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Nonlinear Problems: Analytical and Computational Approach with Applications

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Volume 2012 |Article ID 854517 | https://doi.org/10.1155/2012/854517

Haiyan Yuan, Jingjun Zhao, Yang Xu, "Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations", Abstract and Applied Analysis, vol. 2012, Article ID 854517, 22 pages, 2012. https://doi.org/10.1155/2012/854517

Nonlinear Stability and D-Convergence of Additive Runge-Kutta Methods for Multidelay-Integro-Differential Equations

Academic Editor: Muhammad Aslam Noor
Received30 Dec 2011
Accepted19 Feb 2012
Published04 Jun 2012

Abstract

This paper is devoted to the stability and convergence analysis of the Additive Runge-Kutta methods with the Lagrangian interpolation (ARKLMs) for the numerical solution of multidelay-integro-differential equations (MDIDEs). GDN-stability and D-convergence are introduced and proved. It is shown that strongly algebraically stability gives D-convergence, DA- DAS- and ASI-stability give GDN-stability. A numerical example is given to illustrate the theoretical results.

1. Introduction

Delay differential equations arise in a variety of fields as biology, economy, control theory, electrodynamics (see, e.g., [1โ€“5]). When considering the applicability of numerical methods for the solution of DDEs, it is necessary to analyze the stability of the numerical methods. In the last three decades, many works had dealt with these problems (see, e.g., [6]). For the case of nonlinear delay differential equations, this kind of methodology had been first introduced by Torelli [7] and then developed by [8โ€“12].

In this paper, we consider the following nonlinear multidelay-integro-differential equations (MDIDEs) with ๐‘š delays:๐‘ฆ๎…ž(๐‘ก)=๐‘“[1]๎‚ต๎€ท๐‘ก,๐‘ฆ(๐‘ก),๐‘ฆ๐‘กโˆ’๐œ1๎€ธ,๎€œ๐‘ก๐‘กโˆ’๐œ1๐‘”[1](๎‚ถ๐‘ก,๐‘ ,๐‘ฆ(๐‘ ))๐‘‘๐‘ +๐‘“[2]๎‚ต๎€ท๐‘ก,๐‘ฆ(๐‘ก),๐‘ฆ๐‘กโˆ’๐œ2๎€ธ,๎€œ๐‘ก๐‘กโˆ’๐œ2๐‘”[2]๎‚ถ(๐‘ก,๐‘ ,๐‘ฆ(๐‘ ))๐‘‘๐‘ +โ‹ฏ+๐‘“[๐‘š]๎‚ต๎€ท๐‘ก,๐‘ฆ(๐‘ก),๐‘ฆ๐‘กโˆ’๐œ๐‘š๎€ธ,๎€œ๐‘ก๐‘กโˆ’๐œ2๐‘”[๐‘š]๎‚ถ๎€บ๐‘ก(๐‘ก,๐‘ ,๐‘ฆ(๐‘ ))๐‘‘๐‘ ,๐‘กโˆˆ0๎€ป,๎€บ๐‘ก,๐‘‡๐‘ฆ(๐‘ก)=๐œ‘(๐‘ก),๐‘กโˆˆ0โˆ’๐œ,๐‘ก0๎€ป,(1.1) where ๐œ1โ‰ค๐œ2โ‰คโ‹ฏโ‰ค๐œ๐‘š=๐œ, ๐‘“[๐‘ฃ]โˆถ[๐‘ก0,๐‘‡]ร—๐ถ๐‘ร—๐ถ๐‘ร—๐ถ๐‘โ†’๐ถ๐‘,๐‘”[๐‘ฃ]โˆถ[๐‘ก0,๐‘‡]ร—๐ถ๐‘ร—๐ถ๐‘โ†’๐ถ๐‘๐‘ฃ=1,2,โ€ฆ,๐‘š, and ๐œ‘โˆถ[๐‘ก0โˆ’๐œ,๐‘ก0]โ†’๐ถ๐‘ are continuous functions such that (1.1) has a unique solution. Moreover, we assume that there exist some inner product โŸจโ‹…,โ‹…โŸฉ and the induced norm ||โ‹…|| such that ๎ซ๐‘“Re[๐‘ฃ]๎€ท๐‘ก,๐‘ฆ1,๐‘ข1,๐‘ค1๎€ธโˆ’๐‘“[๐‘ฃ]๎€ท๐‘ก,๐‘ฆ2,๐‘ข2,๐‘ค2๎€ธ,๐‘ฆ1โˆ’๐‘ฆ2๎ฌโ‰ค๐›ผ๐‘ฃโ€–โ€–๐‘ฆ1โˆ’๐‘ฆ2โ€–โ€–2+๐›ฝ๐‘ฃโ€–โ€–๐‘ข1โˆ’๐‘ข2โ€–โ€–2+๐œŽ๐‘ฃโ€–โ€–๐‘ค1โˆ’๐‘ค2โ€–โ€–2,๐‘ฃ=1,2,โ€ฆ,๐‘š,๐‘กโ‰ฅ๐‘ก0,โ€–โ€–๐‘“(1.2)[๐‘ฃ]๎€ท๐‘ก,๐‘ฆ,๐‘ข1๎€ธ,๐‘คโˆ’๐‘“[๐‘ฃ]๎€ท๐‘ก,๐‘ฆ,๐‘ข2๎€ธโ€–โ€–,๐‘คโ‰ค๐‘Ÿ๐‘ฃโ€–โ€–๐‘ข1โˆ’๐‘ข2โ€–โ€–,โ€–โ€–๐‘”(1.3)[๐‘ฃ]๎€ท๐‘ก,๐‘ ,๐‘ค1๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก,๐‘ ,๐‘ค2๎€ธโ€–โ€–โ‰คฬƒ๐‘Ÿ๐‘ฃโ€–โ€–๐‘ค1โˆ’๐‘ค2โ€–โ€–,(๐‘ก,๐‘ )โˆˆ๐ท(1.4)

