ASC Report No. 14/2012
Optimized Imex Runge-Kutta methods for simulations in astrophysics: A detailed study Inmaculada Higueras, Natalie Happenhofer, Othmar Koch, and Friedrich Kupka
Institute for Analysis and Scientific Computing Vienna University of Technology — TU Wien www.asc.tuwien.ac.at ISBN 978-3-902627-05-6
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OPTIMIZED IMEX RUNGE–KUTTA METHODS FOR SIMULATIONS IN ASTROPHYSICS: A DETAILED STUDY
∗
INMACULADA HIGUERAS† , NATALIE HAPPENHOFER‡ , OTHMAR KOCH§ , AND FRIEDRICH KUPKA¶ Abstract. We construct and analyze strong stability preserving implicit–explicit Runge–Kutta methods for the time integration of models of flow and radiative transport in astrophysical applications. It turns out that in addition to the optimization of the region of absolute monotonicity, other properties of the methods are crucial for the success of the simulations as well. The models in our focus dictate to also take into account the step–size limits associated with dissipativity and positivity of the stiff parabolic terms which represent transport by diffusion. Another important property is uniform convergence of the numerical approximation with respect to different stiffness of those same terms. Furthermore, it has been argued that in the presence of hyperbolic terms, the stability region should contain a large part of the imaginary axis even though the essentially non–oscillatory methods used for the spatial discretization have eigenvalues with a negative real part. Hence, we construct several new methods which differ from each other by taking various or even all of these constraints simultaneously into account. It is demonstrated for the problem of double–diffusive convection that the newly constructed schemes provide a significant computational advantage over methods from the literature. They may also be useful for general problems which involve the solution of advectiondiffusion equations, or other transport equations with similar stability requirements. Key words. Runge–Kutta, implicit–explicit, total–variation–diminishing, strong stability preserving, hydrodynamics, stellar convection and pulsation, double–diffusive convection, numerical methods. AMS subject classifications. 65M06, 65M08, 65M20, 65L05.
1. Introduction. In this paper we discuss the construction of strong stability preserving (SSP) implicit–explicit (IMEX) Runge–Kutta (RK) methods optimized for the deployment in radiation hydrodynamical simulations which are common in various fields of astrophysics. Specifically, the associated models of flow and radiative transport inside stars have the structure of advection–diffusion equations which can be discretized in space by dissipative finite difference methods and essentially non– oscillatory (ENO) schemes, and subsequently propagated in time by Runge–Kutta methods. Thus, we consider the ODE initial value problem y(t) ˙ = F (y(t)) + G(y(t)),
y(0) = y0 ,
(1.1)
where we assume that the vector fields F and G have different stiffness properties. For this type of problems, additive Runge–Kutta schemes were demonstrated to improve the efficiency of numerical simulations for the semiconvection problem in astrophysics in [23]. These methods were constructed with the aim to optimize the region of absolute monotonicity. This region characterizes the step–sizes which are admissible in order to ensure that the total variation (or some other suitable sublinear functional) ∗ Supported by the Ministerio de Ciencia e Innovaci´ on, project MTM2011–23203 and by the Austrian Science Fund (FWF), projects P21742–N16 and P20973 † Universidad P´ ublica de Navarra, Departamento de Ingenier´ıa Matem´ atica e Inform´ atica, Campus de Arrosadia, 31006 Pamplona, Spain (
[email protected]). ‡ University of Vienna, Faculty of Mathematics, Nordbergstraße 15, A–1090 Wien, Austria (
[email protected]). § Vienna University of Technology, Institute for Analysis and Scientific Computing, A–1040 Wien, Austria (
[email protected]). ¶ University of Vienna, Faculty of Mathematics, Nordbergstraße 15, A–1090 Wien, Austria (
[email protected]).
1
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Higueras et al.
of the spatial profile does not increase artificially in the course of time integration [11, 33]. In the case of the total–variation seminorm, this property is commonly referred to as total variation diminishing (TVD), or more generally as strong stability preserving (SSP) (see, for example, [5, 6, 28]). In [23], it was demonstrated that for the solution of the semiconvection problem, TVD IMEX methods from the literature provide a significant computational advantage and enhance the stability and accuracy of the simulations. Motivated by these observations the aim of the present paper is to construct new schemes which have additional properties beneficial to astrophysical simulations at the cost of reducing the region of absolute monotonicity from the optimum. We will demonstrate that the new methods are overall more efficient and accurate than methods used previously. An analysis of the properties of TVD IMEX methods and experimental assessment of their performance [23] indicates that the following properties of the methods promise reliable and efficient simulations: • The IMEX scheme should be of second order. Furthermore, the error constant should be small. Since the accuracy of such simulations is generally limited by the spatial resolution, third order methods do not promise further advantages. • The IMEX scheme should be SSP and it should have a large region of absolute monotonicity [11, 12]. This implies that both the explicit and the implicit schemes are SSP. Furthermore, the Kraaijevanger radius (also known as the radius of absolute monotonicity) [21] of both schemes should also be large. • The stability function of the implicit scheme should tend to zero at infinity and the stability region should contain a large subinterval of the negative real axis [−z, 0] with z > 0. This is ensured by L–stability. • For the explicit scheme, the stability region should contain large subintervals of the negative real axis, [−z, 0] with z > 0, and also of the imaginary axis, [−w i, w i, ] with w > 0. The latter requirement is associated with a stable integration of the hyperbolic advection terms (see [24, 36]). • For both schemes, the stability function should be nonnegative for a large interval of the negative real axis, [−z, 0] with z > 0. This condition is directly related to the step–size restrictions associated with the dissipativity of the spatial discretization [34] and should prevent spurious oscillations of the numerical solution. • The region of absolute stability of the IMEX scheme should be large. In order to be robust, it should also contain the curves studied in [2]. • For a convenient and memory–efficient implementation, the coefficients of the scheme should be rational numbers which could enable to recombine the stages in a suitable way. The properties listed above turn out to be more important for a successful simulation than third order accuracy. Still, we cannot use the optimal SSP2(2,2,2) two–stage method [23], because the number of degrees of freedom does not allow to have a positive stability function in conjunction with L–stability. Thus, we will focus on the ˜ bt ) of the form construction of second–order three–stage IMEX RK schemes (A, A, 0
