Optimized Imex Runge-Kutta methods for simulations in astrophysics: A detailed study

Transcription

1 ASC Report No. 4/ Optimied 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.asc.tuien.ac.at ISBN

2 Most recent ASC Reports / H. Woracek Asymptotics of eigenvalues for a class of singular Krein strings / H. Winkler, H. Woracek A groth condition for Hamiltonian systems related ith Krein strings / B. Schörkhuber, T. Meurer, and A. Jüngel Flatness-based trajectory planning for semilinear parabolic PDEs / Michael Karkulik, David Pavlicek, and Dirk Praetorius On D neest vertex bisection: Optimality of mesh-closure and H -stability of L -projection 9/ Joachim Schöberl and Christoph Lehrenfeld Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes 8/ Markus Aurada, Michael Feischl, Thomas Führer, Michael Karkulik, Jens Markus Melenk, Dirk Praetorius Classical FEM-BEM coupling methods: nonlinearities, ell-posedness, and adaptivity 7/ Markus Aurada, Michael Feischl, Thomas Führer, Michael Karkulik, Jens Markus Melenk, Dirk Praetorius Inverse estimates for elliptic integral operators and application to the adaptive coupling of FEM and BEM / J.M. Melenk, A. Parsania, and S. Sauter Generalied DG-Methods for Highly Indefinite Helmholt Problems based on the Ultra-Weak Variational Formulation / J.M. Melenk, H. Reaijafari, B. Wohlmuth Quasi-optimal a priori estimates for fluxes in mixed finite element methods and applications to the Stokes-Darcy coupling 4/ M. Langer, H. Woracek Indefinite Hamiltonian systems hose Titchmarsh-Weyl coefficients have no finite generalied poles of non-negativity type Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstraße 8 4 Wien, Austria admin@asc.tuien.ac.at WWW: FAX: ISBN c Alle Rechte vorbehalten. Nachdruck nur mit Genehmigung des Autors. ASC TU WIEN

3 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 analye strong stability preserving implicit explicit Runge Kutta methods for the time integration of models of flo and radiative transport in astrophysical applications. It turns out that in addition to the optimiation of the region of absolute monotonicity, other properties of the methods are crucial for the success of the simulations as ell. The models in our focus dictate to also take into account the step sie limits associated ith dissipativity and positivity of the stiff parabolic terms hich represent transport by diffusion. Another important property is uniform convergence of the numerical approximation ith 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 discretiation have eigenvalues ith a negative real part. Hence, e construct several ne methods hich 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 nely constructed schemes provide a significant computational advantage over methods from the literature. They may also be useful for general problems hich involve the solution of advectiondiffusion equations, or other transport equations ith similar stability requirements. Key ords. Runge Kutta, implicit explicit, total variation diminishing, strong stability preserving, hydrodynamics, stellar convection and pulsation, double diffusive convection, numerical methods. AMS subject classifications. M, M8, M, L.. Introduction. In this paper e discuss the construction of strong stability preserving (SSP) implicit explicit (IMEX) Runge Kutta (RK) methods optimied for the deployment in radiation hydrodynamical simulations hich are common in various fields of astrophysics. Specifically, the associated models of flo and radiative transport inside stars have the structure of advection diffusion equations hich can be discretied in space by dissipative finite difference methods and essentially non oscillatory (ENO) schemes, and subsequently propagated in time by Runge Kutta methods. Thus, e consider the ODE initial value problem ẏ(t) = F (y(t)) + G(y(t)), y() = y, (.) here e assume that the vector fields F and G have different stiffness properties. For this type of problems, additive Runge Kutta schemes ere demonstrated to improve the efficiency of numerical simulations for the semiconvection problem in astrophysics in []. These methods ere constructed ith the aim to optimie the region of absolute monotonicity. This region characteries the step sies hich are admissible in order to ensure that the total variation (or some other suitable sublinear functional) Supported by the Ministerio de Ciencia e Innovación, project MTM and by the Austrian Science Fund (FWF), projects P74 N and P97 Universidad Pública de Navarra, Departamento de Ingeniería Matemática e Informática, Campus de Arrosadia, Pamplona, Spain (higueras@unavarra.es). University of Vienna, Faculty of Mathematics, Nordbergstraße, A 9 Wien, Austria (natalie.happenhofer@univie.ac.at). Vienna University of Technology, Institute for Analysis and Scientific Computing, A 4 Wien, Austria (othmar@othmar-koch.org). University of Vienna, Faculty of Mathematics, Nordbergstraße, A 9 Wien, Austria (Friedrich.Kupka@univie.ac.at).

