On the construction of second derivative diagonally implicit multistage integration methods for ODEs

We introduce a class of implicit-explicit (IMEX) schemes for the numerical solution of initial value problems of differential equations with both non-stiff and stiff components in which non-stiff and stiff solvers are, respectively, based on the explicit general linear methods (GLMs) and implicit second derivative GLMs (SGLMs). The order conditions of the proposed IMEX schemes are obtained. Linear stability properties of the methods are analyzed and then methods up to order four with a large area of absolute stability region of the pair are constructed assuming that the implicit part of the methods is L-stable. Due to the high stage orders of the constructed methods, they are not marred by order reduction. This is verified by the numerical experiments which demonstrate the efficiency of the proposed methods, too.

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“…For such methods, the order and stage order conditions imply that the matrices W and B take the form (cf. Abdi et al 2014)…”

Section: Construction Of Imex-sdimsims With Desirable Stability Propementioning

confidence: 99%

“…Efficiency of SGLMs has been verified by constructing these methods in some classes with different properties for solving problems with various features. For more details, see (Abdi et al 2014;Hojjati 2011a, b, 2015;Hosseini Nasab et al 2017).…”

Section: Introductionmentioning

confidence: 99%

Implicit–explicit second derivative diagonally implicit multistage integration methods

2020

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“…where f : R m → R m , were introduced by Butcher (1966Butcher ( , 2016 and extended to second derivative general linear methods (SGLMs) to cover second derivative methods (see, for instance, Abdi 2016; Abdi and Behzad 2018;Abdi et al 2014;Abdi andHojjati 2011a, b, 2015). Izzo et al (2010) introduced GLMs for the numerical solution of VIEs (1).…”

Section: Introductionmentioning

confidence: 99%

General linear methods with large stability regions for Volterra integral equations

Abdi

2019

Comp. Appl. Math.

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General linear methods are well-known as a large family of methods for the numerical solution of Volterra integral equations of the second kind. This paper is concerned with the construction of such methods with a large region of order p and high stage order q = p with a large region of absolute stability. Some numerical results are presented which indicate the effectiveness of the proposed schemes.

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“…In [3] Abdi and Hojjati introduced a subclass of SGLMs as SDIMSIMs (second derivative diagonally implicit multistage integration methods) and constructed methods of this subclass with RKS property. They obtained order barriers for some types of SGLMs which have RKS property in [3,4] and constructed SDIMSIMs of all types up to order 4 in [2]. Also, the construction of Nordsieck SGLMs up to order four and their implementation in a variable stepsize environment have been investigated in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Although GLMs include linear multistep methods, Runge-Kutta and many other standard methods, but they don't cover second derivative methods. So, GLMs were extended to second derivative general linear methods (SGLMs) by Butcher and Hojjati in [10] and studied more by Abdi and Hojjati [2,3,4].…”

Section: Introductionmentioning

confidence: 99%

Construction of Nordsieck Second Derivative General Linear Methods With Inherent Quadratic Stability

Movahedinejad

Hojjati

Abdi

2017

Mathematical Modelling and Analysis

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Abstract. This paper describes the construction of second derivative general linear methods in Nordsieck form with stability properties determined by quadratic stability functions. This is achieved by imposing the so-called inherent quadratic stability conditions. After satisfying order and inherent quadratic stability conditions, the remaining free parameters are used to find the methods with L-stable property. Examples of methods with p = q = s = r − 1 up to order four are given.Keywords: stiff differential equations, second derivative methods, Nordsieck methods, inherent quadratic stability, A-and L-stability.

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On the construction of second derivative diagonally implicit multistage integration methods for ODEs

Cited by 41 publications

References 15 publications

Implicit–explicit second derivative diagonally implicit multistage integration methods

Implicit–explicit second derivative diagonally implicit multistage integration methods

General linear methods with large stability regions for Volterra integral equations

Construction of Nordsieck Second Derivative General Linear Methods With Inherent Quadratic Stability

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