0
0
0
c˜1
γ
0
0
c2
a21
0
0
c˜2
a ˜21
γ
0
c3
a31
a32
0
c˜3
a ˜31
a ˜32
γ
A
b1
b2
b3
b1
b2
b3
A˜
(1.2)
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
3
Observe the structural properties of this scheme: the weight vector b is the same for both schemes, and the implicit scheme is a Singly Diagonally Implicit Runge–Kutta method (SDIRK). The first property implies that there are no extra coupling order conditions for the IMEX scheme; that is, if both schemes have second order, the IMEX scheme also has second order [27]. Another advantage of IMEX schemes with b = ˜b is that they preserve linear invariants of the ODE [15]. The second property is also interesting from the computational point of view because it allows to solve, stage to stage, the nonlinear systems that arise when implicit RK methods are used. Actually, the Jacobian matrix may even be frozen throughout the iterations for all the stages (see [8]). Imposing second order convergence leaves a total of seven degrees of freedom for optimization as we explain below. The rest of the paper is organized as follows. In §2, we review some known results that are used throughout the paper. Section 3 is devoted to the construction of a second order 3–stage IMEX RK method with all the properties pointed out in the introduction, amongst them, a nontrivial intersection of the stability region of the explicit RK method and the imaginary axis. It turns out that this property of the explicit scheme leads to an important decreasing of the stability interval, the interval of nonnegativity of the stability function, and the Kraaijevanger coefficient. For this reason, in §4 we construct IMEX RK methods whose implicit scheme is the optimal second order 3–stage SSP RK method. In addition, although the optimal second order 3–stage SSP SDIRK method is not L–stable, we also investigate properties of IMEX RK methods constructed with this optimum SDIRK scheme; this study is done in §5 and §6. In order to test the constructed schemes for problems of our interest, the results of some numerical experiments are given in §7. Some conclusions are given in §8. Finally, in order to help the reader, there is an appendix section that collects the most relevant properties of the methods constructed in this paper; in order to stress the advantages of the new methods, in the appendix section we also include the properties of some methods from the literature. The methods have been obtained with the help of the symbolic computation software Mathematica. 2. Review of some known concepts. In this section we briefly review some known concepts that are used along the paper. 2.1. Order of convergence. As we have pointed out above, the IMEX scheme should achieve second order. To fulfill this requirement, the following conditions should be imposed (see for example [27]), bt e = 1 ,
bt c =
1 , 2
bt c˜ =
1 , 2
(2.1)
where, as usual, e = (1, . . . , 1)t and c = (c1 , . . . , cs ), c˜ = (˜ c1 , . . . , c˜s ); furthermore, we will also assume that Ae = c,
A˜ e = c˜ .
(2.2)
We would like to stress at this point that problem parameters may affect the magnitude of the global error even if the expected convergence order is formally retained. This is the case for problems of the form y 0 = F (y) +
1 G(y) ,
(2.3)
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Higueras et al.
where ε 1. Uniform convergence of IMEX RK methods applied to solve systems of the form (2.3) is studied in [1]. To obtain the results, in [1], the additive ODE (2.3) is transformed into a partitioned system of the form y 0 = f (y, z) , εz 0 = g(y, z) . It turns out that, if bt A˜−1 c = 1 ,
(2.4)
then, for ε ≤ C ∆t, the global error satisfies yn − y(tn ) = O((∆t)p ) + O(ε(∆t)2 ) ,
zn − z(tn ) = O((∆t)2 ) ,
where p is the order of the explicit scheme; if conversely (2.4) is violated, the global error is of the form yn − y(tn ) = O((∆t)p ) + O(ε ∆t) ,
zn − z(tn ) = O(∆t) .
2.2. Radius and regions of absolute monotonicity. For RK and IMEX RK methods, step size restrictions to obtain SSP or TVD schemes are given, respectively, by the Kraaijevanger radius (or radius of absolute monotonicity) and the region of absolute monotonicity. The literature collects an extensive research on TVD and SSP RK methods [3, 4, 10, 11, 12, 13, 14, 17, 18, 21, 22, 26, 27, 28, 30, 31, 32, 33] (see [5, 7, 9, 29] for reviews on the topic). An s–stage RK method (A, bt ) is said to be absolutely monotonic at a given point −r, with r ≥ 0, if I + rA is nonsingular, and (I + rA)−1 A ≥ 0 ,
(I + rA)−1 e ≥ 0 ,
(2.5)
where now e = (1, 1, . . . , 1)t ∈ Rs+1 , A is defined by A 0 A= , bt 0 and the inequalities in (2.5) are understood component–wise. The radius of absolute monotonicity R(A) is defined by R(A) = sup{ r | r ≥ 0 and A is absolutely monotonic on [−r, 0] } . For RK methods, monotonicity can be ensured under a stepsize restriction of the form ∆t ≤ τ0 · R(A), where τ0 is the step size restriction for monotonicity when the explicit Euler method is used. For details see [21]. For additive RK methods, the concept of radius of absolute monotonicity is extended to the region of absolute monotonicity [11, Definition 2.3] (see also [33]). An ˜ is said to be absolutely monotonic (a.m.) at a s–stage additive RK method (A, A) ˜ is invertible, given point (−r1 , −r2 ) with r1 , r2 ≥ 0, if the matrix I + r1 A + r2 A −1 ˜ (I + r1 A + r2 A) e ≥ 0, and ˜ (I + r1 A + r2 A)
−1
A ≥ 0,
(2.6)
˜ −1 A ˜ ≥ 0. (I + r1 A + r2 A)
(2.7)
5
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
˜ is defined by The region of absolute monotonicity, R(A, A), ˜ = { (r1 , r2 ) | r1 ≥ 0 , r2 ≥ 0 and (A, A) ˜ is a.m. on [−r1 , 0] × [−r2 , 0] } . R(A, A) ˜ under the Numerical monotonicity can be ensured for the additive RK method (A, A) stepsize restriction ∆t ≤ min {r1 τ0 , r2 τ˜0 }, where r1 and r2 are such that the point ˜ and τ0 , τ˜0 > 0 are the step size restrictions for monotonicity when (r1 , r2 ) ∈ R(A, A), the explicit Euler method is used for F and G, respectively (see [11] for details). Consequently, in order to obtain nontrivial step size restrictions for RK and additive RK methods, we should have, respectively, R(A) > 0, and points (r1 , r2 ) ∈ ˜ with r1 > 0 and r2 > 0. In [11, 21], algebraic criteria for nontrivial radius R(A, A) and regions of absolute monotonicity are given in terms of sign conditions of the coef˜ ≥ 0), and some inequalities ficient matrix (or matrices), namely, A ≥ 0 (or A ≥ 0, A of the incidence matrix of certain matrices. A trivial way to ensure these properties for IMEX schemes is to impose aij , a ˜ij > 0 for i < j ,
bj , ˜bj > 0 ,
and γ > 0 .
