Family Of Explicit Research Articles

On a generalization of time-accurate and highly-stable explicit operators for stiff problems

In this work we propose a generalization of the family of Time-Accurate and highly-Stable Explicit (TASE) operators recently introduced by Calvo, Montijano and Rández (2021). In this family the TASE operator of order p depends on p free real parameters. Here we consider operators that can also be defined by complex parameters occurring in conjugate pairs. Despite this choice the calculations continue to be done only in real arithmetic, thus not burdening the computational cost of the previous version of the family. Conversely, this generalization leads to improve both the accuracy and stability properties of explicit Runge-Kutta schemes supplemented with TASE operators. Numerical experiments showing the competitiveness of the methods proposed in this paper with respect to classical integrators for stiff problems are also presented.

Generalized TASE-RK methods for stiff problems

A family of Time-Accurate and Stable Explicit (TASE) methods for the numerical integration of Initial Value Problems in stiff Ordinary Differential Equations (ODEs) y′(t)=f(t,y) was recently introduced in [1]. The main idea was to make local extrapolation of a stabilized Euler method. More recently, in [3] a similar approach by considering the stabilization of arbitrary explicit Runge-Kutta methods (TASE-RK) was analyzed. In this case the explicit Runge-Kutta method integrates a transformed ODE obtained by multiplying the vector field f(t,y) by a certain operator which approximates the identity mapping up to a given order p. The main inconvenience of both approaches is that to reach order p the solution of p2 linear systems plus the evaluation of p derivatives are required per integration step.In order to substantially reduce the computational costs of the former approaches in the linear system solution, but maintaining the good accuracy and stability properties, a new family of TASE-RK methods which allow to introduce a few more free parameters are considered. The formulation of the methods was conceived to be implemented not only in sequential mode but it admits parallelism in a straightforward way. Furthermore, since these methods are linearly implicit, connections to the class of W-methods [19] are properly established. The order conditions for the new class of methods are widely studied by using the rooted tree theory. For p=3,4, new methods with p sequential stages and order p are derived and compared on semidiscrete 1D and 2D Partial Differential Equations (PDEs) to those in [1,3] and other standard Rosenbrock and W-methods in the literature.

Time-accurate and highly-stable explicit peer methods for stiff differential problems

We derive a new class of parallelizable two-step peer methods for the numerical solution of stiff systems of Ordinary Differential Equations (ODEs), inspired by a technique introduced in Bassenne et al., (2021) and Calvo et al., (2021). It consists in the improvement of the stability properties of explicit Runge–Kutta (RK) methods by premultiplying the vector field of the problem to solve by a carefully chosen family of operators, called Time-Accurate and Highly-Stable Explicit (TASE) operators.In this work, we show that for suitable choices of the free parameters of the family of TASE operators proposed in Calvo et al., (2021), it is possible to improve the stability properties of explicit parallelizable peer methods. Furthermore, we also introduce a new family of TASE operators that fits better than the existing ones on some problems of interest in applications. Numerical tests on Partial Differential Equations (PDEs) are carried out to show that the new TASE-peer schemes are competitive with TASE-RK methods and other well-known stiff integrators, and are potentially more advantageous.

A note on the stability of time–accurate and highly–stable explicit operators for stiff differential equations

A family of Time-Accurate and highly-Stable Explicit (TASE) operators for the numerical solution of stiff IVPs that includes those proposed by Bassenne et al. (2021) [1] is proposed. In this family the TASE operator of order k depends on k free parameters in contrast with Bassenne's family in which it depends only on one parameter to be chosen for stability and accuracy requirements. A complete study of A–stability properties is carried out for explicit RK schemes supplemented with TASE operators with order k≤4. For orders 2, 3 and 4, particular schemes that are nearly strongly A–stable and therefore suitable for stiff problems are given. Some numerical experiments showing the behaviour of the methods are presented.

The Value of Considering Autocorrelation Between Inflows in the Stochastic Planning of Water Resource Systems

Reliability programming formulations offer a family ofexplicit stochastic models for planning the operationof complex water resources systems. These models usecumulative probability distributions of the sum ofinflows to characterize their variability in theplanning period. Applicability of these models for avariety of problems has been limited, mainly due tothe assumption of independence between inflows indifferent time periods that leads to the derivation ofconservative operating policies. This paper presentsthree new approaches to overcome this limitation. Theperformance of the proposed approaches is demonstratedthrough comparison of the operating policies derivedfrom these approaches and the Independent Approach. Operational planning of the Manitoba Hydro energygeneration system, a predominantly hydro-based utilitycompany in Manitoba (Canada), is used as the case study.