Explicit Runge–Kutta Methods with Estimates of the Local Truncation Error

1978 ◽  
Vol 15 (4) ◽  
pp. 772-790 ◽  
Author(s):  
J. H. Verner
Keyword(s):  
Truncation Error ◽  
Local Truncation Error ◽  
Runge Kutta

scholarly journals A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

2021 ◽  
Vol 26 (3) ◽  
pp. 59
Author(s):  
Musa Ahmed Demba ◽  
Higinio Ramos ◽  
Poom Kumam ◽  
Wiboonsak Watthayu
Keyword(s):  
Stability Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Truncation Error ◽  
Local Truncation Error ◽  
Nyström Methods ◽  
Oscillating Systems ◽  
The Common ◽  
Runge Kutta ◽  
The Stability

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

ACCURATE FINITE DIFFERENCE REPRESENTATION OF NEUMANN BOUNDARY DATA FOR TIME-HARMONIC WAVE PROPAGATION

1996 ◽  
Vol 04 (04) ◽  
pp. 425-432 ◽  
Author(s):  
ISAAC HARARI
Keyword(s):  
Finite Difference ◽  
Boundary Data ◽  
Truncation Error ◽  
Harmonic Wave ◽  
Neumann Boundary ◽  
Local Truncation Error ◽  
Matrix Equations ◽  
Order Accuracy ◽  
Time Harmonic ◽  
Global Accuracy

Finite difference stencils for inhomogeneous Neumann boundary conditions in acoustic problems with arbitrary source distributions are constructed and analyzed. Boundary stencils are compatible with corresponding interior stencils, preserving symmetry of matrix equations without degrading global accuracy. Higher-order accuracy is attained within the compact support of lower-order methods. Results are verified by local truncation error analysis.

A Loading Correction Scheme for a Structure-Dependent Integration Method

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Shuenn-Yih Chang
Keyword(s):  
Steady State ◽  
High Frequency ◽  
Truncation Error ◽  
Integration Method ◽  
Frequency Mode ◽  
Local Truncation Error ◽  
Steady State Response ◽  
High Frequency Mode ◽  
State Response ◽  
Correction Scheme

A structure-dependent integration method may experience an unusual overshooting behavior in the steady-state response of a high frequency mode. In order to explore this unusual overshooting behavior, a local truncation error is established from a forced vibration response rather than a free vibration response. As a result, this local truncation error can reveal the root cause of the inaccurate integration of the steady-state response of a high frequency mode. In addition, it generates a loading correction scheme to overcome this unusual overshooting behavior by means of the adjustment the difference equation for displacement. Apparently, these analytical results are applicable to a general structure-dependent integration method.

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