4 Higueras et al. of the spatial profile does not increase artificially in the course of time integration [, ]. 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, [,, 8]). In [], it as 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 ne schemes hich have additional properties beneficial to astrophysical simulations at the cost of reducing the region of absolute monotonicity from the optimum. We ill demonstrate that the ne 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 [] indicates that the folloing 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 [, ]. This implies that both the explicit and the implicit schemes are SSP. Furthermore, the Kraaijevanger radius (also knon as the radius of absolute monotonicity) [] of both schemes should also be large. The stability function of the implicit scheme should tend to ero at infinity and the stability region should contain a large subinterval of the negative real axis [, ] ith >. This is ensured by L stability. For the explicit scheme, the stability region should contain large subintervals of the negative real axis, [, ] ith >, and also of the imaginary axis, [ i, i, ] ith >. The latter requirement is associated ith a stable integration of the hyperbolic advection terms (see [4, ]). For both schemes, the stability function should be nonnegative for a large interval of the negative real axis, [, ] ith >. This condition is directly related to the step sie restrictions associated ith the dissipativity of the spatial discretiation [4] 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 []. For a convenient and memory efficient implementation, the coefficients of the scheme should be rational numbers hich could enable to recombine the stages in a suitable ay. The properties listed above turn out to be more important for a successful simulation than third order accuracy. Still, e cannot use the optimal SSP(,,) to stage method [], because the number of degrees of freedom does not allo to have a positive stability function in conjunction ith L stability. Thus, e ill focus on the construction of second order three stage IMEX RK schemes (A, Ã, bt ) of the form c a c a a A b b b c γ c ã γ c ã ã γ Ã b b b (.)

5 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study Observe the structural properties of this scheme: the eight 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 [7]. Another advantage of IMEX schemes ith b = b is that they preserve linear invariants of the ODE []. The second property is also interesting from the computational point of vie because it allos to solve, stage to stage, the nonlinear systems that arise hen implicit RK methods are used. Actually, the Jacobian matrix may even be froen throughout the iterations for all the stages (see [8]). Imposing second order convergence leaves a total of seven degrees of freedom for optimiation as e explain belo. The rest of the paper is organied as follos. In, e revie some knon results that are used throughout the paper. Section is devoted to the construction of a second order stage IMEX RK method ith 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 e construct IMEX RK methods hose implicit scheme is the optimal second order stage SSP RK method. In addition, although the optimal second order stage SSP SDIRK method is not L stable, e also investigate properties of IMEX RK methods constructed ith this optimum SDIRK scheme; this study is done in and. 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 ne methods, in the appendix section e also include the properties of some methods from the literature. The methods have been obtained ith the help of the symbolic computation softare Mathematica.. Revie of some knon concepts. In this section e briefly revie some knon concepts that are used along the paper... Order of convergence. As e have pointed out above, the IMEX scheme should achieve second order. To fulfill this requirement, the folloing conditions should be imposed (see for example [7]), b t e =, b t c =, bt c =, (.) here, as usual, e = (,..., ) t and c = (c,..., c s ), c = ( c,..., c s ); furthermore, e ill also assume that A e = c, Ã e = c. (.) We ould 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 = F (y) + G(y), (.) ɛ

6 4 Higueras et al. here ε. Uniform convergence of IMEX RK methods applied to solve systems of the form (.) is studied in []. To obtain the results, in [], the additive ODE (.) is transformed into a partitioned system of the form It turns out that, if y = f(y, ), ε = g(y, ). then, for ε C t, the global error satisfies b t à c =, (.4) y n y(t n ) = O(( t) p ) + O(ε( t) ), n (t n ) = O(( t) ), here p is the order of the explicit scheme; if conversely (.4) is violated, the global error is of the form y n y(t n ) = O(( t) p ) + O(ε t), n (t n ) = O( t)... Radius and regions of absolute monotonicity. For RK and IMEX RK methods, step sie 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 [, 4,,,,, 4, 7, 8,,,, 7, 8,,,, ] (see [, 7, 9, 9] for revies on the topic). An s stage RK method (A, b t ) is said to be absolutely monotonic at a given point r, ith r, if I + ra is nonsingular, and (I + ra) A, (I + ra) e, (.) here no e = (,,..., ) t R s+, A is defined by ( ) A A = b t, and the inequalities in (.) are understood component ise. The radius of absolute monotonicity R(A) is defined by R(A) = sup{ r r and A is absolutely monotonic on [ r, ] }. For RK methods, monotonicity can be ensured under a stepsie restriction of the form t τ R(A), here τ is the step sie restriction for monotonicity hen the explicit Euler method is used. For details see []. For additive RK methods, the concept of radius of absolute monotonicity is extended to the region of absolute monotonicity [, Definition.] (see also []). An s stage additive RK method (A, Ã) is said to be absolutely monotonic (a.m.) at a given point ( r, r ) ith r, r, if the matrix I + r A + r à is invertible, (I + r A + r Ã) e, and (I + r A + r Ã) A, (.) (I + r A + r Ã) Ã. (.7)