(2.8)
For this reason, for the IMEX RK schemes constructed in this paper we will assume the positivity conditions (2.8). 2.3. Amplification function for second order 3–point and fourth order 5–point spatial discretization. In order to study the stability of numerical schemes, we can study the dissipativity of time integrators in conjunction with spatial discretizations by means of Fourier analysis [16]. For the dissipativity analysis of advection–diffusion equations, it is sufficient to consider only the diffusion term since the advection term becomes negligible in the limit where the spatial discretization parameter tends to zero [34]. We thus consider the heat equation ut + a uxx = 0 and the second order 3–point spatial discretization uxx (xj , tn ) ≈
unj+1 − 2 unj + unj−1 , (4x)2
and the fourth order 5–point spatial discretization uxx (xj , tn ) ≈
−unj+2 + 16 unj+1 − 30 unj + 16 unj−1 − unj−2 . (4x)2
These are two of the spatial discretizations actually implemented in the ANTARES simulation code, which numerically solves the equations of hydrodynamics and various generalizations thereof [25, 23]. When these spatial discretizations are used to solve the heat equation, the stability function R(z) of the RK method (A, bt ), defined by R(z) = 1 + z bt (I − z A)−1 e , is evaluated at a point of the form z = −µ h(θ), where µ ≥ 0, and h : [−π, π] → R, depends on the discretization considered. The function h satisfies h(z) > 0 for z ∈ [−π, π], z 6= 0, and, due to the consistency of the spatial discretization, it also has the property h(0) = 0. We thus obtain the amplification function g(µ, θ) = R(−µ h(θ)) .
(2.9)
6
Higueras et al.
Observe that R(0) = 1 implies g(µ, 0) = 1. In particular, for the second order 3–point and for the fourth order 5–point spatial discretizations we obtain, respectively, the following functions h3 and h5 , θ 2 −i θ iθ , (2.10) h3 (θ) = −(e − 2 + e ) = 4 sin 2 1 h5 (θ) = − 16e−i θ + 16ei θ − e−2iθ − e2i θ − 30 12 2 θ 2 = (7 − cos(θ)) sin . (2.11) 3 2 Since the functions h in (2.10) and (2.11) are even, h(θ) = h(−θ), we subsequently restrict the values of θ to θ ∈ [0, π]. In the dissipativity analysis we are interested in the values: a) µ0 such that, for µ ∈ [0, µ0 ], it holds that g(µ, θ) > 0 for θ ∈ [0, π], and b) µ1 such that, for µ ∈ [0, µ1 ], it holds that |g(µ, θ)| ≤ 1 for θ ∈ [0, π] . If µi = ∞, i = 0, 1, we will understand that the interval [0, µi ] is [0, µi ). It turns out that, if z0 is the first negative zero of R(z), from the definition of g(µ , θ) in (2.9) we obtain that h(θ) µ = −z0 ,
θ ∈ [0, π] .
The lowest value µ0 is given by µ0 = −
z0 . maxθ∈[0,π] h(θ)
In a similar way, if z1 is the first negative zero of |R(z)| − 1, we obtain that h(θ) µ = −z1 ,
θ ∈ [0, π] .
The lowest value µ1 is given by µ1 = −
z1 maxθ∈[0,π] h(θ)
.
Observe that [z1 , 0] is the stability interval of the RK method, that is, the intersection of the stability region with the real axis. In particular, for the 3–points and for the 5–points discretizations, as hi is monotonic, we get max h3 (θ) = h3 (π) = 4 ,
θ∈[0,π]
max h5 (θ) = h5 (π) =
θ∈[0,π]
16 . 3
Consequently, for each RK method, in the dissipativity analysis, it is enough to compute z0 , the first negative zero of R(z), and z1 , the first strictly negative zero of the function |R(z)| − 1. For each spatial discretization, the values µ0 and µ1 are simply scaled values of |z0 | and |z1 |, respectively, and thus, for our purpose, the larger |z0 | and |z1 |, the better. 3. A new second order 3 stages SSP IMEX scheme. In this section we construct a new second order 3 stages SSP IMEX scheme of the form (1.2) with all the requirements explained in §1. We begin constructing the explicit scheme and, in a second step, we will deal with the implicit one. Finally, we will fix the remaining free parameters to obtain the desired properties for the IMEX scheme.
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
7
3.1. Explicit scheme. Condition (2.2) and second order conditions (2.1) for the explicit scheme lead to b1 = 1 − b2 − b3 ,
c3 =
1 − 2 b2 c2 , 2 b3
a21 = c2 ,
a31 = c3 − a32 .
(3.1)
Hence, four degrees of freedom of the explicit scheme are left to be determined after the previous considerations. For a second order 3–stage explicit scheme, the stability function is given by [8] R(z) = 1 + z +
z2 + b3 a32 c2 z 3 . 2
Thus, defining a32 =
α , b3 c 2
(3.2)
we obtain the stability function R(z) = 1 + z +
z2 + α z3. 2
(3.3)
To express the four degrees of freedom, in the following we choose α, c2 , b2 , and b3 . In the next subsection, we study the stability function (3.3) and the stability region S in terms of α. 3.1.1. Stability function for the explicit scheme. We study the stability function (3.3) to determine the values of α such that the method fulfills all the desired requirements. Intersection of the stability region with the imaginary axis. To obtain the intersection of the stability region S with the imaginary axis, we study the values of α such that |R(i w)| ≤ 1. In our case, from R(i w) = 1 −
w2 + i (w − w3 α) , 2
we obtain that, for α > 1/8, √ the interval [−i w(α), i w(α)] is contained in the stability region S, where w(α) = 8 α − 1/(2 α). The plot for w(α) can be seen in Figure 3.1 (continuous line). The maximum value is w(1/4) = 2. In particular, we obtain that the interval [−i, i] is contained in the stability region S for √ 1 √ 1 2− 3 , 2 + 3 ≈ [0.133975, 1.86603] . α∈ (3.4) 2 2 √ In the Figure 3.2, we show the stability regions for α = 21 2 − 3 (continuous √ line) and α = 21 2 + 3 (dashed line), and a zoom in the area close to the imaginary √ axis. It can be seen that for α = 12 2 − 3 , although [−i, i] is contained in the stability region, the border of the stability region is very close to the the imaginary axis. In the Figure 3.3, we show the regions of absolute stability for different values of α. Intersection of the stability region with the real axis. To obtain the intersection of the stability region S with the real axis, we study the values of w > 0 such that |R(−w)| ≤ 1. In Figure 3.1 (dashed line), we see the values of w(α) for α > 1/8.
