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.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of IMACS. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by-nc -nd /4 .0/).
Generalized TASE-RK methods for stiff problems
Pagano G.;
2023-01-01
Abstract
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.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of IMACS. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by-nc -nd /4 .0/).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.