7 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study The region of absolute monotonicity, R(A, Ã), is defined by R(A, Ã) = { (r, r ) r, r and (A, Ã) is a.m. on [ r, ] [ r, ] }. Numerical monotonicity can be ensured for the additive RK method (A, Ã) under the stepsie restriction t min {r τ, r τ }, here r and r are such that the point (r, r ) R(A, Ã), and τ, τ > are the step sie restrictions for monotonicity hen the explicit Euler method is used for F and G, respectively (see [] for details). Consequently, in order to obtain nontrivial step sie restrictions for RK and additive RK methods, e should have, respectively, R(A) >, and points (r, r ) R(A, Ã) ith r > and r >. In [, ], algebraic criteria for nontrivial radius and regions of absolute monotonicity are given in terms of sign conditions of the coefficient matrix (or matrices), namely, A (or A, Ã ), and some inequalities of the incidence matrix of certain matrices. A trivial ay to ensure these properties for IMEX schemes is to impose a ij, ã ij > for i < j, b j, b j >, and γ >. (.8) For this reason, for the IMEX RK schemes constructed in this paper e ill assume the positivity conditions (.8)... Amplification function for second order point and fourth order point spatial discretiation. In order to study the stability of numerical schemes, e can study the dissipativity of time integrators in conjunction ith spatial discretiations by means of Fourier analysis []. 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 here the spatial discretiation parameter tends to ero [4]. We thus consider the heat equation u t + a u xx = and the second order point spatial discretiation u xx (x j, t n ) un j+ un j + un j ( x), and the fourth order point spatial discretiation u xx (x j, t n ) un j+ + un j+ un j + un j un j ( x). These are to of the spatial discretiations actually implemented in the ANTARES simulation code, hich numerically solves the equations of hydrodynamics and various generaliations thereof [, ]. When these spatial discretiations are used to solve the heat equation, the stability function R() of the RK method (A, b t ), defined by R() = + b t (I A) e, is evaluated at a point of the form = µ h(θ), here µ, and h : [ π, π] R, depends on the discretiation considered. The function h satisfies h() > for [ π, π],, and, due to the consistency of the spatial discretiation, it also has the property h() =. We thus obtain the amplification function g(µ, θ) = R( µ h(θ)). (.9)

8 Higueras et al. Observe that R() = implies g(µ, ) =. In particular, for the second order point and for the fourth order point spatial discretiations e obtain, respectively, the folloing functions h and h, ( ) θ h (θ) = (e i θ + e i θ ) = 4 sin, (.) h (θ) = ( e i θ + e i θ e iθ e i θ ) = ( ) θ (7 cos(θ)) sin. (.) Since the functions h in (.) and (.) are even, h(θ) = h( θ), e subsequently restrict the values of θ to θ [, π]. In the dissipativity analysis e are interested in the values: a) µ such that, for µ [, µ ], it holds that g(µ, θ) > for θ [, π], and b) µ such that, for µ [, µ ], it holds that g(µ, θ) for θ [, π]. If µ i =, i =,, e ill understand that the interval [, µ i ] is [, µ i ). It turns out that, if is the first negative ero of R(), from the definition of g(µ, θ) in (.9) e obtain that The loest value µ is given by h(θ) µ =, θ [, π]. µ = max θ [,π] h(θ). In a similar ay, if is the first negative ero of R(), e obtain that The loest value µ is given by h(θ) µ =, θ [, π]. µ = max θ [,π] h(θ). Observe that [, ] is the stability interval of the RK method, that is, the intersection of the stability region ith the real axis. In particular, for the points and for the points discretiations, as h i is monotonic, e get max h (θ) = h (π) = 4, θ [,π] max h (θ) = h (π) = θ [,π]. Consequently, for each RK method, in the dissipativity analysis, it is enough to compute, the first negative ero of R(), and, the first strictly negative ero of the function R(). For each spatial discretiation, the values µ and µ are simply scaled values of and, respectively, and thus, for our purpose, the larger and, the better.. A ne second order stages SSP IMEX scheme. In this section e construct a ne second order stages SSP IMEX scheme of the form (.) ith all the requirements explained in. We begin constructing the explicit scheme and, in a second step, e ill deal ith the implicit one. Finally, e ill fix the remaining free parameters to obtain the desired properties for the IMEX scheme.

9 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study 7.. Explicit scheme. Condition (.) and second order conditions (.) for the explicit scheme lead to b = b b, c = b c b, a = c, a = c a. (.) Hence, four degrees of freedom of the explicit scheme are left to be determined after the previous considerations. For a second order stage explicit scheme, the stability function is given by [8] Thus, defining e obtain the stability function R() = b a c. a = α b c, (.) R() = α. (.) To express the four degrees of freedom, in the folloing e choose α, c, b, and b. In the next subsection, e study the stability function (.) and the stability region S in terms of α.... Stability function for the explicit scheme. We study the stability function (.) to determine the values of α such that the method fulfills all the desired requirements. Intersection of the stability region ith the imaginary axis. To obtain the intersection of the stability region S ith the imaginary axis, e study the values of α such that R(i ). In our case, from R(i ) = + i ( α), e obtain that, for α > /8, the interval [ i (α), i (α)] is contained in the stability region S, here (α) = 8 α /( α). The plot for (α) can be seen in Figure. (continuous line). The maximum value is (/4) =. In particular, e obtain that the interval [ i, i] is contained in the stability region S for [ ( α ), ( + ) ] [.97,.8]. (.4) ( ) In the Figure., e sho the stability regions for α = (continuous ( ) line) and α = + (dashed line), and a oom in the area close to the imaginary ( ) axis. It can be seen that for α =, 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., e sho the regions of absolute stability for different values of α. Intersection of the stability region ith the real axis. To obtain the intersection of the stability region S ith the real axis, e study the values of > such that R( ). In Figure. (dashed line), e see the values of (α) for α > /8.