8
Higueras et al. w 3.0 2.5 2.0 1.5 1.0 0.5
0.5
1.0
1.5
2.0
2.5
3.0
Α
Fig. 3.1. For α ∈ [1/8, 3]. Continuous line: w > 0 such that [−w i, w i] ∈ S; dashed line: z > 0 such that [−z, 0] ∈ S; dotted line: z > 0 such that R(−z) ≥ 0; dashed–dotted line: RLin (A). ImHzL
ImHzL
3
0.2
2 0.1 1
ReHzL
0
ReHzL
0.0
-1 -0.1 -2
-3 -3
-2
-1
0
-0.2 -0.010
1
Fig. 3.2. Left: stability regions for α =
1 2
-0.005
0.000
0.005
0.010
√ 2 − 3 (continuous line) and α =
1 2
2+
√ 3
(dashed line). Right: zoom in the area close to the imaginary axis.
Nonnegativity of the stability function. We study the points w > 0 such that R(−w) ≥ 0. In Figure 3.1 (dotted line), we show the values of w(α) such that R(z) ≥ 0 for z ∈ [−w(α), 0], when α > 1/8. 3.1.2. Radius of absolute monotonicity for linear problems. In order to obtain the radius of absolute monotonicity for linear problems (or the threshold factor), we compute the Taylor expansion of the stability function (3.3) at x = −p to obtain p2 − p3 α + (p + z) 1 − p + 3 p2 α 2 1 2 +(p + z) − 3 p α + α (p + z)3 . 2
R(z) = 1 − p +
The largest value p such that all the coefficients are nonnegative is the radius of absolute monotonicity of the method [31, 32]. For α ≥ 1/8, we observe that all the coefficients are nonnegative if p ≤ 1/(6α), and hence the radius of absolute monotonicity for the linear problem is given by RLin (A) =
1 . 6α
(3.5)
In Figure 3.1 (dashed-dotted line) we show the radius of absolute monotonicity as a function of α. Recall that, for explicit second order 3–stage RK methods, the optimal radius of absolute monotonicity for linear problems RLin (A) = 2 is obtained
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
9
ImHzL 3
2
1
ReHzL
0
-1
-2
-3 -3
-2
-1
0
1
Fig. 3.3. Regions of absolute stability for different values of α: dashed α = 0.17; brown: √ continuous magenta: α = 0.25; dashed pink: α = 0.5; continuous pink: α = 12 2 + 3 .
for α = 1/12 [22]. However, for α = 1/12 there is not any interval of the imaginary axis contained in the stability region S, as this requires α > 1/8. 3.1.3. Radius of absolute monotonicity for nonlinear problems. For nonlinear problems, the radius of absolute monotonicity R(A) satisfies R(A) ≤ RLin (A) [21]. In our case, if b2 =
1 , 36 α
c2 = 6 α ,
1 18 α − 1 ≤ b3 ≤ , 36 α 18α
(3.6)
after some computations, from (2.5) we obtain that R(A) = RLin (A) =
1 . 6α
(3.7)
3.1.4. Choice of α. So far, after imposing (3.1)–(3.2) and (3.6), we have two free parameters left, namely b3 and α, that should satisfy √ 1 √ 1 1 18 α − 1 ≤ b3 ≤ , α∈ 2− 3 , 2 + 3 ≈ [0.133975, 1.86603] . 36 α 18 α 2 2 Taking into account the stability interval, the interval of the imaginary axis contained in the stability region, and the radius of absolute monotonicity as functions of α (see Figure 3.1), √ and expressions (3.4) and (3.7), we can conclude that α should be close to 12 2 − 3 ≈ 0.133975. A choice that is compatible with this requirement is α = 5/36 ≈ 0.138889. For this value, the coefficients for the explicit scheme obtained so far are 0
0
0
0
5 6 1 3 b3
5 6 1 6 b3
0
0
1 6 b3
0
1 5
b3
A
4 5
− b3
(3.8)
where b3 is such that 1 3 ≤ b3 ≤ . 5 5
(3.9)
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Higueras et al.
For scheme (3.8), expression (3.7) is RLin (A) = R(A) =
6 . 5
Furthermore, the interval [−w i, w i] ∈ S for w = 1.2; the interval [−z, 0] ∈ S for z = 2.84745, and R(−z) ≥ 0 for z ∈ [0, 1.81803]. The stability region, and a zoom of the region close to the imaginary axis is given in Figure 3.7. 3.2. Implicit scheme. For the implicit scheme, the weight vector b is the same as the one for the explicit scheme. Thus, from the previous section, see (3.8) and (3.9), we have that b1 =
4 − b3 , 5
b2 =
1 , 5
1 3 ≤ b3 ≤ . 5 5
(3.10)
Condition (2.2) and second order conditions (2.1) for the implicit scheme lead to c˜3 =
−2 b2 c˜2 + 2 b2 γ + 2 b3 γ − 2 γ + 1 , 2 b3
a ˜21 = c˜2 − γ ,
c˜1 = γ ,
a ˜31 = c˜3 − a ˜32 − γ .
(3.11) (3.12)
Thus, the implicit scheme introduces three additional degrees of freedom to the combined scheme (1.2). We choose these as c˜2 , a ˜32 , and γ in the following. Since b1 is constrained by the second order condition (2.1) with (2.2) and with (3.1), and since b2 is constrained by (3.6), it remains to constrain b3 and thus a total of four degrees of freedom is left to be optimized for the implicit scheme. In the next subsection, we study the stability function of the implicit scheme and impose conditions to obtain L–stability. 3.2.1. Stability function for the implicit scheme. As it has been pointed out in §1, we aim at constructing an L–stable method, that is, an A–stable method such that limz→∞ R(z) = 0. A detailed study of stability issues for SDIRK methods is given in [8, Chapter IV.6]. For SDIRK methods with a11 = · · · = ass = γ, the stability function is of the form R(z) =
P (z) , Q(z)
where P (z) is a polynomial of degree at most s, and Q(z) = (1 − γz)s . For these schemes, in the case γ > 0, A–stability is equivalent to E(y) = |Q(i y)|2 − |P (i y)|2 ≥ 0
for all y ∈ R .
