On approximate implicit Taylor methods for ordinary differential equations

Abstract

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

Jacobian-free explicit multiderivative Runge-Kutta methods for hyperbolic conservation laws

“…Presumably, also implicit MDRK schemes need to be considered to take care of the diffusive effects. A possible starting point could be the implicit variant of the approximate Taylor methods, which have been recently developed for ODEs in [2]. Concerning the stability properties of the scheme one could think of exploring more types of MDRK schemes, possibly with SSP properties [6,11].…”

“…1.0619 − 1 120 σ 5 P (1) P (2) P (2) + 1 600 σ 6 P (2) P (2) P (2) 3DRKCAT5-2 1 − σ P (1) + 1 2 σ 2 P (2) − 1 6 σ 3 P (3) + 1 24 σ 4 P (1) P (3) 0.4275 − 1 120 σ 5 P (2) P (3) + 1 900 σ 6 P (3) P (3) 3DRKCAT7-3 1 − σ P (1) + 1 2 σ 2 P (2) − 1 6 σ 3 P (3) + 1 24 σ 4 P (1) P (3) 0.2300 − 1 120 σ 5 P (2) P (3) + 1 720 σ 6 P (3) P (3) − 1 5040 σ 7 P (1) P (2) P (3) + 4DRKCAT6-2 1 − σ P (1) + 1 2 σ 2 P (2) − 1 6 σ 3 P (3) + 1 24 σ 4 P (4) 0.8563 − 1 120 σ 5 P (1) P (4) + 1 720 σ 6 P (2) P (4) − 1 6480 σ 7 P (3) P (4) + 1 77760 σ 8 P (4) P (4) The critical CFL values σ * up to four decimals are shown in Tbl. 2.…”