10 8 Higueras et al Α Fig... For α [/8, ]. Continuous line: > such that [ i, i] S; dashed line: > such that [, ] S; dotted line: > such that R( ) ; dashed dotted line: R Lin (A) ( Fig... Left: stability regions for α = ) ( (continuous line) and α = + ) (dashed line). Right: oom in the area close to the imaginary axis. Nonnegativity of the stability function. We study the points > such that R( ). In Figure. (dotted line), e sho the values of (α) such that R() for [ (α), ], hen α > /8.... Radius of absolute monotonicity for linear problems. In order to obtain the radius of absolute monotonicity for linear problems (or the threshold factor), e compute the Taylor expansion of the stability function (.) at x = p to obtain R() = p + p p α + (p + ) ( p + p α ) +(p + ) ( p α ) + α (p + ). The largest value p such that all the coefficients are nonnegative is the radius of absolute monotonicity of the method [, ]. For α /8, e observe that all the coefficients are nonnegative if p /(α), and hence the radius of absolute monotonicity for the linear problem is given by R Lin (A) = α. (.) In Figure. (dashed-dotted line) e sho the radius of absolute monotonicity as a function of α. Recall that, for explicit second order stage RK methods, the optimal radius of absolute monotonicity for linear problems R Lin (A) = is obtained

11 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study 9 Fig... Regions of absolute stability for different values of α: dashed ( bron: α =.7; continuous magenta: α =.; dashed pink: α =.; continuous pink: α = + ). for α = / []. Hoever, for α = / there is not any interval of the imaginary axis contained in the stability region S, as this requires α > /8.... Radius of absolute monotonicity for nonlinear problems. For nonlinear problems, the radius of absolute monotonicity R(A) satisfies R(A) R Lin (A) []. In our case, if b = α, c = α, after some computations, from (.) e obtain that α b 8 α 8α, (.) R(A) = R Lin (A) = α. (.7)..4. Choice of α. So far, after imposing (.) (.) and (.), e have to free parameters left, namely b and α, that should satisfy α b 8 α [ ( 8 α, α ), ( + ) ] [.97,.8]. 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 (.), and ) expressions (.4) and (.7), e can conclude that α should be close to.97. A choice that is compatible ith this requirement is α = / For this value, the coefficients for the explicit scheme obtained so far are b b b (.8) A 4 b b here b is such that b. (.9)

12 Higueras et al. For scheme (.8), expression (.7) is R Lin (A) = R(A) =. Furthermore, the interval [ i, i] S for =.; the interval [, ] S for =.8474, and R( ) for [,.88]. The stability region, and a oom of the region close to the imaginary axis is given in Figure.7... Implicit scheme. For the implicit scheme, the eight vector b is the same as the one for the explicit scheme. Thus, from the previous section, see (.8) and (.9), e have that b = 4 b, b =, b. (.) Condition (.) and second order conditions (.) for the implicit scheme lead to c = b c + b γ + b γ γ + b, c = γ, (.) ã = c γ, ã = c ã γ. (.) Thus, the implicit scheme introduces three additional degrees of freedom to the combined scheme (.). We choose these as c, ã, and γ in the folloing. Since b is constrained by the second order condition (.) ith (.) and ith (.), and since b is constrained by (.), it remains to constrain b and thus a total of four degrees of freedom is left to be optimied for the implicit scheme. In the next subsection, e study the stability function of the implicit scheme and impose conditions to obtain L stability.... Stability function for the implicit scheme. As it has been pointed out in, e aim at constructing an L stable method, that is, an A stable method such that lim R() =. A detailed study of stability issues for SDIRK methods is given in [8, Chapter IV.]. For SDIRK methods ith a = = a ss = γ, the stability function is of the form R() = P () Q(), here P () is a polynomial of degree at most s, and Q() = ( γ) s. For these schemes, in the case γ >, A stability is equivalent to E(y) = Q(i y) P (i y) for all y R. (.) For L stable methods, R( ) = and thus the highest coefficient of P is ero. Furthermore, if the method is knon to be of order p, ith p s, then the rest of the coefficients are uniquely determined in terms of s degree Laguerre polynomials (see [8, Chapter IV., p. ]). These results can be applied to our method (p = and s = ). In this case, according to [8, Chapter IV.], expression (.) is given by ( E(y) = y 4 γ γ γ + γ ) + y γ, (.4) 4

13 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study... Fig..4. Error constant for second order stage SDIRK methods. and the stability function can be computed as R() = ( γ γ + ) + ( γ) + ( γ ). (.) The regions of γ for L stability can be obtained imposing nonnegativity to the coefficients of E(y) in (.4); in this ay e obtain that the parameter γ should satisfy ( 9 + ( + 7 ) ) ( γ + ) , 4 hose numerical approximations (see [8, Chapter IV., Table.4]) are given by.84 γ.8. (.) For second order stage L stable SDIRK schemes, the error constant is also knon in terms of the degree Laguerre polynomial L (x) (see [8, Chapter IV., p. ]), C = L ( γ ) γ = γ + γ γ +, and third order is obtained for γ =.48. In Figure.4 e sho the values of log C. If e compute the function E(y) from the Butcher tableau of the SDIRK method, and e use the values of b, b, c, c, ã, and ã given by (.) (.), e obtain that the function E(y) is of the form (.4) if c = ã b γ + γ 4 γ + γ ã b. (.7) For this value, the stability function computed from the Butcher tableau is (.). At this point, e recall that the coefficients of the SDIRK method should be positive. With the value of c given by (.7), the coefficient ã in (.) is ã = γ ( γ 4 γ + ) ã b. If e impose that ã > and take into account (.), e obtain that γ must satisfy.84 γ ( ) or ( + ) γ.8.