(3.13)
For L–stable methods, R(∞) = 0 and thus the highest coefficient of P is zero. Furthermore, if the method is known to be of order p, with p ≥ s − 1, then the rest of the coefficients are uniquely determined in terms of s–degree Laguerre polynomials (see [8, Chapter IV.6, p. 105]). These results can be applied to our method (p = 2 and s = 3). In this case, according to [8, Chapter IV.6], expression (3.13) is given by 1 4 4 3 2 E(y) = y −6 γ + 18 γ − 12 γ + 3 γ − + y6 γ 6 , (3.14) 4
11
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study Log ÈCÈ 1
0.1
0.01
0.001
0.5
1.0
1.5
2.0
Γ
Fig. 3.4. Error constant for second order 3–stage SDIRK methods.
and the stability function can be computed as z 2 6 γ 2 − 6 γ + 1 + z (2 − 6 γ) + 2 . R(z) = 2 (1 − γ z)3
(3.15)
The regions of γ for L–stability can be obtained imposing nonnegativity to the coefficients of E(y) in (3.14); in this way we obtain that the parameter γ should satisfy 1 12
! r √ 1 9 + 3 3 − 6 12 + 7 3 ≤γ≤ 4 √
3+
√
s 3+
14 8+ √ 3
! ,
whose numerical approximations (see [8, Chapter IV.6, Table 6.4]) are given by 0.18042531 ≤ γ ≤ 2.18560010 .
(3.16)
For second order 3–stage L–stable SDIRK schemes, the error constant is also known in terms of the 3–degree Laguerre polynomial L3 (x) (see [8, Chapter IV.6, p. 105]), 1 1 3γ C = −L3 + , γ 3 = −γ 3 + 3 γ 2 − γ 2 6 and third order is obtained for γ = 0.43586652. In Figure 3.4 we show the values of log |C|. If we compute the function E(y) from the Butcher tableau of the SDIRK method, and we use the values of b1 , b2 , c˜3 , c˜1 , a ˜21 , and a ˜31 given by (3.10)–(3.12), we obtain that the function E(y) is of the form (3.14) if c˜2 =
2a ˜32 b3 γ + 2 γ 3 − 4 γ 2 + γ . 2a ˜32 b3
(3.17)
For this value, the stability function computed from the Butcher tableau is (3.15). At this point, we recall that the coefficients of the SDIRK method should be positive. With the value of c˜2 given by (3.17), the coefficient a ˜21 in (3.12) is γ 2 γ2 − 4 γ + 1 a ˜21 = . 2a ˜32 b3 If we impose that a ˜21 > 0 and take into account (3.16), we obtain that γ must satisfy 0.180425 ≤ γ ≤
√ 1 2− 2 2
or
√ 1 2 + 2 ≤ γ ≤ 2.1856 . 2
12
Higueras et al.
√ √ As 12 2 − 2 ≈ 0.29289322 and 21 2 + 2 ≈ 1.70710678, taking into account the size of the error constant (see Figure 3.4), we restrict the values of γ for L–stability in (3.16) to 0.18042531 ≤ γ ≤ 0.29289322 .
(3.18)
Note that this interval does not contain the value of γ for which the scheme would be of third order. With regard to the intersection of the real and imaginary axis with the stability region, observe that, as the method is L–stable, the imaginary axis as well as the real axis is contained in the stability region S. It remains to study the nonnegativity of the stability function. We analyze now if there is any value of γ such that R(z) ≥ 0 for all z < 0. We obtain a positive answer for √ 1 (3.19) 3 − 6 ≈ (0, 0.18350342] . γ ∈ 0, 3 Combining this result with (3.18), we get that adequate values for γ are 0.18042531 ≤ γ ≤ 0.18350342 .
(3.20)
3.2.2. Radius of absolute monotonicity for linear problems. For linear problems, the computation of the radius of absolute monotonicity for implicit schemes is not an easy task. To obtain it, we have to find the points z such that the stability function R is absolutely monotonic at z. Recall that a function f is said to be absolutely monotonic at a given point x ∈ R if f (k) (x) exist and f (k) (x) ≥ 0 for k ≥ 0. For explicit methods, the stability function R is a polynomial, and thus, we can analyze a finite number of derivatives to check if the function is absolutely monotonic at a given point x. However, for implicit schemes, R is a rational function, and we cannot use directly the definition of absolute monotonicity. In this section, we apply Theorem 4.4 in [35] to obtain the radius of absolute monotonicity for linear problems for our SDIRK method. Roughly speaking, this result allows us to restrict the analysis to a finite number of derivatives, that is, R(k) (x) ≥ 0 for 0 ≤ k < K(x), and the key point is to obtain the integer K(x). In the following proposition we compute it for the stability function√(3.15). Proposition 3.1. Assume that γ satisfies (3.20), γ 6= 3 − 6 /3. If R(k) (x) ≥ 0 for k = 0, 1, 2, 3, then the stability function (3.15) is absolutely monotonic at x. Proof. In order to apply [35, Theorem 4.4], we construct all the elements (sets, intervals, functions, etc.) involved in this result. Following the notation in [35], the stability function (3.15) is of the form R(z) = P (z)/Q(z) with 1 2 + z (1 − 3 γ) + 1 , P (z) = z 3 γ − 3 γ + 2 2
Q(z) = γ
3
1 −z γ
3 .
The polynomials P and Q have degrees m = 2 and n = 3, respectively, and there is a unique pole α0 = 1/γ > 0 with multiplicity µ(α0 ) = 3. Thus the set A = A+ = {α0 }, I(α0 ) = R, and B(R(z)) = ∞. Next, the stability function should be decomposed in partial fractions of the form R(z) = c(α0 , 1)
1 1 1 + c(α0 , 2) + c(α0 , 3) . 2 α0 − z (α0 − z) (α0 − z)3
13
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study x 6
5
4
3
2
1
0.1810
0.1815
0.1820
0.1825
0.1830
0.1835
Γ
Fig. 3.5. From bottom to top, values of −x such that condition (3.21) holds for k = 1, 2, 3, 4.
In our case, 6 γ2 − 6 γ + 1 −3 γ 2 + 5 γ − 1 2 γ2 − 4 γ + 1 , c(α0 , 2) = , c(α0 , 3) = . 3 4 2γ γ 2 γ5 √ For γ satisfying (3.20), γ 6= 3 − 6 /3, these functions have constant sign. More precisely, c(α0 , 1) ≥ 0, c(α0 , 2) ≤ 0, and c(α0 , 3) > 0. Now, we construct the function (see [35, Formula (4.2)]) 2 2 c(α0 , 1) k! 1 2 c(α0 , 2) (k + 1)! 1 F (k, x) = −x − − x , (k + 2)! γ (k + 2)! γ c(α0 , 1) =
where we have used the sign conditions of c(α0 , 1) and c(α0 , 2). Another function needed to construct K(x) in [35] is L(x); in our case, L(x) = 0, see [35, Formula (4.3)]. We can now construct the integer K(x) in [35, Definition 4.1]), defined as the smallest integer k such that k ≥ L(x) and F (k, x) < |c(α0 , 3)| .
