On the local error and the local truncation error of linear multistep methods

1990 ◽  
Vol 30 (4) ◽  
pp. 673-681
Author(s):  
J. D. Lambert
Keyword(s):  
Truncation Error ◽  
Multistep Methods ◽  
Linear Multistep Methods ◽  
Local Error ◽  
Local Truncation Error

scholarly journals Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 950
Author(s):  
Denis Butusov
Keyword(s):  
Truncation Error ◽  
Multistep Methods ◽  
Applied Mathematics ◽  
Simulation Software ◽  
Van Der Pol Oscillator ◽  
Linear Multistep Methods ◽  
Local Truncation Error ◽  
Stepsize Control ◽  
Adaptive Stepsize ◽  
Implicit And Explicit

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

scholarly journals Local error estimation and step size control in adaptive linear multistep methods

2020 ◽  
Author(s):  
Carmen Arévalo ◽  
Gustaf Söderlind ◽  
Yiannis Hadjimichael ◽  
Imre Fekete
Keyword(s):  
Error Estimation ◽  
Multistep Methods ◽  
Size Control ◽  
Linear Multistep Methods ◽  
Local Error ◽  
Step Size ◽  
Local Error Estimation ◽  
Step Size Control

A correction method for milling stability analysis based on local truncation error

2021 ◽  
Author(s):  
Yi Wu ◽  
Youpeng You ◽  
Anmin Liu ◽  
Bin Deng ◽  
Tuo Ye ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Truncation Error ◽  
Correction Method ◽  
Local Truncation Error ◽  
Milling Stability

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

scholarly journals The stability problem for linear multistep methods: Old and new results

2007 ◽  
Vol 210 (1-2) ◽  
pp. 2-12 ◽  
Author(s):  
L. Aceto ◽  
D. Trigiante
Keyword(s):  
Stability Problem ◽  
Multistep Methods ◽  
Linear Multistep Methods ◽  
The Stability

High-order linear multistep methods with general monotonicity and boundedness properties

2005 ◽  
Vol 209 (1) ◽  
pp. 226-248 ◽  
Author(s):  
Steven J. Ruuth ◽  
Willem Hundsdorfer
Keyword(s):  
Multistep Methods ◽  
High Order ◽  
Linear Multistep Methods ◽  
Order Linear

A novel family of P-stable symmetric extended linear multistep methods for oscillators

2014 ◽  
Vol 249 ◽  
pp. 597-610
Author(s):  
Xiong You ◽  
Yuanmei Zhou ◽  
Xiaohao Cheng
Keyword(s):  
Multistep Methods ◽  
Linear Multistep Methods
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