 + c2
ϕ3
)K2
+( −c
c2 + 1ϕ2 − 2
c2+1ϕ3
)Nn−1
]
and we can see that both Nn and Nn−1 are multiplied by a linear combination of ϕ-functions. If we then consider the EARK3221c2 (5.5) scheme, when written explicitlyin the form of (5.2), we obtain
K2 = N(tn + c2h, ec2hLyn + h
[c2ϕ21Nn − c2
2ϕ22hN ′n
])(5.7)
yn+1 = ehLyn + h
[(ϕ1 − 2
c22
ϕ3
)Nn + 2
c22
ϕ3K2 +(
ϕ2 − 2
c2ϕ3
)hN ′
n
](5.8)
If we then approximate N ′n by 3
2Nn − 2Nn−1 + 12Nn−2, we can see that (5.7) and
(5.8) become
Y2 = ec2hLyn + h
[c2ϕ21Nn − c2
2ϕ22
(3
2Nn − 2Nn−1 + 1
2Nn−2
)]
= ec2hLyn + h
[(c2ϕ21 − 2
3c2
2ϕ22
)Nn + 2ϕ22c
22Nn−1 − 1
2ϕ22c
22Nn−2
]
K2 = N(tn + c2h,Y2)
yn+1 = ehLyn + h
[(ϕ1 + 3
2ϕ2 − 3c2 + 2
c22
ϕ3
)Nn + 2
c22
ϕ3K2
+(
4
c2ϕ3 − 2ϕ2
)Nn−1 +
(1
2ϕ2 − 1
c2ϕ3
)Nn−2
]
Exponential almost Runge-Kutta methods for semilinear problems 581
This is a 3rd order EGLM, albeit, in this case, one which uses two previous stepswithin the method rather than just one as (3.5) does.
6 Stability analysis
6.1 Stability of ERK methods
Cox and Matthews [11] studied the stability of several 2nd-order schemes and in par-ticular, some linearly-implicit schemes such as an Adams-Moulton/Adams-Bashforthmethod, and the standard Integrating Factor methods [22]. These are compared witha number of ETD and ERK methods. Following [5], we begin with the model non-linear, scalar, autonomous ODE,
u′ = αu + f (u) (6.1)
and linearise (6.1) about a fixed point u0, such that αu0 + f (u0) = 0 leading to
u′ = αu + λu (6.2)
where u is now the perturbation to u0 and λ = f ′(u0). The fixed point u0 is stable if
Re(α + λ) < 0
for all eigenvalues λ. It must be noted that this technique can only provide a compar-ison of the relative stabilities of various schemes. In addition, the stability analysisfor a scalar test equation is of limited significance for parabolic PDEs.
In general, both α and λ are complex so the resulting stability region is four-dimensional. Following [11], in order to plot two-dimensional stability regions, weconsider two cases. First, by assuming that λ is complex and α is fixed, negative andreal, we can plot the resulting stability regions in the complex plane. Second, we lookat the case where α is not fixed and both α and λ are real.
The analysis is performed by applying a scheme to the test problem (6.2) withλu regarded as the nonlinear term. For ERK22 (3.3) with c2 = 1, this results in thefollowing expression:
un+1 = un ehα + h
(λ (ehα−1
hα− 1) (un ehα + λun (ehα−1)
α)
hα
)
+ h
(λun
(ehα − 1
hα−
ehα−1hα
− 1
hα
))
By setting r = un+1/un, x = hλ and y = hα, we then obtain
r = (x y + x2) e2y + (y3 + (−x2 − 2x)y − 2x2) ey + (x2 + x)y + x2
y3(6.3)
In the first case, we will fix y < 0 ∈ R and plot the boundary of the stability regionwhich occurs when r = 1. To plot this in the complex plane, we set r = eiθ and solve
582 J. Carroll, E. O’Callaghan
Fig. 2 ERK22 (3.3) stability boundaries
Fig. 3 ETD2 (3.2) stability boundaries
for x on the interval θ ∈ [0,2π). Figure 2(a) shows the stability region boundariesfor ERK22 (3.3) when y = −1,−2,−5 respectively.