(3.21)
For γ satisfying (3.20), in Figure 3.5 we show, from bottom to top, the values of −x such that condition (3.21) holds for k = 1, 2, 3, 4. We do not consider k = 0 because, as we have seen in the previous subsection (see (3.19)), for the values of γ satisfying (3.20), it holds that R(z) ≥ 0 for z ≤ 0. Furthermore, we restrict the values to −x ∈ [0, 6] because, by numerical search, it has been found that the nonlinear optimum radius of absolute monotonicity for second order 3–stage SDIRK schemes is 6 [4, 19]. Consequently, for −x ∈ [0, 6], K(x) ≤ 4. We can now apply Theorem 4.4 in [35] to ensure that R(z) is absolutely monotonic in [x, 0] if and only if c(α0 , 3) > 0 and R(k) (x) ≥ 0 for 0 ≤ k < K(x). Thus, the next step is to analyze the values of x such that R(k) (x) ≥ 0 for k = 0, 1, 2, 3. For each γ satisfying (3.20), the radius of absolute monotonicity is given by s 1 − 4γ −12 γ 3 + 28 γ 2 − 10 γ + 1 R(γ) = − , (3.22) 2 γ (6 γ 2 − 6 γ + 1) γ 2 (6 γ 2 − 6 γ + 1) and varies in the interval [4.25512, 4.44949]. In Figure 3.6 we show, for each γ, the interval of absolute monotonicity, and a zoom of the value range of the radius of absolute monotonicity. We observe that, for the different values of γ, the radius of absolute monotonicity does not change significantly.
14
Higueras et al. x 5
4
R 3 4.4
2
4.3
4.2 1 4.1
0.1810
0.1815
0.1820
0.1825
0.1830
0.1835
Γ
0.1810
0.1815
0.1820
0.1825
0.1830
0.1835
Γ
Fig. 3.6. Left: Interval of absolute monotoniciy. Right: zoom of the range values of the radius of absolute monotonicity.
3.3. Uniform convergence for IMEX Runge–Kutta methods. Condition (2.4) seems to be of interest when stiff systems of the form (2.3) are solved. For this reason, we impose the implied relationship a ˜32 =
3 γ − 6 γ2 . 5 b3
(3.23)
3.4. Choice of γ. Based on the analysis done so far, now we choose the value of γ satisfying (3.20). If we rationalize the mid point, 0.1819643628, we obtain γ=
2 ≈ 0.181818 . 11
(3.24)
At this point, it remains to obtain the parameter b3 ; recall from (3.10) that this value should satisfy 1 3 ≤ b3 ≤ . 5 5
(3.25)
We still have to deal with the radius of absolute monotonicity for nonlinear problems for the SDIRK method, and with the region of absolute monotonicity for the IMEX method. 3.5. Absolute monotonicity for the SDIRK and IMEX schemes. In this section we determine b3 satisfying (3.25) to obtain a large radius of absolute monotonicity for the SDIRK method, and a large region of absolute monotonicity for the IMEX scheme. ˜ for the SDIRK From (2.5), a study of the radius of absolute monotoninicy R(A) ˜ method yields that the largest value possible is R(A) = 42/11 ≈ 3.8181, and this value is attained for any b3 such that 4066 4463 ≤ b3 ≤ 11275 11275
(0.360621 ≤ b3 ≤ 0.395831) .
(3.26)
˜ Considering the points (r1 , r2 ) in the region of absolute monotonicity R(A, A), when r1 = 0 (or r2 = 0), we obtain the point (0, r2 ) (or (r1 , 0)), that is, the intersection of the region of absolutely monotonicity with the axis r1 = 0 (axis r2 = 0). These
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
15
˜ r1 ≤ R(A). Quite often, due to the mixed conditions (2.6) values satisfy r2 ≤ R(A), for r1 = 0, and (2.7) for r2 = 0, that is, ˜ −1 A ≥ 0 , (I + r2 A)
˜ ≥ 0, (I + r1 A)−1 A
˜ and r1 < R(A). We aim for determining the largest values we obtain r2 < R(A) of r2 and r1 such that the points (r1 , 0) and (0, r2 ) belong to the region of absolute monotonicity. A detailed study yields the result that the largest value of r2 such that (0, r2 ) ∈ ˜ is r2 = 66/43 ≈ 1.53488, provided that R(A, A) 1 1462 ≤ b3 ≤ 5 3025
(0.2 ≤ b3 ≤ 0.483306) .
(3.27)
˜ is r1 = 126/55 ≈ In a similar way, the largest value of r1 such that (r1 , 0) ∈ R(A, A) 2.29091, provided that 109616 1 ≤ b3 ≤ 5 213565
(0.2 ≤ b3 ≤ 0.513268) .
(3.28)
Taking into account (3.26), (3.27), and (3.28), we consider b3 =
4 ≈ 0.363636 . 11
(3.29)
By fixing this value, we have finished the construction of the IMEX scheme. 3.6. Second order 3–stage SSP IMEX scheme (summary). In this section, we give the coefficients of the IMEX scheme obtained from our earlier considerations, that is 0
0
0
0
5 6 11 12
5 6 11 24
0
0
11 24
0
24 55
1 5
4 11
A
2 11 289 462 751 924
A˜
2 11 205 462 2033 4620
0
0
2 11 21 110
0 2 11
24 55
1 5
4 11
(3.30)
and summarize its properties. This is a second order IMEX scheme such that the implicit method is L–stable. The stability functions for the explicit and implicit schemes are 11 13 z 2 + 110 z + 242 z2 5 3 RA (z) = 1 + z + + z , RA˜ (z) = . (3.31) 2 36 2 (11 − 2 z)3 In Figure 3.7, we show the stability regions for the explicit and the implicit schemes in (3.30), as well as a zoom of the region in a neighborhood of the origin. In Figure 3.8, we show the stability region of the IMEX scheme and a zoom of the region at the origin. For the explicit scheme, we obtain R(z) ≥ 0 for z ∈ [−1.81803, 0], and |R(z)| ≤ 1 for z ∈ [−2.84745, 0]. Furthermore, |R(iw)| ≤ 1 for w ∈ [−1.2, 1.2]. For the implicit scheme, R(z) ≥ 0 and |R(z)| ≤ 1 for z ≤ 0. Furthermore, |R(iw)| ≤ 1 for w ∈ R. This IMEX method satisfies condition (2.4) for uniform convergence.
