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In this subproject, we propose three lines of research. The first focuses on the convergence and implementation of the so-called SAFERK (Stiffly Accurate First Explicit Runge-Kutta) methods. This one-parameter family of methods provides Runge-Kutta methods with the same order as the classical RAdau IIA methods and the same number of implicit stages, while exhibiting lower error coefficients. The free parameter of the family can be selected to minimize error coefficients or maximize the damping of the rigid components of the problem under consideration. The methods allow for adaptive implementation, as in the refined RADAU5 and RADAU codes, which are based on the Radau IIA methods. In fact, this family of methods has proven to be competitive with other standard numerical methods in the numerical integration of rigid systems and algebraic differential equations—that is, differential equations with algebraic constraints. We now intend to extend the convergence analysis to large systems of Ordinary Differential Equations and Algebraic Differential Equations derived from the spatial discretization of Partial Differential Equations and Algebraic Partial Differential Equations, respectively, using the Line Method. Particularly interesting is the phenomenon of order reduction associated with time-dependent boundary conditions, and a key objective of this first line of research is to describe how SAFERK methods are affected by such boundary conditions (either Dirichlet or Neumann) compared to methods of the Radau IIA family. The second line of research is dedicated to the time integration of differential equations arising from the spatial discretization of partial differential equations in several spatial dimensions, considering ROW- and W-class methods in combination with Approximate Matrix Factorization. In particular, we will be interested in obtaining pairs of such methods for the purpose of stabilizing some well-known pairs of explicit Runge-Kutta methods that have been widely used in the numerical integration of stiff or moderately stiff problems, such as the Bogachi and Shampine pairs (of orders 2 and 3) or the Dormand and Prince pairs (of orders 4 and 5). These pairs have been considered for producing commercial codes in Matlab and are the base integrators of the ode23 and ode45 codes in Matlab, respectively. However, such codes cannot handle stiff problems due to their explicit nature. Consequently, it is our goal to obtain efficient W-methods that extend pairs of explicit Runge-Kutta methods to make them suitable, through stabilization, for the time integration of large stiff systems. We will explore efficient pairs of W-methods not only by studying stability but also by trying to avoid the phenomenon of order reduction due to time-dependent boundary conditions. Finally, in the third line of research, we will explore the use of W-methods for the time integration of parabolic partial differential equations involving mixed second-order derivatives. These problems arise in a wide variety of applications, although we will focus on practical models in financial mathematics (Heston models). [/vc_column_text][/vc_tta_section][vc_tta_section title=»Abstract» tab_id=»abstract»][vc_column_text] In this subproject we propose three lines of research. The first one focuses on the convergence and implementation of the so-called SAFERK methods (standing for Stiffly Accurate First Explicit Runge-Kutta). These one-parameter family of methods provides Runge-Kutta methods with the same order as the classical Radau IIA methods for the same number of implicit stages, while providing smaller error coefficients. The free parameter of the family can be selected in order to either minimize error coefficients or maximize damping for the stiff components of the differential problem under consideration. The methods allow an analogous adaptive implementation as it is done in the perfected stiff codes RADAU5 and RADAU, based on the Runge-Kutta-Radau IIA methods. In fact, this family of methods has proven to be competitive to other state-of-the-art numerical methods for the numerical integration of stiff systems and Differential Algebraic Equations, that is, differential equations which are also endowed with algebraic constraint. We now intend to extend the convergence analysis to large systems of Ordinary Differential Equations and Differential Algebraic Equations coming from the spatial discretization of Partial Differential Equations and Partial Differential-Algebraic Equations, respectively, by means of the Method of Lines. Particularly interesting is the order reduction phenomenon associated to time dependent boundary conditions, and it is key goal of this first line of research to describe how SAFERK methods are affected by such boundary conditions (either of Dirichlet or Neumann type) in comparison to methods of the Radau IIA family. The second line of research is devoted to the time integration of differential equations arising in the spatial discretization of partial differential equations in several spatial dimensions by considering the classes of ROW- and W-methods in combination with the Approximate Matrix Factorization. In particular, we will be interested in deriving pairs of such methods in order to stabilize some well-known pairs of explicit Runge-Kutta methods which have been extensively used in the numerical integration of non-stiff or mildly stiff problems, like the pairs by Bogacki and Shampine (with orders 2 and 3) or by Dormand and Prince (with orders 4 and 5). These pairs have been considered in order to produce commercial codes in Matlab and are the base integrators of the codes ode23 and ode45 in Matlab, respectively. However such codes cannot cope with very stiff problems due to their explicit nature. Hence, we aim at obtaining efficient Wmethods by extending relevant pairs of explicit Runge-Kutta so as to make them suitable through stabilization for the time integration of very large stiff systems. Efficient pairs of W-methods will be explored not only looking at the stability issues, but also at avoiding the order reduction phenomenon for time dependent boundary conditions. Finally, in the third line of research we will explore the use of W-methods for the time integration of parabolic Partial Differential Equations where mixed second order derivatives are present. These problems arise in a number of applications, although we will concentrate on practical models from financial mathematics (Heston models).
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