For the second case, we fix r = 1, and plot the growth of the real extents of thestability regions against varying y. Under this restriction, the solutions to (6.3) are
x = − y2
ey − y − 1x = −y
and Fig. 2(b) shows a graph of those solutions. We can see that, as y grows in magni-tude, the stability region’s real extents also grow approximately linearly. Figure 3(a)plots boundary curves for ETD2 (3.2). The region of stability is significantly smallerthan that of ERK22 (3.3) and Fig. 3(b) highlights the slower growth of the real extentsof that stability region as y grows.
Krogstad [17] investigated the stability regions of a number of 4th order schemes,notably that of ERK44 (3.4), and some multi-step generalisation of IF methods, lead-ing to the conclusion that (3.4) had the largest stability region. Maset and Zennaro[20] studied the requirements for unconditional stability in ERK methods. Focusing
Exponential almost Runge-Kutta methods for semilinear problems 583
Fig. 4 The stability boundaries for ERK methods (3.3) and (6.4)
Fig. 5 EGLM322c2 (3.5) stability boundaries
on ERK22 (3.3) and a companion scheme, ERK12:
0 I
c2 c2ϕ1,2 ec2hα
(1 − 12c2
)ϕ11
2c2ϕ1
(6.4)
they demonstrated that the schemes are unconditionally stable when c2 ≥ 1 andc2 ≥ 1
2 respectively. Looking at the stability plots of both schemes for varying c2at a fixed α = 10 we can that see the stability regions in Figs. 4(a) and 4(b) exceed α
when the conditions on c2, as derived by Maset and Zennaro, are satisfied.
6.2 Stability of EGLMs
EGLMs can also be analysed with this approach. Studying the boundary plots of onesuch method, EGLM322c2 (3.5), we can see that, in Fig. 5(a), the stability regionextends back into the negative complex plane. This is a property observed for ERK2,but not for ETD2 (see Figs. 2(a) and 3(a)).
Figure 5(b) shows the desired linear growth of the EGLM322c2 (3.5) stability re-gion’s real extents.
584 J. Carroll, E. O’Callaghan
Fig. 6 EARK3221c2 (5.5) stability boundaries
Fig. 7 EGLM322c2 (3.5) and EARK3221c2 (5.5) side-by-side stability boundaries
6.3 Stability of EAGLMs
This method of analysis can not be applied directly to EARK methods due to thecomplexity arising from the presence of the derivatives of N . Instead we performthe analysis on the EAGLM formulation of EARK methods where we use values ofN at previous timesteps to approximate the derivatives of N needed by individualmethods. Arising from this, some clear inferences may be made relating to the corre-sponding stability properties of EARK methods. Figure 6 plots the stability regionsfor EARK3221c2 (5.5) with c2 = 1. Note that the growth of the stability regions isagain linear with α. In this particular case, the value of N ′
n was approximated by−2Nn−2 + 1
2Nn−1 + 32Nn = N ′
n + O(h3).To gain a better understanding of the comparative stabilities between EGLMs
and EAGLMs, we plot the stability regions together. Figure 7(a) shows the stabil-ity regions of the two primary 3rd order schemes, namely EGLM322c2 (3.5) andEARK3221c2 (5.5) for fixed hα = y = 20. It is evident that the EAGLM scheme hasa smaller stability region than the EGLM. However, we will see in the next section,that EAGLMs have a lower computational cost per step over EGLMs when measur-ing ϕ× vector operations (see Tables 2 and 3). Therefore, we also plot both schemes
Exponential almost Runge-Kutta methods for semilinear problems 585
with the EARK3221c2 region scaled by the savings in ϕ-vector products. This is simi-lar to a technique used by Butcher [7] where the stability regions were scaled relativeto the number of internal stages of the classical schemes being compared. Figure 7(b)shows the new stability region and we can see now that the real axis stability extentsof the EAGLM scheme exceed those of the EGLM scheme.
7 Computational cost
The number of distinct ϕ-functions required by exponential integrator methods is akey parameter when making a relative comparison of their computational efficien-cies. In Sect. 2, we noted some important algorithms developed for ϕ-function evalu-ation and, in particular, identified an approach common to many of those algorithms,namely the operation of the ϕ-function upon a vector.
Ostermann, Thalhammer and Wright [27] counted separately the numbers of ϕ-functions and ϕ(A) × v operations for a number of EI schemes [27, Table 5.1] toprovide a relative indication of where the EGLM schemes ranked in terms of com-putational efficiency. We will follow this approach when comparing the relative effi-ciencies of the EARK methods and EAGLMs.
