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Higueras Sanz, Inmaculada

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Higueras Sanz

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Inmaculada

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Ingeniería Matemática e Informática

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0000-0003-3860-3360

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288

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2014 - 20194
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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
    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
    Optimized strong stability preserving IMEX Runge-Kutta methods
    (2014) Higueras Sanz, Inmaculada; Happenhofer, Natalie; Koch, Othmar; Kupka, Friedrich; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza
    We construct and analyze robust strong stability preserving IMplicit-EXplicit Runge-Kutta (IMEX RK) methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise. It turns out that besides the optimization of the region of absolute monotonicity, some other properties of the methods are crucial for the success of such simulations. In particular, the models in our focus dictate to also take into account the step size limits associated with dissipativity, positivity and the stiff parabolic terms which represent transport by diffusion, the uniform convergence with respect to different stiffness properties of those same terms, etc. Furthermore, in the literature, some other properties, like the inclusion of a part of the imaginary axis in the stability region, have been argued to be relevant. In this paper, we construct several new IMEX RK methods which differ from each other by taking various or even all of these constraints simultaneously into account. It is demonstrated for some simple examples as well as for the problem of double-diffusive convection, that the newly constructed schemes provide a significant computational advantage over other methods from the literature. Due to their accumulation of different stability properties, the optimized IMEX RK methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection-diffusion equations, or other transport equations with similar stability requirements.