16
Higueras et al. ImHzL
ImHzL
3
2
2 1 1
-5
-4
-3
-2
1
-1
ReHzL
-0.04
-0.02
0.02
0.04
0.02
0.04
ReHzL
-1 -1 -2
-3
-2
ImHzL
ImHzL
2 20
1
10
-20
10
-10
20
ReHzL
-0.04
-0.02
-10
ReHzL
-1
-20 -2
Fig. 3.7. Stability region and a zoom of the stability region of the explicit scheme in (3.30) (top) and the implicit scheme in (3.30) (bottom)
. w
w
r2
2
15
1.5 10 1 5 1.0
-50
-40
-30
-20
z
-10
-0.04
0.02
-0.02
0.04
z
-5 0.5 -1 -10
-15
-2
0.5
1.0
1.5
2.0
r1
Fig. 3.8. For IMEX (3.30): stability region (right), a zoom of the region at the origin (center) and region of absolute monotonicity (right).
With regard to the radius of absolute monotonicity, for linear problems we have that √ ˜ = 11 33 − 517 ≈ 4.34177 . RLin (A) = 6/5 = 1.2, RLin (A) (3.32) 26 ˜ = 42/11 ≈ 3.81818, and For nonlinear problems, we have R(A) = 6/5 = 1.2, R(A) ˜ = (r1 , r2 ) ∈ R2 : 0 ≤ r1 < 6 , 0 ≤ r2 ≤ 1 (66 − 55 r1 ) . R(A, A) 5 43 In Figure 3.8 we show the region of absolute monotonicity for the IMEX method (3.30).
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study
17
Due to the properties listed above, we will refer to scheme (3.30) as SSP2(3,3,2)– LSPUM, where the letters have the following meanings: ’L’: L–stable; ’S’: the stability region for the explicit part contains an interval on the imaginary axis; ’P’: the amplification factor g is always positive; ’U’: the IMEX method features uniform convergence (2.4); ’M’: the IMEX method has a nontrivial region of absolute monotonicity.
(3.33)
To assess the merits of the method just constructed, in the next section we will construct alternative schemes based on optimal methods from the literature. These will be compared experimentally in §7. 4. IMEX methods with second order 3–stage optimum SSP explicit RK scheme as explicit method. In this section, we construct a number of additional SSP IMEX methods, based on the optimal explicit three–stage second–order method in conjunction with a compatible implicit scheme. These will be compared to the scheme constructed in Section 3. Thus, we consider IMEX schemes of the form 0
0
0
0
c˜1
γ
0
0
1 2
0
0
c˜2
a ˜21
γ
0
1
1 2 1 2
1 2
0
c˜3
a ˜31
a ˜32
γ
A
1 3
1 3
1 3
1 3
1 3
1 3
A˜
(4.1)
The explicit method coincides with the explicit scheme for the SSP2(3,3,2) schemes in [11, 27]. Observe that now, the weight vector b for the implicit scheme is already fixed. Recalling the discussion from §3.2, we note that this leaves only three degrees of freedom for the scheme (4.1), namely, c˜2 , a ˜32 , and γ. For the implicit schemes considered in this section, we will impose L–stability and the nonnegativity of the stability function R(z) for all z ≤ 0. As the stability function for an L–stable second order 3–stage SDIRK method only depends on the diagonal elements γ, the study conducted in §3.2.1 is valid and, therefore, we choose again γ = 2/11. Furthermore, after imposing second order conditions for the implicit scheme, and a stability function of the form (3.15), c˜2 is also determined, and we obtain that there is one free parameter left: a ˜32 . In this section, we construct three schemes by choosing this value as follows: 1. In the first one, we impose condition (2.4) for uniform convergence, see §4.1. 2. In the second one, we optimize the radius of absolute monotonicity of the SDIRK scheme, see §4.2. 3. In the third one, we optimize the region of absolute monotonicity of the IMEX scheme, see §4.3. 4.1. IMEX method with uniform convergence (2.4). In this case, the IMEX scheme obtained by imposing (2.4) is given by the coefficient tableaux 0
0
1 2
0
0
1
1 2 1 2
1 2
0
A
1 3
1 3
1 3
0
0
2 11 69 154 67 77
A˜
2 11 41 154 289 847
0
0
2 11 42 121
0 2 11
1 3
1 3
1 3
(4.2)
18
Higueras et al. w
w
r2
2
15
1.5 10 1 5 1.0
-50
-40
-30
-20
z
-10
-0.04
0.02
-0.02
z
0.04
-5 0.5 -1 -10
-2
-15
0.5
1.0
1.5
2.0
r1
Fig. 4.1. For the IMEX method (4.2): Left: Stability region; center: zoom of the stability region; right: region of absolute monotonicity.
This method satisfies condition (2.4) for uniform convergence. Note that (3.23) must not be reused to derive (4.2), because (3.23) has been derived assuming (3.10) where b2 = 1/5 instead of b2 = 1/3. Recalling (3.2) and (3.3), the stability function for the explicit scheme is 1 1 2 z + z3 , (4.3) 2 12 whereas for the implicit scheme it is given by RA˜ (z) in (3.31), since it depends only on γ (see (3.15)). For linear problems, the radius of absolute monotonicity for the explicit scheme ˜ is given by (3.32). For is RLin (A) = 2, whereas for the implicit scheme RLin (A) nonlinear problems, the explicit scheme is the optimal second order 3–stage explicit SSP method and thus R(A) = 2; for the implicit method, from (2.5), we get RA (z) = 1 + z +
1694 √ ≈ 3.08947 . 275 + 74701
˜ = R(A) For the IMEX scheme, ˜ = R(A, A)
(r1 , r2 ) ∈ R2 : 0 ≤ r1
1 for all w ∈ R.
21
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study ImHzL
ImHzL
3
2
2 1 1
-5
-4
-3
-2
1
-1
ReHzL
-0.04
-0.02
0.02
0.04
0.02
0.04
ReHzL
-1 -1 -2
-3
-2
ImHzL
ImHzL
2 20
1
10
-20
-15
-10
ReHzL
-5
-0.04
-0.02
-10
ReHzL
-1
-20 -2
Fig. 5.1. Stability region and a zoom of the stability region of the explicit scheme in (5.1) (top) and the implicit scheme in (5.1) (bottom). w
w
r2
2
15
1.5 10 1 5 1.0
-50
-40
-30
-20
z
-10
-0.04
0.02
-0.02
0.04
z
-5 0.5 -1 -10
-2
-15
0.5
1.0
1.5
2.0
r1
Fig. 5.2. For the IMEX method (5.1): Left: Stability region; center: zoom of the stability region; right: region of absolute monotonicity.