Niesen and Wright [25] generalised the operation of a ϕ-function upon a vectorinto the implementation of a linear combination of ϕ-functions upon input vectorsv0, . . . , vp:
eAv0 + ϕ1(A)v1 + ϕ2(A)v2 + · · · + ϕp(A)vp (7.1)
in an single operation. To take this approach into account, we include a third count inour efficiency comparison, namely the number of linear combination function callsnecessary to implement each scheme.
As an illustration, we present in detail the evaluation of these different efficiencymeasures when applied to the ERK scheme, ERK33 [15, Scheme 5.8]:
0 I
c2 c2ϕ1,2 ec2hL
23
23ϕ1,3 − 4
9c2ϕ2,3
49c2
ϕ2,3 e23 hL
ϕ1 − 32ϕ2 0 3
2ϕ2
(7.2)
– Explicit Evaluation. This is the approach used by the EXPINT package [4], whichemploys Padé approximations. When the ϕ-functions are computed explicitly, theycan be used for a number of matrix-vector products at negligible cost. As such, acount of the number of distinct matrix exponentials and ϕ-functions provides agood relative indicator of a scheme’s complexity. Also, in what follows, we exploita key optimisation applicable to the majority of EI schemes, using the identity:
ecihL × yn + hciϕ1,i (hL) × v = yn + hciϕ1,i (hL) × (Lyn + v) (7.3)
to eliminate the need to perform the matrix-exponential by a vector operation,(expm × vector) and this gives a significant performance boost. Under this mea-sure, ERK33 (7.2) requires 5 ϕ-function evaluations.
586 J. Carroll, E. O’Callaghan
– ϕ× vector Operations. We noted in Sect. 2 that a number of techniques have beendeveloped which work with the operation of the ϕ-function upon a vector. Forsome schemes, it may be necessary to compute a ϕ× vector product with the sameϕ-function at two different stages within a step. Therefore, the number of distinctϕ-functions represents only a lower bound on the estimated number of operationsnecessary to compute one step. In practice, each scheme must be considered sep-arately to determine the true and optimal number of operations needed. A pseudocode example illustrates this procedure:
This algorithm performs a single step for ERK33 (7.2). There are a total of sevenmatrix-vector operations. By incorporating the (7.3) optimisation, we can reducethis to 5 operations in total:
– ϕ× vector Linear Combinations. Finally, this is the technique introduced by thePHIPM code [25] whereby each stage can be completed in a single operation. ForERK33 (7.2), the pseudo code algorithm is:
1. M1 ← (yn Nn )
2. K2 ← N(tn + c2h,phipm(M1))
3. M2 ← (yn Nn−19c2
N2 + 19c2
K2 )
4. K3 ← N(tn + c2h,phipm(M2))
5. M3 ← (yn Nn−32 N2 + 3
2K2 )
6. yn+1 ← phipm(M3)
The operations to construct the Mi matrices, whose dimensions we denote as(n,mi), are computationally negligible. In practice, any of the ϕ× vector tech-niques could be modified to take Mi as a parameter. However, each vector wouldneed to be processed individually and therefore the cost of processing the matrixwould scale linearly by the number of vectors, mi . The phipm(Mi) operation costis independent of mi and therefore, we simply count the number of stages in thescheme. Under this measurement, ERK33 (7.2) receives a count of 3 operations.
Table 1 summarises the computational cost analysis for the ETD schemes (3.1)and (3.2) and the ERK schemes (7.2) and (3.4), under each of the three categoriesof ϕ-function evaluation. ETD Euler is 1-stage and requires only one ϕ-function. It’scomputational cost, therefore, is 1 under each of the categories and it serves as auseful baseline. The cost should be interpreted as a relative measure of the CPU timeneeded to take a time step. For example, under the explicit evaluation of ϕ-functions,we would expect ERK44 (3.4) to take 3 times longer to complete the same number oftime steps as ETD2 (3.2) (the ratio of distinct ϕ’s is 6:2).
