Second-order stabilized explicit Runge–Kutta methods for stiff problems

Martín-Vaquero, J.; Janssen, Bill

doi:10.1016/j.cpc.2009.05.006

Cited by 30 publications

(24 citation statements)

References 41 publications

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“…We also observed that higher-order methods can have more difficulties with the propagation of errors with several numerical tests [16]. In Ref.…”

Section: Introductionmentioning

confidence: 55%

“…Existing codes based on stabilized explicit Runge-Kutta methods have difficulty in problems where some of the eigenvalues are very large (especially the algorithms with order higher than two), as we showed in Ref. [16]. We compared several excellent methods such as ROCK2, RKC, DUMKA3, and ROCK4 and showed that they each have difficulties in solving some very stiff problems: higher-order methods require small steps to solve problems where any of the eigenvalues are larger than 10 6 in absolute value, and second-order methods such as ROCK2 and RKC obtain larger errors than permitted in some problems with small tolerances.…”

Section: Introductionmentioning

confidence: 92%

“…Lebedev and Medovikov used these ideas and the composition method with the code DUMKA. For example, DUMKA3 used stability polynomials of degrees s up to s max = 243; however, DUMKA3 has some difficulties when some of the eigenvalues of the problem are more negative than −10 5 or −10 6 [16]. DUMKA3 is available at the web page http://www.math.tulane.edu/∼amedovik/, but we are not using it here for comparison since it is a third-order code.…”

Section: Introductionmentioning

confidence: 99%

“…In Ref. [16], we showed a new way to build stabilized explicit Runge-Kutta methods: we used sixth-order polynomials to derive second-order methods to show that the new procedure to construct the methods makes it possible to obtain good stability properties. In this article, we shall use a very similar technique, but with second-order polynomials instead of the sixth-order ones we used in the previous article.…”

Section: Introductionmentioning

confidence: 99%

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SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Kleefeld

Martín-Vaquero

2012

Numerical Methods Partial

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Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge-Kutta methods (also called Runge-Kutta-Chebyshev methods) were proposed to try to avoid these difficulties. The Runge-Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín-Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802-1810, we showed that previous codes based on stabilized Runge-Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth-order polynomials. Here, we develop a new method based on second-order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well-known secondorder methods such as RKC and ROCK2.

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“…We also observed that higher-order methods can have more difficulties with the propagation of errors with several numerical tests [16]. In Ref.…”

Section: Introductionmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Kleefeld

Martín-Vaquero

2012

Numerical Methods Partial

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“…For example, the first-and second-order Runge-Kutta-Chebyshev (RKC) methods [19], the second-and fourth-order Orthogonal-Runge-Kutta-Chebyshev (ROCK) methods [4,1], the third-order DUMKA method [10,13,11], and the second-order Stabilized Explicit Runge-Kutta (SERK2) methods [12]. To be able to treat very stiff reaction terms, [22] proposes to combine the RKC approach with the IMEX idea.…”

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confidence: 99%

Partitioned Runge–Kutta–Chebyshev Methods for Diffusion-Advection-Reaction Problems

Zbinden¹

2011

SIAM J. Sci. Comput.

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Abstract. An integration method based on RKC (Runge-Kutta-Chebyshev) methods is discussed which has been designed to treat moderately stiff and non-stiff terms separately. The method, called PRKC (Partitioned Runge-Kutta-Chebyshev), is a one-step, partitioned Runge-Kutta method of second-order. It belongs to the class of stabilized methods, viz. explicit Runge-Kutta methods possessing extended real stability intervals. The aim of the PRKC method is to reduce the number of function evaluations of the non-stiff terms and to get a non-zero imaginary stability boundary.Key words. Numerical integration of differential equations, Runge-Kutta-Chebyshev methods, Stabilized second-order integration method, Partitioned Runge-Kutta methods AMS subject classifications. 65L20, 65M12, 65M201. Introduction. Stabilized Runge-Kutta methods are explicit methods with extended stability domain along the negative real axis. Their real stability interval has a length proportional to the square of the number of stages. Therefore, these methods are especially suited for the time integration of moderately stiff systems, of large dimension and with eigenvalues known to lie in a long narrow strip along the negative axis. For instance, two-and three-dimensional parabolic partial differential equations (PDE) converted by the method of lines (MOL) give rise to this type of systems. Compared to implicit or IMEX (IMplicit EXplicit) methods, stabilized methods do not require the solution of large linear or nonlinear systems and have a low storage demand. Compared to general explicit methods, they avoid a too severe step size restriction.There exist several stabilized methods. For example, the first-and second-order Runge-Kutta-Chebyshev (RKC) methods [19], the second-and fourth-order Orthogonal-Runge-Kutta-Chebyshev (ROCK) methods [4,1], the third-order DUMKA method [10,13,11], and the second-order Stabilized Explicit Runge-Kutta (SERK2) methods [12]. To be able to treat very stiff reaction terms, [22] proposes to combine the RKC approach with the IMEX idea. A two-step method of this type is discussed in [17], which provides better stability on the imaginary axis. We also wish to mention the essentially optimal explicit Runge-Kutta methods [18] with stability domain containing a given set.In this paper we discuss a one-step, stabilized method of second-order based on the RKC method. The method, called PRKC (Partitioned Runge-Kutta-Chebyshev), treats stiff and non-stiff terms separately. It is devoted to solve systems of ordinary differential equations (ODE) representing space discretization of PDEs aṡ

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An Efficient Parallel Algorithm for Numerical Solution of Low Dimension Dynamics Problems

Orlov

Kuzin

Shabrov

2018

Communications in Computer and Information Science

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Second-order stabilized explicit Runge–Kutta methods for stiff problems

Cited by 30 publications

References 41 publications

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Partitioned Runge–Kutta–Chebyshev Methods for Diffusion-Advection-Reaction Problems

An Efficient Parallel Algorithm for Numerical Solution of Low Dimension Dynamics Problems

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