forall๐‘กโˆˆ[๐‘ก0,๐‘‡],forall๐‘ฆ,๐‘ฆ1,๐‘ฆ2,๐‘ข,๐‘ข1,๐‘ข2,๐‘ค,๐‘ค1,๐‘ค2โˆˆ๐ถ๐‘, (โˆ’๐›ผ๐‘ฃ),๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,ฬƒ๐‘Ÿ๐‘ฃ are all nonnegative constants. Throughout this paper, we assume that the problem (1.1) has unique exact solution ๐‘ฆ(๐‘ก). Space discretization of some time-dependent delay partial differential equations give rises to such delay differential equations containing additive terms with different stiffness properties. In these situations, additive Runge-Kutta (ARK) methods are used. Some recent works about ARK can refer to [13, 14]. For the additive MDIDEs (1.1), similar to the proof of Theoremโ€‰2.1 in [7], it is straightforward to prove that under the conditions (1.2)~(1.4), the analytic solutions satisfyโ€–๐‘ฆ(๐‘ก)โˆ’๐‘ง(๐‘ก)โ€–โ‰คmax๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–,(1.5) where ๐‘ง(๐‘ก) is the solution of the perturbed problem to (1.1).

To demand the discrete numerical solutions to preserve the convergence properties of the analytic solutions, Torelli [7] introduced a concept of RN-, GRN-stability for numerical methods applied to dissipative nonlinear systems of DDEs such as (1.1) when ๐‘”[๐‘ฃ](๐‘ก,๐‘ ,๐‘ฆ(๐‘ ))=0,๐‘ฃ=1,2,โ€ฆ,๐‘š, which is the straightforward generalization of the well-known concept of BN-stability of numerical methods with respect to dissipative systems of ODEs (see also [9]). More recently, one has noticed a growing interesting the analysis of delay integro-differential equations (DIDEs). This type of equations have been investigated in various fields, such as mathematical biology and control theory (see [15โ€“17]). The theory of computational methods for delay integro-differential equations (DIDEs) has been studied by many authors, and a great deal of interesting results have been obtained (see [18โ€“22]). Koto [23] dealt with the linear stability of Runge-Kutta (RK) methods for systems of DIDEs; Huang and Vandewalle [24] gave sufficient and necessary stability conditions for exact and discrete solutions of linear Scalar DIDEs. However, little attention has been paid to nonlinear multidelay-integro-differential equations (MDIDEs).

So, the aim of this paper is the study of stability and convergence properties for ARK methods when they are applied to nonlinear multidelay-integro-differential equations (MDIDEs) with ๐‘š delays.