14 Higueras et al. ( ) ( ) As.989 and , taking into account the sie of the error constant (see Figure.4), e restrict the values of γ for L stability in (.) to.84 γ.989. (.8) Note that this interval does not contain the value of γ for hich the scheme ould be of third order. With regard to the intersection of the real and imaginary axis ith the stability region, observe that, as the method is L stable, the imaginary axis as ell as the real axis is contained in the stability region S. It remains to study the nonnegativity of the stability function. We analye no if there is any value of γ such that R() for all <. We obtain a positive anser for ( γ, ( ) ] (,.84]. (.9) Combining this result ith (.8), e get that adequate values for γ are.84 γ.84. (.)... 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, e have to find the points such that the stability function R is absolutely monotonic at. 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) for k. For explicit methods, the stability function R is a polynomial, and thus, e can analye a finite number of derivatives to check if the function is absolutely monotonic at a given point x. Hoever, for implicit schemes, R is a rational function, and e cannot use directly the definition of absolute monotonicity. In this section, e apply Theorem 4.4 in [] to obtain the radius of absolute monotonicity for linear problems for our SDIRK method. Roughly speaking, this result allos us to restrict the analysis to a finite number of derivatives, that is, R (k) (x) for k < K(x), and the key point is to obtain the integer K(x). In the folloing proposition e compute it for the stability function (.). Proposition.. Assume that γ satisfies (.), γ ( ) /. If R (k) (x) for k =,,,, then the stability function (.) is absolutely monotonic at x. Proof. In order to apply [, Theorem 4.4], e construct all the elements (sets, intervals, functions, etc.) involved in this result. Folloing the notation in [], the stability function (.) is of the form R() = P ()/Q() ith ( P () = γ γ + ) ( ) + ( γ) +, Q() = γ γ. The polynomials P and Q have degrees m = and n =, respectively, and there is a unique pole α = /γ > ith multiplicity µ(α ) =. Thus the set A = A + = {α }, I(α ) = R, and B(R()) =. Next, the stability function should be decomposed in partial fractions of the form R() = c(α, ) α + c(α, ) (α ) + c(α, ) (α ).

15 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study x Γ Fig... From bottom to top, values of x such that condition (.) holds for k =,,, 4. In our case, c(α, ) = γ γ + γ, c(α, ) = γ + γ γ 4, c(α, ) = γ 4 γ + γ. For γ satisfying (.), γ ( ) /, these functions have constant sign. More precisely, c(α, ), c(α, ), and c(α, ) >. No, e construct the function (see [, Formula (4.)]) F (k, x) = c(α, ) k! (k + )! ( ) γ x c(α, ) (k + )! (k + )! γ x, here e have used the sign conditions of c(α, ) and c(α, ). Another function needed to construct K(x) in [] is L(x); in our case, L(x) =, see [, Formula (4.)]. We can no construct the integer K(x) in [, Definition 4.]), defined as the smallest integer k such that k L(x) and F (k, x) < c(α, ). (.) For γ satisfying (.), in Figure. e sho, from bottom to top, the values of x such that condition (.) holds for k =,,, 4. We do not consider k = because, as e have seen in the previous subsection (see (.9)), for the values of γ satisfying (.), it holds that R() for. Furthermore, e restrict the values to x [, ] because, by numerical search, it has been found that the nonlinear optimum radius of absolute monotonicity for second order stage SDIRK schemes is [4, 9]. Consequently, for x [, ], K(x) 4. We can no apply Theorem 4.4 in [] to ensure that R() is absolutely monotonic in [x, ] if and only if c(α, ) > and R (k) (x) for k < K(x). Thus, the next step is to analye the values of x such that R (k) (x) for k =,,,. For each γ satisfying (.), the radius of absolute monotonicity is given by 4 γ R(γ) = γ ( γ γ + ) γ + 8 γ γ + γ ( γ γ + ), (.) and varies in the interval [4., ]. In Figure. e sho, for each γ, the interval of absolute monotonicity, and a oom 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.

16 4 Higueras et al. x 4 R Γ Γ Fig... Left: Interval of absolute monotoniciy. Right: oom of the range values of the radius of absolute monotonicity... Uniform convergence for IMEX Runge Kutta methods. Condition (.4) seems to be of interest hen stiff systems of the form (.) are solved. For this reason, e impose the implied relationship ã = γ γ b. (.).4. Choice of γ. Based on the analysis done so far, no e choose the value of γ satisfying (.). If e rationalie the mid point,.8948, e obtain γ =.888. (.4) At this point, it remains to obtain the parameter b ; recall from (.) that this value should satisfy b. (.) We still have to deal ith the radius of absolute monotonicity for nonlinear problems for the SDIRK method, and ith the region of absolute monotonicity for the IMEX method... Absolute monotonicity for the SDIRK and IMEX schemes. In this section e determine b satisfying (.) to obtain a large radius of absolute monotonicity for the SDIRK method, and a large region of absolute monotonicity for the IMEX scheme. From (.), a study of the radius of absolute monotoninicy R(Ã) for the SDIRK method yields that the largest value possible is R(Ã) = 4/.88, and this value is attained for any b such that 4 7 b 44 7 (. b.98). (.) Considering the points (r, r ) in the region of absolute monotonicity R(A, Ã), hen r = (or r = ), e obtain the point (, r ) (or (r, )), that is, the intersection of the region of absolutely monotonicity ith the axis r = (axis r = ). These