Exponential almost Runge-Kutta methods for semilinear problems 587
Table 1 Relative performance measure for the 2- and 3-stage ERKs
Scheme Explicit evaluation ϕ× vector Linear combination
ETD Euler (3.1) 1 1 1
ETD2 (3.2) 2 2 1
ERK33 (7.2) 5 5 3
ERK44 Krogstad (3.4) 6 6 4
Table 2 Relative performance measure for the four EGLMs
Scheme c2 Explicit evaluation ϕ× vector Linear combination
EGLM322c2 (3.5) c2 = 1 5 5 2
EGLM322c2 (3.5) c2 = 1 3 4 2
EGLM423c2 (3.6) c2 = 1 7 7 2
EGLM423c2 (3.6) c2 = 1 4 6 2
EGLMp2(p−1)c2 c2 = 1 2p − 1 2p − 1 2
EGLMp2(p−1) c2 = 1 p 2p − 2 2
Table 3 Relative performance measure for the four EAGLMs
Scheme c2 Explicit evaluation ϕ× vector Linear combination
EARK3221c2 (5.5) c2 = 1 5 5 2
EARK3221c2 (5.5) c2 = 1 3 3 2
EARK4232c2 (5.6) c2 = 1 7 7 2
EARK4232c2 (5.6) c2 = 1 4 4 2
acEARKp2(p−1)(p−2)c2 c2 = 1 2p − 1 2p − 1 2
EARKp2(p−1)(p−2) c2 = 1 p p + 2 2
We provide similar summaries for the EGLM and EARK/EAGLM schemes re-spectively. In addition, we can deduce the costs for the higher order 2-stage schemesin both families.
The conclusion which we can draw from Tables 2 and 3 is that, under the ϕ× vec-tor operation count, the EARK methods are more efficient that their EGLMs coun-terparts for the c2 = 1 case. In addition, the efficiency gap widens with higher-orderschemes since, for a single increase in order, the EGLMs incur two additional ϕ×vector operations while the EARK methods incur only one.
8 Numerical experiments
To highlight the competitiveness of EARK methods, we present experimental bench-marks against a number of established parabolic PDE test problems. All experimentswere performed in Matlab 2011b 64bit running on Windows 7 × 64. The CPU wasan Intel Core 2 Quad Q9450 clocked at 2.66 GHz and the system had 8GB of RAM.
588 J. Carroll, E. O’Callaghan
For each problem, we perform two numerical experiments, one using the 3rd or-der schemes (ERK33 (7.2), EGLM322 (3.5) with c2 = 1 and EARK3221 (5.5) withc2 = 1) and the second using the 4th order schemes (ERK44 (3.4), EGLM423c2 (3.5)with c2 = 1 and EARK4232c2 (3.6) with c2 = 1). In each case, we plot two mea-surements. The first pair of figures plot step-size against accuracy which illustratesthe differences in the per-step accuracy differences between schemes of the sameconvergence order. The second set of figures plot computational cost against accu-racy. First, we use the phipm implementation which provides a relative picture ofthe cost of each scheme when implemented with the linear combination approachto ϕ-function evaluations (7.1). Secondly we plot the same measurements using theReal Leja Points Method (ReLPM) [9]. This illustrates a scheme’s efficiency whenimplemented with ϕ× vector operations.
8.1 Brusselator system
The Brusselator System [14]
u′i = 1 + u2
i vi − 4ui + α(N + 1)2(ui−1 − 2ui + ui+1)
v′i = 3ui + u2
i vi + α(N + 1)2(vi−1 − 2vi + vi+1)
on the interval x ∈ [0,1] and t ∈ [0,15], with α = 1/50 and initial conditions,
ui(0) = 1 + sin(2πx)
vi(0) = 3
models diffusion in a chemical reaction. The system is discretised in space, withM = 127 resulting in 2M equations. The Jacobian is banded with a constant width5 if the equations are ordered as u1, v1, u2, v2, . . . . As M increases, the problembecomes increasingly stiff.
8.2 The Allen-Cahn equation
The Allen-Cahn equation [10] is a 1D parabolic PDE
yt = λyxx + y − y3,
on the interval x ∈ [−1,1] and t ∈ [0,50], with initial condition,
y(0, x) = 0.53x + 0.47 sin(−1.5πx)
and boundary conditions,
y(t,−1) = −1, y(t,1) = 1
We make use of the implementation supplied by the EXPINT Matlab package [4,Sect. 4.2.6] which discretises the linear part, λwxx , using a Chebyshev differentiationmatrix to produce a dense L matrix.