2. The GDN-Stability of the Additive Runge-Kutta Methods

An additive Runge-Kutta method with the Lagrangian interpolation (ARKLM) of ๐‘  stages and ๐‘š levels can be organized in the Butcher tableau:๐ถ๐ด[1]๐ด[2]โ‹ฏ๐ด[๐‘š]๐‘[1]๐‘‡๐‘[2]๐‘‡โ‹ฏ๐‘[๐‘š]๐‘‡=๐‘1๐‘Ž[1]11๐‘Ž[1]12โ‹ฏ๐‘Ž[1]1๐‘ ๐‘Ž[๐‘š]11๐‘Ž[๐‘š1]12โ‹ฏ๐‘Ž[๐‘š]1๐‘ ๐‘2๐‘Ž[1]21๐‘Ž[1]22โ‹ฏ๐‘Ž[1]2๐‘ ๐‘Ž[๐‘š]21๐‘Ž[๐‘š]22โ‹ฏ๐‘Ž[๐‘š]2๐‘ โ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฏ๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘ ๐‘Ž[1]๐‘ 1๐‘Ž[1]๐‘ 2โ‹ฏ๐‘Ž[1]๐‘ ๐‘ ๐‘Ž[๐‘š]๐‘ 1๐‘Ž[๐‘š]๐‘ 2โ‹ฏ๐‘Ž[๐‘š]๐‘ ๐‘ ๐‘1[1]๐‘2[1]โ‹ฏ๐‘๐‘ [1]โ‹ฏ๐‘1[๐‘š]๐‘2[๐‘š]โ‹ฏ๐‘๐‘ [๐‘š],(2.1) where ๐ถ=[๐‘1,๐‘2,โ€ฆ,๐‘๐‘ ]๐‘‡, ๐‘[๐‘ฃ]=[๐‘1[๐‘ฃ],๐‘2[๐‘ฃ],โ€ฆ,๐‘๐‘ [๐‘ฃ]], and ๐ด[๐‘ฃ]=(๐‘Ž[๐‘ฃ]๐‘–๐‘—)๐‘ ๐‘–,๐‘—=1.

The adoption of the method (2.1) for solving the problem (1.1) leads to๐‘ฆ๐‘›+1=๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘๐‘—[๐‘ฃ]๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚,๐‘ฆ๐‘–(๐‘›)=๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚,(2.2) where ๐‘ก๐‘›=๐‘ก0+๐‘›โ„Ž,๐‘ก๐‘—(๐‘›)=๐‘ก๐‘›+๐‘๐‘—โ„Ž, ๐‘ฆ๐‘›, and ๐‘ฆ๐‘—(๐‘›), ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›) are approximations to the analytic solution ๐‘ฆ(๐‘ก๐‘›), ๐‘ฆ(๐‘ก๐‘›+๐‘๐‘—โ„Ž), ๐‘ฆ(๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ) of (1.1), respectively, and the argument ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›) is determined byฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)=โŽงโŽชโŽจโŽชโŽฉ๐œ‘๎€ท๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ๎€ธ๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค0๐‘Ÿ๎“๐‘ƒ๐‘ฃ=โˆ’๐‘‘๐ฟ๐‘ƒ๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธ๐‘ฆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘ƒ๐‘ฃ)๐‘—๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ>0,(2.3) with ๐œ๐‘ฃ=(๐‘š๐‘ฃโˆ’๐›ฟ๐‘ฃ)โ„Ž, ๐›ฟ๐‘ฃโˆˆ[0,1), integer ๐‘š๐‘ฃโ‰ฅ๐‘Ÿ+1, ๐‘Ÿ,๐‘‘โ‰ฅ0, and๐ฟ๐‘ƒ๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธ=๐‘Ÿ๎‘๐‘˜=โˆ’๐‘‘๐‘˜โ‰ ๐‘ƒ๐‘ฃ๎‚ต๐›ฟ๐‘ฃโˆ’๐‘˜๐‘ƒ๐‘ฃ๎‚ถโˆ’๐‘˜,๐‘ƒ๐‘ฃ=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,๐‘Ÿ.(2.4) We assume ๐‘š๐‘ฃโ‰ฅ๐‘Ÿ+1 is to guarantee that no (unknown) values ๐‘ฆ๐‘—(๐‘–) with ๐‘–โ‰ฅ๐‘› are used in the interpolation procedure๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚€๐‘กisanapproximationto๐‘ค๐‘—(๐‘›)๎‚๎€œโˆถ=๐‘ก๐‘—(๐‘›)๐‘ก๐‘ฃ)๐‘—(๐‘›โˆ’๐‘š๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›)๎‚,๐‘ ,๐‘ฆ(๐‘ )๐‘‘๐‘ ,(2.5) which can be computed by a appropriate compound quadrature rule:๐‘ค๐‘—[๐‘ฃ](๐‘›)=โ„Ž๐‘š๐‘ฃ๎“๐‘ž=0๐‘‘๐‘ž๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ก๐‘—(๐‘›โˆ’๐‘ž),๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚,๐‘ฃ=1,2,โ€ฆ,๐‘š,๐‘—=1,2,โ€ฆ,๐‘ .(2.6) As for the quadrature rule (2.6), we usually adopt the compound trapezoidal rule, the compound Simpsons rule or the compound Newton-Cotes rule, and so forth according to the requirement of the convergence of the method (see [19]) and denote ๐‘€=max1โ‰ค๐‘ฃโ‰ค๐‘š{๐‘š๐‘ฃ} and ๐œ‚=max1โ‰ค๐‘ฃโ‰ค๐‘š{๐œ‚๐‘ฃ} with ๐œ‚๐‘ฃ satisfing โˆ‘๐‘š๐‘ฃ๐‘ž=0|๐‘‘๐‘ž|<๐œ‚๐‘ฃ, ๐‘ฃ=1,2,โ€ฆ,๐‘š.

