scholarly journals Variable order nested hybrid multistep methods for stiff ODEs

2020 ◽  
Keyword(s):  
Multistep Methods ◽  
Variable Order ◽  
Stiff Odes

A variable-step variable-order algorithm for systems of stiff odes

1997 ◽  
Vol 63 (1-2) ◽  
pp. 149-157 ◽  
Author(s):  
T. Van Hecke ◽  
G. Vanden G Berghe ◽  
M. Van Daele ◽  
H De Meyer
Keyword(s):  
Variable Order ◽  
Stiff Odes ◽  
Variable Step ◽  
Order Algorithm

scholarly journals Solving Directly Higher Order Ordinary Differential Equations by Using Variable Order Block Backward Differentiation Formulae

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1289
Author(s):  
Asnor ◽  
Mohd Yatim ◽  
Ibrahim
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Computational Effort ◽  
Variable Order ◽  
Numerical Efficiency ◽  
Stiff Odes ◽  
Backward Differentiation Formulae ◽  
Bdf Method ◽  
Selection Of

Variable order block backward differentiation formulae (VOHOBBDF) method is employedfor treating numerically higher order Ordinary Differential Equations (ODEs). In this respect, the purpose of this research is to treat initial value problem (IVP) of higher order stiff ODEs directly. BBDF method is symmetrical to BDF method but it has the advantage of producing more than one solutions simultaneously. Order three, four, and five of VOHOBBDF are developed and implemented as a single code by applying adaptive order approach to enhance the computational efficiency. This approach enables the selection of the least computed LTE among the three orders of VOHOBBDF and switch the code to the method that produces the least LTE for the next step. A few numerical experiments on the focused problem were performed to investigate the numerical efficiency of implementing VOHOBBDF methods in a single code. The analysis of the experimental results reveals the numerical efficiency of this approach as it yielded better performances with less computational effort when compared with built-in stiff Matlab codes. The superior performances demonstrated by the application of adaptive orders selection in a single code thus indicate its reliability as a direct solver for higher order stiff ODEs.

Symmetric Multistep Methods Revisited: II. Numerical Experiments

1999 ◽  
Vol 173 ◽  
pp. 309-314 ◽  
Author(s):  
T. Fukushima
Keyword(s):  
Multistep Methods ◽  
Computational Time ◽  
Integration Time ◽  
Implicit Methods ◽  
Symmetric Methods ◽  
Celestial Bodies ◽  
The Stability ◽  
Predictor Corrector ◽  
Symmetric Multistep Methods ◽  
Integration Errors

AbstractBy using the stability condition and general formulas developed by Fukushima (1998 = Paper I) we discovered that, just as in the case of the explicit symmetric multistep methods (Quinlan and Tremaine, 1990), when integrating orbital motions of celestial bodies, the implicit symmetric multistep methods used in the predictor-corrector manner lead to integration errors in position which grow linearly with the integration time if the stepsizes adopted are sufficiently small and if the number of corrections is sufficiently large, say two or three. We confirmed also that the symmetric methods (explicit or implicit) would produce the stepsize-dependent instabilities/resonances, which was discovered by A. Toomre in 1991 and confirmed by G.D. Quinlan for some high order explicit methods. Although the implicit methods require twice or more computational time for the same stepsize than the explicit symmetric ones do, they seem to be preferable since they reduce these undesirable features significantly.

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