Higueras Sanz, Inmaculada
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Higueras Sanz
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Inmaculada
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Estadística, Informática y Matemáticas
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InaMat2. Instituto de Investigación en Materiales Avanzados y Matemáticas
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Publication Open 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 IngeniaritzaIn 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.Publication Open 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 IngeniaritzaSpace 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.Publication Open 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 MatematikaWhen 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.