In addition, we always put๐‘ฆ๐‘—(๐‘›)=๐œ‘(๐‘ก๐‘›+๐‘๐‘—โ„Ž), ๐‘ฆ๐‘›=๐œ‘(๐‘ก๐‘›) whenever ๐‘›โ‰ค0.

In order to write (2.2), (2.3), (2.5), and (2.6) in a more compact way, we introduce some notations. The ๐‘ร—๐‘ identity matrix will be denoted by ๐ผ๐‘, ๐‘’=(1,1,โ€ฆ,1)๐‘‡โˆˆ๐‘…๐‘†, ๎‚๐บ=๐บโŠ—๐ผ๐‘ is the Kronecker product of matrix ๐บ and ๐ผ๐‘. For ๐‘ข=(๐‘ข1,๐‘ข2,โ€ฆ,๐‘ข๐‘ )๐‘‡, ๐‘ฃ=(๐‘ฃ1,๐‘ฃ2,โ€ฆ,๐‘ฃ๐‘ )๐‘‡โˆˆ๐ถ๐‘๐‘†, we define the inner product and the induced norm in ๐ถ๐‘๐‘† as follows: โŸจ๐‘ข,๐‘ฃโŸฉ=๐‘ ๎“๐‘–=1โŸจ๐‘ข๐‘–,๐‘ฃ๐‘–๎„ถ๎„ต๎„ตโŽทโŸฉ,โ€–๐‘ขโ€–=๐‘ ๎“๐‘–=1โ€–โ€–๐‘ข๐‘–โ€–โ€–2.(2.7)

Moreover, we also adopt that๐‘ฆ(๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ฆ1(๐‘›)๐‘ฆ2(๐‘›)โ‹ฎ๐‘ฆ๐‘ (๐‘›)โŽคโŽฅโŽฅโŽฅโŽฆ,ฬƒ๐‘ฆ[๐‘ฃ](๐‘›)=โŽกโŽขโŽขโŽขโŽฃฬƒ๐‘ฆ1[๐‘ฃ](๐‘›)ฬƒ๐‘ฆ2[๐‘ฃ](๐‘›)โ‹ฎฬƒ๐‘ฆ๐‘ [๐‘ฃ](๐‘›)โŽคโŽฅโŽฅโŽฅโŽฆ,๐‘ค[๐‘ฃ](๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ค1[๐‘ฃ](๐‘›)๐‘ค2[๐‘ฃ](๐‘›)โ‹ฎ๐‘ค๐‘ [๐‘ฃ](๐‘›)โŽคโŽฅโŽฅโŽฅโŽฆ,๐‘‡(๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ก1(๐‘›)๐‘ก2(๐‘›)โ‹ฎ๐‘ก๐‘ (๐‘›)โŽคโŽฅโŽฅโŽฅโŽฆ,๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ๐‘“[๐‘ฃ]๎‚€๐‘ก1(๐‘›),๐‘ฆ1(๐‘›),ฬƒ๐‘ฆ1[๐‘ฃ](๐‘›),๐‘ค1[๐‘ฃ](๐‘›)๎‚๐‘“[๐‘ฃ]๎‚€๐‘ก2(๐‘›),๐‘ฆ2(๐‘›),ฬƒ๐‘ฆ2[๐‘ฃ](๐‘›),๐‘ค2[๐‘ฃ](๐‘›)๎‚โ‹ฎ๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘ (๐‘›),๐‘ฆ๐‘ (๐‘›),ฬƒ๐‘ฆ๐‘ [๐‘ฃ](๐‘›),๐‘ค๐‘ [๐‘ฃ](๐‘›)๎‚โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.(2.8)