17 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study values satisfy r R(Ã), r R(A). Quite often, due to the mixed conditions (.) for r =, and (.7) for r =, that is, (I + r Ã) A, (I + r A) Ã, e obtain r < R(Ã) and r < R(A). We aim for determining the largest values of r and r such that the points (r, ) and (, r ) belong to the region of absolute monotonicity. A detailed study yields the result that the largest value of r such that (, r ) R(A, Ã) is r = /4.488, provided that b 4 (. b.48). (.7) In a similar ay, the largest value of r such that (r, ) R(A, Ã) is r = /.99, provided that b 9 Taking into account (.), (.7), and (.8), e consider (. b.8). (.8) b = 4.. (.9) By fixing this value, e have finished the construction of the IMEX scheme... Second order stage SSP IMEX scheme (summary). In this section, e give the coefficients of the IMEX scheme obtained from our earlier considerations, that is (.) A 4 4 Ã 4 and summarie 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 R A () = + + +, RÃ() = ( ) ( ). (.) In Figure.7, e sho the stability regions for the explicit and the implicit schemes in (.), as ell as a oom of the region in a neighborhood of the origin. In Figure.8, e sho the stability region of the IMEX scheme and a oom of the region at the origin. For the explicit scheme, e obtain R() for [.88, ], and R() for [.8474, ]. Furthermore, R(i) for [.,.]. For the implicit scheme, R() and R() for. Furthermore, R(i) for R. This IMEX method satisfies condition (.4) for uniform convergence. 4

18 Higueras et al. Im Im 4 Re Re Im Im Re Re Fig..7. Stability region and a oom of the stability region of the explicit scheme in (.) (top) and the implicit scheme in (.) (bottom). r r Fig..8. For IMEX (.): stability region (right), a oom of the region at the origin (center) and region of absolute monotonicity (right). With regard to the radius of absolute monotonicity, for linear problems e have that ( R Lin (A) = / =., R Lin (Ã) = ) (.) For nonlinear problems, e have R(A) = / =., R(Ã) = 4/.888, and { R(A, Ã) = (r, r ) R : r <, r } 4 ( r ). In Figure.8 e sho the region of absolute monotonicity for the IMEX method (.).

19 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study 7 Due to the properties listed above, e ill refer to scheme (.) as SSP(,,) LSPUM, here the letters have the folloing 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 alays positive; U : the IMEX method features uniform convergence (.4); M : the IMEX method has a nontrivial region of absolute monotonicity. (.) To assess the merits of the method just constructed, in the next section e ill construct alternative schemes based on optimal methods from the literature. These ill be compared experimentally in IMEX methods ith second order stage optimum SSP explicit RK scheme as explicit method. In this section, e construct a number of additional SSP IMEX methods, based on the optimal explicit three stage second order method in conjunction ith a compatible implicit scheme. These ill be compared to the scheme constructed in Section. Thus, e consider IMEX schemes of the form A c γ c ã γ c ã ã γ Ã (4.) The explicit method coincides ith the explicit scheme for the SSP(,,) schemes in [, 7]. Observe that no, the eight vector b for the implicit scheme is already fixed. Recalling the discussion from., e note that this leaves only three degrees of freedom for the scheme (4.), namely, c, ã, and γ. For the implicit schemes considered in this section, e ill impose L stability and the nonnegativity of the stability function R() for all. As the stability function for an L stable second order stage SDIRK method only depends on the diagonal elements γ, the study conducted in.. is valid and, therefore, e choose again γ = /. Furthermore, after imposing second order conditions for the implicit scheme, and a stability function of the form (.), c is also determined, and e obtain that there is one free parameter left: ã. In this section, e construct three schemes by choosing this value as follos:. In the first one, e impose condition (.4) for uniform convergence, see 4... In the second one, e optimie the radius of absolute monotonicity of the SDIRK scheme, see 4... In the third one, e optimie the region of absolute monotonicity of the IMEX scheme, see IMEX method ith uniform convergence (.4). In this case, the IMEX scheme obtained by imposing (.4) is given by the coefficient tableaux (4.) A Ã

20 8 Higueras et al. r r Fig. 4.. For the IMEX method (4.): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity. This method satisfies condition (.4) for uniform convergence. Note that (.) must not be reused to derive (4.), because (.) has been derived assuming (.) here b = / instead of b = /. Recalling (.) and (.), the stability function for the explicit scheme is R A () = + + +, (4.) hereas for the implicit scheme it is given by RÃ() in (.), since it depends only on γ (see (.)). For linear problems, the radius of absolute monotonicity for the explicit scheme is R Lin (A) =, hereas for the implicit scheme R Lin (Ã) is given by (.). For nonlinear problems, the explicit scheme is the optimal second order stage explicit SSP method and thus R(A) = ; for the implicit method, from (.), e get R(Ã) = For the IMEX scheme, { R(A, Ã) = (r, r ) R : r < ( 8 7 r )} 7 r 4 r + 9. The points (,.8) and (, ) are included in the region of absolute monotonicity R(A, Ã). In Figure 4. e sho the stability region and the region of absolute monotonicity of the IMEX scheme. Due to its properties, e ill henceforth refer to method (4.) as SSP(,,) LPUM (see (.)). 4.. IMEX scheme ith large radius of absolute monotonicity R(Ã). A detailed study shos that the largest value for R(Ã) is.884, and that this value is attained for ã =.9. We rationalie this number to obtain ã = We thus obtain the IMEX scheme (4.4) A Ã