Exponential almost Runge-Kutta methods for semilinear problems 589
Table 4 Memory usage in megabytes (MBs)
Dimension 32 64 128 256 512
ODE15s 52 595 3950 N/A N/A
PHIPM 53 71 170 585 1382
ReLPM 25 63 147 533 1094
8.3 The RDA 2D equation
The 2D RDA is a stiff problem presented by Caliari and Ostermann [9]. The standardsemi-discretisation using finite differences on a uniform spatial mesh gives rise toan L matrix which is pentadiagonal in structure. The equation describes reaction-diffusion-advection
ut = ε(uxx + uyy) − α(ux + uy) + ρu
(u − 1
2
)(1 − u)
with x ∈ [0,1]2 and t ∈ [0,0.3], subject to homogeneous Neumann boundary condi-tions and initial condition,
u(t = 0, x, y) = 0.3 + 256(x(1 − x)y(1 − y)
)2.
We will run our experiments for ε = 0.05, α = −1 and ρ = 100, the same combi-nation of parameters as those used by Caliari and Ostermann [9]. The problem isdiscretised in space, with M = 31. Being a 2D problem, the L matrix can be verylarge, even for moderately coarse mesh discretisation and, depending on the integra-tor, can have excessive memory requirements. For example, in Table 4 we list thememory required for different problem discretisations, comparing EIs to Matlab’sODE15s. It is clear that memory usage is significantly lower for both the PHIPM andReLPM approaches to implementing an EI.
8.4 Interpreting the results
Looking at the plots, we can draw a number of conclusions relating to the EARKmethods. Figures 8, 10 and 12 plot accuracy against stepsize for each scheme whenapplied to the three test problems respectively. The EARK schemes demonstrategreater accuracy over the EGLMs, and equal or surpass the performance of the ERKs.Figures 9, 11 and 13 plot accuracy against CPU timings using firstly PHIPM andsecondly ReLPM. With PHIPM, the competitive accuracy demonstrated by the ERKmethods is offset by their higher computational costs and cause them to significantlyunder-perform the EARK methods. The EGLMs, like the EARK methods, are 2-stageschemes and so the computational costs of the two families are nearly identical. Thesuperior accuracy, therefore, of the EARK methods remains their key differentiatingfactor in outperforming EGLMs. For the ReLPM experiments, the fewer ϕ× vectoroperations needed to implement the EARK methods results in additional performanceimprovements when compared to both the ERK methods and EGLMs. Tables 5, 6 and7 provide global error and cpu measurements from the experiments which reinforcethe conclusions drawn from the figures.
590 J. Carroll, E. O’Callaghan
Fig. 8 Step-size against accuracy for the Brusselator system
Fig. 9 Brusselator system CPU timing
9 Conclusions
The ARK motivation of including the derivatives of N in the construction of expo-nential integrator Almost Runge-Kutta (EARK) schemes results in a very compactstructure and efficient use of ϕ-functions while ensuring good stability propertiesand high order. Although one-step in derivation and analysis, it proved necessary toutilise past values of N when implementation the EARK schemes in order to ensurethat the required derivatives of N are approximated to a sufficiently-high accuracy.
Exponential almost Runge-Kutta methods for semilinear problems 591
Fig. 10 Step-size against accuracy for the Allen-Cahn problem
Fig. 11 Allen-Cahn system CPU timing
This will also be a key competitive feature of local error estimation when implement-ing variable step-size EARK algorithms (to be reported in future work). The inclu-sion of past values of N gave rise to the development of the exponential integratorAlmost General Linear Methods (EAGLMs) which provided a convenient unifyingframework for ETD methods, ERK methods, EARK methods and EGLMs. The re-sults of some numerical experiments provide evidence that the EARK methods arecompetitive with both their ERK and EGLM counterparts in terms of accuracy and
592 J. Carroll, E. O’Callaghan
Fig. 12 Step-size against accuracy for the RDA2 problem
Fig. 13 RDA2 system CPU timing
computational efficiency when solving semilinear problems using fixed integrationstepsizes. In a future paper, we will show that such an EARK/EAGLM combinationprovides a highly-efficient algorithm for variable stepsize integration of semilinearproblems in which the local error estimate becomes available without any additionalcomputation (e.g. ϕ-function evaluations) unlike the ERK and EGLM counterparts,resulting in a significant saving in computational expense.
Exponential almost Runge-Kutta methods for semilinear problems 593
Table 5 Brusselator system: global errors and cpu timings for phipm and ReLPM
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