For the implicit scheme, R(z) ≥ 0 for z ∈ [−6, 0] and |R(z)| ≤ 1 for z ≤ 0. Moreover, |R(iw)| = 1 for w ∈ R. It can be checked that this IMEX method satisfies condition (2.4) for uniform convergence. With regard to the radius of absolute monotonicity, for linear problems we have that [3, 21] RLin (A) = 2
˜ = 6. RLin (A)
˜ = 6, and For nonlinear problems, we have R(A) = 2, R(A) q 3 3 2 2 ˜ R(A, A) = (r1 , r2 ) ∈ R : 0 ≤ r1 < 2 , 0 ≤ r2 ≤ (8 − 3r1 ) − 5r1 − 32r1 + 48 . 2 2
22
Higueras et al.
In Figure 5.2, we show the region of absolute monotonicity. The points (2, 0) and ˜ (0, 1.6077) are included in the region of absolute monotonicity R(A, A). 6. IMEX method with second order 3–stage optimum SSP SDIRK method. In this section we consider IMEX schemes such that the implicit method is the optimal 3–stage second order SSP SDIRK method [3], 0
0
0
c2
a21
0
0
c3
a31
a32
0
1 3
1 3
1 3
A
1 6 1 2 5 6
0
A˜
1 6 1 3 1 3
0
0
1 6 1 3
0
1 3
1 3
1 3
1 6
(6.1)
We aim at constructing an explicit scheme such that there is a nontrivial intersection between the stability region and the imaginary axis. Second order conditions (2.1) yield c3 =
1 (3 − 2 c2 ) , 2
a31 = c3 − a32 ,
(6.2)
and with a32 =
3α , c2
(6.3)
we obtain the stability function (3.3). Therefore, the analysis conducted in subsection 3.1.1 in terms of α is valid. We take α = 5/36, the parameter chosen in §3.1.1, and, after this choice, since b2 = b3 = 31 , there is only one free parameter, namely c2 . It remains to analyze the radius of absolute monotonicity of the explicit scheme for nonlinear problems, and the region of absolute monotonicity of the IMEX method; it also remains to impose condition (2.4) to ensure uniform convergence. We will use these conditions to fix the value of c2 . If we impose condition (2.4) for uniform convergence, we obtain c2 = 1/2. Another possibility is to use this parameter to obtain a large radius of absolute monotonicity. For the linear case, as the stability function is given by (3.3), we obtain RLin = 6/5. Remember that for the scheme constructed in §3, we were able to obtain for the nonlinear case R(A) = 6/5; however, with (6.2) and (6.3) and α = 5/36, this cannot be achieved. In fact, unfortunately, the radius for the nonlinear case is rather small. It can be computed that R(A) =
−12 c22 + 18 c2 − 5 , 5 c2
√ whose maximum value, R(A) = 18 − 4 15 /5 ≈ 0.501613, is attained for c2 = p 1 5/3 ≈ 0.645497; this value is much smaller than 2, the optimum value possible for 2 second order 3–stage RK methods [21]. In the next sections, we give the coefficients and the properties of this family of methods for different values of c2 .
23
Optimized IMEX Methods for Simulations in Astrophysics: a detailed study w
w
r2
2
15
1.5 10 1 5 1.0
-50
-40
-30
-20
z
-10
-0.04
0.02
-0.02
z
0.04
-5 0.5 -1 -10
-2
-15
0.5
1.0
1.5
2.0
r1
Fig. 6.1. For the IMEX method (6.4): Left: Stability region; center: zoom of the stability region; right: region of absolute monotonicity.
6.1. IMEX with c2 = 1/2. We obtain the IMEX method 0
0
0
1 2
0
0
1
1 2 1 6
5 6
0
A
1 3
1 3
1 3
1 6 1 2 5 6
0
A˜
1 6 1 3 1 3
0
0
1 6 1 3
0
1 3
1 3
1 3
(6.4)
1 6
This is a second order IMEX scheme such that b = ˜b. As α = 5/36, the stability function of the explicit method is the same as for the scheme (3.30). The implicit method is the optimum 3–stage second order SSP SDIRK scheme. The stability functions are given by 1 2 5 3 z + z , 2 36
RA (z) = 1 + z +
RA˜ (z) =
(z + 6)3 . (6 − z)3
This IMEX method satisfies condition (2.4) for uniform convergence. For the radius of absolute monotonicity for nonlinear problems, we obtain R(A) = 2/5 = 0.4, and ˜ = (r1 , r2 ) ∈ R2 : 0 ≤ r1 < 2 , 0 ≤ r2 ≤ 1 (6 − 15 r1 ) . R(A, A) 5 5 The points (0, 1.2) and (0.4, 0) are included in the region of absolute monotonicity ˜ In Figure 6.1 we show the stability region for the IMEX scheme and the R(A, A). region of absolute monotonicity. q 6.2. IMEX method with c2 = 12 53 . The IMEX method is
1 2 1 6
0 q
9− A
5 3
√
1 2
15
3 2
0 q
−
0 5 3
q
1 3
5 3
1 2
0 q 1 3
0 0 5 3
0 1 3
1 6 1 2 5 6
A˜
1 6 1 3 1 3
0
0
1 6 1 3
0
1 3
1 3
1 3
1 6
(6.5)
24
Higueras et al. ImHzL
ImHzL
2
5 1
-100
-80
-60
-40
ReHzL
-20
-0.04
0.02
-0.02
ReHzL
0.04
-1 -5
-2
Fig. 6.2. Stability regions for the IMEX scheme when c1 = 1/2 (solid line), and c2 = (dotted line). w
w
q
5 3
r2
2
15
1 2
1.5 10 1 5 1.0
-50
-40
-30
-20
z
-10
-0.04
0.02
-0.02
0.04
z
-5 0.5 -1 -10
-2
-15
0.5
1.0
1.5
2.0
r1
Fig. 6.3. For the IMEX method (6.5): Left: Stability region; center: zoom of the stability region; right: region of absolute monotonicity.
This is a second order IMEX method such that b = ˜b. As α = 5/36, the stability function is the same as for scheme (3.30). The implicit method is the optimum 3–stage second order SSP SDIRK scheme. The stability functions are given by RA (z) = 1 + z +
1 2 5 3 z + z , 2 36
RA˜ (z) =
(z + 6)3 . (6 − z)3
In Figure 6.2, we compare q the stability regions for the IMEX scheme when c1 = 1 1/2, solid line, and c2 = 2 53 , dotted line. As expected (the stability function for the explicit schemes is the same), there is no difference for values close to the origin. This IMEX method does not satisfy condition (2.4) for uniform convergence. For the of absolute monotonicity for nonlinear problems, we obtain R(A) = √ radius 1 5 18 − 4 15 ≈ 0.501613 , and ˜ = R(A, A)
(r1 , r2 ) ∈ R2 : 0 ≤ r1
