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Roldán Marrodán, Teodoro

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Roldán Marrodán

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Teodoro

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Ingeniería Matemática e Informática

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0000-0001-8773-3554

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2016 - 20195
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    PublicationOpen Access
    Efficient SSP low-storage Runge-Kutta methods
    (2019) Roldán Marrodán, Teodoro; Higueras Sanz, Inmaculada; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
    In this paper we study the efficiency of Strong Stability Preserving (SSP) Runge-Kutta methods that can be implemented with a low number of registers using their Shu-Osher representation. SSP methods have been studied in the literature and stepsize restrictions that ensure numerical monotonicity have been found. However, for some problems, the observed stepsize restrictions are larger than the theoretical ones. Aiming at obtaining additional properties of the schemes that may explain their efficiency, in this paper we study the influence of the local error term in the observed stepsize restrictions. For this purpose, we consider the family of 5-stage third order SSP explicit Runge-Kutta methods, namely SSP(5,3), and the Buckley-Leverett equation. We deal with optimal SSP(5,3) schemes whose implementation requires at least 3 memory registers, and non-optimal 2-register SSP(5,3) schemes. The numerical experiments done show that small error constants improve the efficiency of the method in the sense that larger observed SSP coefficients are obtained.
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    PublicationOpen Access
    Construction of additive semi-implicit Runge-Kutta methods with low-storage requirements
    (Springer US, 2016) Higueras Sanz, Inmaculada; Roldán Marrodán, Teodoro; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
    Space discretization of some time-dependent partial differential equations gives rise to systems of ordinary differential equations in additive form whose terms have different stiffness properties. In these cases, implicit methods should be used to integrate the stiff terms while efficient explicit methods can be used for the non-stiff part of the problem. However, for systems with a large number of equations, memory storage requirement is also an important issue. When the high dimension of the problem compromises the computer memory capacity, it is important to incorporate low memory usage to some other properties of the scheme. In this paper we consider Additive Semi-Implicit Runge-Kutta (ASIRK) methods, a class of implicitexplicit Runge-Kutta methods for additive differential systems. We construct two second order 3-stage ASIRK schemes with low-storage requirements. Having in mind problems with stiffness parameters, besides accuracy and stability properties, we also impose stiff accuracy conditions. The numerical experiments done show the advantages of the new methods.
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    PublicationOpen Access
    Strong stability preserving properties of composition Runge-Kutta schemes
    (Springer, 2019) Higueras Sanz, Inmaculada; Roldán Marrodán, Teodoro; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
    In this paper Strong Stability Preserving (SSP) properties of Runge Kutta methods obtained by com- posing k different schemes with different step sizes are studied. The SSP coefficient of the composition method is obtained and an upper bound on this coefficient is given. Some examples are shown. In par- ticular, it is proven that the optimal n2-stage third order explicit Runge-Kutta methods obtained by D.I. Ketcheson [SIAM J. Sci. Comput. 30(4), 2008] are composition of first order SSP schemes.
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    PublicationOpen Access
    New third order low-storage SSP explicit Runge-Kutta methods
    (Springer, 2019) Higueras Sanz, Inmaculada; Roldán Marrodán, Teodoro; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
    When a high dimension system of ordinary differential equations is solved numerically, the computer memory capacity may be exhausted. Thus, for such systems, it is important to incorporate low memory usage to some other properties of the scheme. In the context of strong stability preserving (SSP) schemes, some low-storage methods have been considered in the literature. In this paper we study 5-stage third order 2N low-storage SSP explicit Runge-Kutta schemes. These are SSP schemes that can be implemented with 2N memory registers, where N is the dimension of the problem, and retain the previous time step approximation. This last property is crucial for a variable step size implementation of the scheme. In this paper, first we show that the optimal SSP methods cannot be implemented with 2N memory registers. Next, two non-optimal SSP 2N low-storage methods are constructed; although their SSP coefficients are not optimal, they achieve some other interesting properties. Finally, we show some numerical experiments.
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    PublicationOpen Access
    Order barrier for low-storage DIRK methods with positive weights
    (Springer, 2018) Higueras Sanz, Inmaculada; Roldán Marrodán, Teodoro; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
    In this paper we study an order barrier for low-storage diagonally implicit Runge-Kutta (DIRK) methods with positive weights. The Butcher matrix for these schemes, that can be implemented with only two memory registers in the van der Houwen implementation, has a special structure that restricts the number of free parameters of the method. We prove that third order low-storage DIRK methods must contain negative weights, obtaining the order barrier p ≤ 2 for these schemes. This result extends the well known one for symplectic DIRK methods, which are a particular case of low-storage DIRK methods. Some other properties of second order low-storage DIRK methods are given.