21 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study 9 r r Fig. 4.. For the IMEX method (4.4): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity. We remark that this method does not satisfy condition (.4) for uniform convergence. The stability functions for the explicit and implicit RK methods in (4.4) are given by (4.) and (.), respectively. For linear problems, the radius of absolute monotonicity for the explicit scheme is R Lin (A) =, hereas for the implicit scheme R Lin (Ã) is given by (.). For nonlinear problems, the explicit scheme is the optimal second order stage explicit SSP method and thus R(A) = ; for the implicit method, e have R(Ã) = ( 8749 ) For the IMEX scheme, R(A, Ã) = { (r, r ) R : r < 7 } (r 44) 47r 7 7r The points (,.849) and (, ) are included in the region of absolute monotonicity R(A, Ã). In Figure 4. e sho the stability region and the region of absolute monotonicity for the IMEX scheme. 4.. IMEX scheme ith large region of absolute monotonicity. A detailed study shos that the largest value of r such that (, r ) R(A, Ã) is r =.4; this value is attained for ã =.47. We rationalie this number to obtain that for ã =.479, e have that the point (, r ) is in the region of absolute monotonicity for r = ( ) We thus obtain the IMEX scheme (4.) A Ã

22 Higueras et al.. r r Fig. 4.. For the IMEX method (4.): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity. We remark that this method does not satisfy conditon (.4) for uniform convergence. The stability functions for the explicit and implicit RK methods in (4.) are given by (4.) and (.), respectively. In Figure 4. e sho the stability region for the IMEX scheme. For the implicit scheme, R(Ã) = ( ) For the IMEX scheme, the region of absolute monotonicity is shon in Figure 4.. The points (,.4) and (, ) are included in the region of absolute monotonicity R(A, Ã). We ill later refer to this scheme as SSP(,,) LPM (see (.)).. IMEX scheme ith optimal explicit SSP RK and optimal SSP SDIRK SSP methods. In this section e analye the IMEX method constructed ith the optimal explicit and implicit stage SSP methods, (.) A Ã This is a second order IMEX method such that b = b. The explicit part is the optimal three stage explicit SSP method of second order [], hile the implicit part represents the optimal three stage scheme [, 4, 9]. The stability function for the explicit scheme is given by (4.), hile for the implicit scheme it is (see, e.g., []) RÃ() = ( + ) ( ). (.) In Figure. e sho the stability regions for the explicit and the implicit methods in the IMEX scheme (.). Observe that the implicit scheme is A stable. Hoever, as lim R() =, the method is not L stable. For the explicit scheme, e obtain R() for [.874, ], and R() for [ 4.984, ]. Furthermore, R(i) > for all R.

23 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study Im Im 4 Re Re Im Im Re Re Fig... Stability region and a oom of the stability region of the explicit scheme in (.) (top) and the implicit scheme in (.) (bottom). r r Fig... For the IMEX method (.): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity. For the implicit scheme, R() for [, ] and R() for. Moreover, R(i) = for R. It can be checked that this IMEX method satisfies condition (.4) for uniform convergence. With regard to the radius of absolute monotonicity, for linear problems e have that [, ] R Lin (A) = R Lin (Ã) =. For nonlinear problems, e have R(A) =, R(Ã) =, and { R(A, Ã) = (r, r ) R : r <, r (8 r ) } r r + 48.

24 Higueras et al. In Figure., e sho the region of absolute monotonicity. The points (, ) and (,.77) are included in the region of absolute monotonicity R(A, Ã).. IMEX method ith second order stage optimum SSP SDIRK method. In this section e consider IMEX schemes such that the implicit method is the optimal stage second order SSP SDIRK method [], c a c a a (.) A Ã We aim at constructing an explicit scheme such that there is a nontrivial intersection beteen the stability region and the imaginary axis. Second order conditions (.) yield c = ( c ), a = c a, (.) and ith a = α c, (.) e obtain the stability function (.). Therefore, the analysis conducted in subsection.. in terms of α is valid. We take α = /, the parameter chosen in.., and, after this choice, since b = b =, there is only one free parameter, namely c. It remains to analye 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 (.4) to ensure uniform convergence. We ill use these conditions to fix the value of c. If e impose condition (.4) for uniform convergence, e obtain c = /. 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 (.), e obtain R Lin = /. Remember that for the scheme constructed in, e ere able to obtain for the nonlinear case R(A) = /; hoever, ith (.) and (.) and α = /, this cannot be achieved. In fact, unfortunately, the radius for the nonlinear case is rather small. It can be computed that R(A) = c + 8 c c, hose maximum value, R(A) = ( 8 4 ) /., is attained for c = /.4497; this value is much smaller than, the optimum value possible for second order stage RK methods []. In the next sections, e give the coefficients and the properties of this family of methods for different values of c.

25 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study r r Fig... For the IMEX method (.4): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity... IMEX ith c = /. We obtain the IMEX method (.4) A Ã This is a second order IMEX scheme such that b = b. As α = /, the stability function of the explicit method is the same as for the scheme (.). The implicit method is the optimum stage second order SSP SDIRK scheme. The stability functions are given by R A () = ( + ), RÃ() = ( ). This IMEX method satisfies condition (.4) for uniform convergence. For the radius of absolute monotonicity for nonlinear problems, e obtain R(A) = / =.4, and R(A, {(r Ã) =, r ) R : r <, r } ( r ). The points (,.) and (.4, ) are included in the region of absolute monotonicity R(A, Ã). In Figure. e sho the stability region for the IMEX scheme and the region of absolute monotonicity... IMEX method ith c =. The IMEX method is ( 9 ) A Ã (.)

