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In this paper Extrapolated Stabilized Explicit Runge-Kutta methods (ESERK) are proposed to solve nonlinear partial differential equations (PDEs) in right triangles. These algorithms evaluate more times the function than a standard explicit Runge–Kutta scheme (nt times per step), and these extra evaluations do not increase the order of convergence but the stability region grows with O(nt2). Hence, the total computational cost is O(nt) times lower than with a traditional explicit algorithm. Thus, these algorithms have been traditionally considered to solve stiff PDEs in squares/rectangles or cubes. In this paper, for the first time, ESERK methods are considered in a right triangle. It is demonstrated that such type of codes keep the convergence and the stability properties under certain conditions. This new approach would allow to solve nonlinear parabolic PDEs with stabilized explicit Runge–Kutta schemes in complex domains, that would be decomposed in rectangles and right triangles.

A novel family of optical line codes to counteract the effects of a dispersive fiber is presented. The performance of the first code in the family, referred to as order-1 code, is analytically evaluated and compared to that of the duobinary and phased amplitude-shift signaling (PASS) codes, which are a modified form of duobinary proposed by Stark et al. (1999). The order-1 line code turns out to be very robust to chromatic dispersion and, for a given penalty, allows the bridging of a distance 1.5 times (or more) greater than duobinary. The novel family of codes was conceived by exploiting the finding that, approximately, a very dispersive fiber turns an input pulse into the form of its Fourier transform seen in the time domain. Higher-order codes allow the bridging of larger distances if combined with appropriate chirping [ABSTRACT FROM PUBLISHER]

• 2 new families with over 100 explicit schemes are built for stiff huge systems of ODEs. • 4 parallelized codes are provided with excellent results for nonlinear PDEs. • Comparison of the methods, discussion on the optimum order of convergence. • Discussion on how to combine the different families in one algorithm. There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s , hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost. Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different. [ABSTRACT FROM AUTHOR]