Locally Linearized Runge Kutta method of Dormand and Prince

Abstract

In this paper, the effect that produces the local linearization of the embedded Runge-Kutta formulas of Dormand and Prince for initial value problems is studied. For this, embedded Locally Linearized Runge-Kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations. For a variety of well-known physical equations with different dynamics, the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones, which implies a substantial reduction of the number of time steps and, consequently, a sensitive reduction of the overall computation cost of their adaptive implementation.

Local Linearization—Runge–Kutta methods: A class of A-stable explicit integrators for dynamical systems

“…In summary, results of Table II clearly indicate that the non adaptive implementation of the LLRK4 scheme provides similar or much better accuracy than the Matlab codes with equal or lower number of steps in the integration of variety of equations. This suggests that adaptive implementations the LLRK discretizations might archive similar accuracy than the Matlab codes with lower or much lower number of steps, a subject that has been already studied in [58,44].…”

“…where ϕ(z) = (e z − 1)/z, the LLRK4 scheme (44) can easily modified to defined an order 4 LLRK scheme for high dimensional ODEs. Indeed, such scheme can be defined by the same expression (44), but replacing the formulas of φ(t n , y n ; hn 2 ) and φ(t n , y n ; h n ) by φ(t n , y n ; h n 2 ) = ϕ( h n 2 f x ( y n ))f ( y n ) Table II. The "exact" path of x 1 is computed with the Matlab code ode15s with RT = RA = 10 −13 on a very thin partition.…”

“…In this section, the performance of the LLRK4 scheme (44) is illustrated by means of numerical simulations. To do so, a variety of ODEs were selected.…”

Construcción y estudio de códigos adaptativos de linealización local para ecuaciones diferenciales ordinarias

“…Teorema 6 [24,16] El error relativo y absoluto de la aproximación de Padé con escalamiento y potenciación (18) de e A está dado por:…”

“…En general, los códigos LLRK4 adaptivos son los más confiables en el sentido de tener un aceptablemente bajo error relativo en todos los ejemplos aunque su costo computacional es mayor que el de los esquemas Runge Kutta considerados. Estos resultados motivan el estudio de algoritmos adaptativos alternativos que permitan mantener la alta precision de los esquemas LL mejorando su eficiencia computacional, como por ejemplo los algoritmos con esquemas "embedding" considerados en [18].…”