26 4 Higueras et al Fig... Stability regions for the IMEX scheme hen c = / (solid line), and c = (dotted line). r r Fig... For the IMEX method (.): Left: Stability region; center: oom of the stability region; right: region of absolute monotonicity. This is a second order IMEX method such that b = b. As α = /, the stability function is the same as for scheme (.). The implicit method is the optimum stage second order SSP SDIRK scheme. The stability functions are given by R A () = ( + ), RÃ() = ( ). In Figure., e compare the stability regions for the IMEX scheme hen c = /, solid line, and c =, 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 (.4) for uniform convergence. ( For the radius ) of absolute monotonicity for nonlinear problems, e obtain R(A) = 8 4., and { R(A, Ã) = (r, r ) R : r < ( 8 4 ), r r } In Figure. e sho the stability region and the region of absolute monotonicity of the IMEX scheme. The points (,.9) and (., ) are included in the region of absolute monotonicity R(A, Ã). 7. Numerical experiments. To demonstrate the capabilities and limitations of the time integrators discussed in the previous sections, e tested the methods on

27 Optimied IMEX Methods for Simulations in Astrophysics: a detailed study Method t max t mean CFL max CFL mean CFL start Time SSP(,,) PM, γ=.4 7. s 9.4 s...4 :4:4 SSP(,,) LUM 9. s 7. s..8. :4:7 SSP(,,) LSPUM 88.9 s 7. s :9: SSP(,,) LPUM. s s ::4 SSP(,,) LPM.48 s 8. s... :4: SSP(,,) LPM 9. s. s..7. Table 7. Simulation A: Time steps, CFL numbers and allclocktimes over the first 8 scrt. simulations of double diffusive convection in an idealied setting performed ith the hydrodynamics code ANTARES []. Since the ANTARES code and the setting are described in detail in [], e ill restrict ourselves to the results. The challenge of these simulations is that they require the time integration method to ork for quite different conditions ithin a single simulation run: only a method hich can handle stiff terms and is also SSP can succeed ith acceptable time steps. In this section e test the schemes named SSP(,,) LSPUM, SSP(,,) LPUM, and SSP(,,) LPM, hose coefficients are given by (.), (4.), and (4.), respectively. For the meaning of the notation L S P U M, e refer to (.). 7.. Simulation A. This setting corresponds to Simulation in [], namely, e assume a Prandtl number P r =., a Leis number Le =., a modified Rayleigh number Ra =., and a stability parameter R ρ =.. The simulations ere run on the Vienna Scientific Cluster (VSC) on 4 CPUs. To demonstrate the merits of the methods constructed in this paper, e also give results for the method IMEX SSP(,,), hich is a combination of the forard and backard Euler methods (see (9.)). It becomes clear that such a simple lo order scheme is not advisable for our purposes. For a better comparison, e also list the best performing schemes of [], namely the SSP(,,) scheme ith γ =.4 and the SSP(,,) method, hose coefficients are given in (9.), and (9.), respectively. According to the notation introduced in this paper, see (.), e ill refer to these methods as SSP(,,) LPM, SSP(,,) PM, and SSP(,,) LUM. In Table 7. e sho the time steps, CFL numbers and allclocktimes over the first 8 scrt. The attentive reader ill notice that there is no allclock time given for the simulation performed ith IMEX SSP(,,) LPM. The reason for this is that the time steps achieved by this scheme ere so small (note that CFL start =.!) that there as no point in asting computational resources for an excessively long comparative simulation. For IMEX SSP(,,) LPM, the time steps observed in the course of the simulation are shon in Figure 7.. Figure 7. compares the evolution of the time steps for the different schemes used. 7.. Simulation B. As for Simulation A, this simulation shos the idealied evolution of a single double diffusive layer. Hoever, the parameters differ and are

28 Higueras et al. τ τ diff τ fluid τ visc t in [s] scrt Fig. 7.. Simulation A: Time step evolution ith time integrator IMEX SSP(,,) LPM. 9 SSP(,,)-LUM co=. SSP(,,)-LSPUM co=. SSP(,,)-LPUM co=.4 SSP(,,)-LPM co=. 8 7 t in [s] 4 scrt Fig. 7.. Simulation A: Time step evolution over the entire scrt. chosen such that the full capabilities of the IMEX schemes in the diffusive region are tested. The parameters are P r =., Le =., Ra =., R ρ =.. The resolution is 4x4 grid points. The simulations ere run on the VSC on 4 cores. For better comparison, e also tested the best performing scheme IMEX SSP(,,) LUM from [] (see (9.)). Note that for stability reasons, the adaptive time step criterion of [] as slightly modified. In Simulation A the time step as readjusted at the occurence of n y. to point oscillations (n y denoting the number of grid points in the horiontal direction), and, to permit the system to readjust itself, no modifications on the time step ere alloed in the subsequent time steps. Hoever, for Simulation B, this criterion as chosen to be more restrictive, namely, the time step is diminished at the occurence of n y. to point instabilities and if the system has not adjusted itself after 8 time steps, the time step is further reduced. In Table 7. e sho the time steps, the CFL numbers, and allclocktimes over