With the above notation, method (2.2),(2.3), (2.5), and (2.6) can be written as๐‘ฆ๐‘›+1=๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ,๐‘ฆ(๐‘›)=ฬƒ๐‘’๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ,ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)=โŽงโŽชโŽจโŽชโŽฉ๎€ท๐‘กฬƒ๐‘’๐œ‘๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ๎€ธ,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค๐‘ก0,๐‘Ÿ๎“๐‘ƒ๐‘ฃ=โˆ’๐‘‘๐ฟ๐‘ƒ๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธ๐‘ฆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘ƒ๐‘ฃ)๐‘—,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ>๐‘ก0,๐‘ค๐‘—[๐‘ฃ](๐‘›)=โ„Ž๐‘š๐‘ฃ๎“๐‘ž=0๐‘‘๐‘ž๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›+๐‘๐‘—โ„Ž,๐‘ก๐‘›โˆ’๐‘ž+๐‘๐‘—โ„Ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚.(2.9) In 1997, Zhang and Zhou [25] introduced the extension of RN-stability to GDN-stability as follows.

Definition 2.1. An ARKLM (2.1) for DDEs is called GDN-stable if, numerical approximations ๐‘ฆ๐‘› and ๐‘ง๐‘› to the solution of (1.1) and its perturbed problem, respectively, satisfy โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–โ‰ค๐ถmax๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰บ๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–,๐‘›โ‰ฅ0,(2.10) where constant ๐ถ>0 depends only on the method, the parameter ๐›ผ๐‘ฃ,๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,ฬƒ๐‘Ÿ๐‘ฃ, and the interval length ๐‘‡โˆ’๐‘ก0, ๐œ“(๐‘ก) is the initial function to the perturbed problem of (1.1).

Definition 2.2. An ARKLM (2.1) is called strongly algebraically stable if matrices ๐‘€๐›พ๐œ‡ are nonnegative definite, where ๐‘€๐›พ๐œ‡=๐ต[๐›พ]๐ด[๐œ‡]+๐ด๐‘‡[๐›พ]๐ต[๐œ‡]โˆ’๐‘[๐›พ]๐‘๐‘‡[๐œ‡],๐ต[๐›พ]๎‚€๐‘=diag1[๐›พ],๐‘2[๐›พ],โ€ฆ,๐‘๐‘ [๐›พ]๎‚,(2.11) for ๐œ‡,๐›พ=1,2,โ€ฆ,๐‘š.
Let ๎‚†๐‘ฆ๐‘›,๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[1](๐‘›),ฬƒ๐‘ฆ๐‘—[2](๐‘›),โ€ฆ,ฬƒ๐‘ฆ๐‘—[๐‘š](๐‘›),๐‘ค๐‘—[1](๐‘›),๐‘ค๐‘—[2](๐‘›),โ€ฆ,๐‘ค๐‘—[๐‘š](๐‘›)๎‚‡๐‘ ๐‘—=1,๎‚†๐‘ง๐‘›,๐‘ง๐‘—(๐‘›),ฬƒ๐‘ง๐‘—[1](๐‘›),ฬƒ๐‘ง๐‘—[2](๐‘›),โ€ฆ,ฬƒ๐‘ง๐‘—[๐‘š](๐‘›),๎๐‘ค๐‘—[1](๐‘›),๎๐‘ค๐‘—[2](๐‘›)๎๐‘ค,โ€ฆ,๐‘—[๐‘š](๐‘›)๎‚‡๐‘ ๐‘—=1(2.12)be two sequences of approximations to problems (1.1) and its perturbed problem, respectively. From method (2.1) with the same step size โ„Ž, and write ๐‘ˆ๐‘–(๐‘›)=๐‘ฆ๐‘–(๐‘›)โˆ’๐‘ง๐‘–(๐‘›),๎‚๐‘ˆ๐‘–[๐‘ฃ](๐‘›)=ฬƒ๐‘ฆ๐‘–[๐‘ฃ](๐‘›)โˆ’ฬƒ๐‘ง๐‘–[๐‘ฃ](๐‘›),๐‘ˆ0(๐‘›)=๐‘ฆ๐‘›โˆ’๐‘ง๐‘›,๐‘„๐‘–[๐‘ฃ](๐‘›)๎‚ƒ๐‘“=โ„Ž[๐‘ฃ]๎‚€๐‘ก๐‘–(๐‘›),๐‘ฆ๐‘–(๐‘›),ฬƒ๐‘ฆ๐‘–[๐‘ฃ](๐‘›),๐‘ค๐‘–[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘–(๐‘›),๐‘ง๐‘–(๐‘›),ฬƒ๐‘ง๐‘–[๐‘ฃ](๐‘›),๎๐‘ค๐‘–[๐‘ฃ](๐‘›),๎‚๎‚„๐‘–=1,2,โ€ฆ,๐‘ ,๐‘ฃ=1,2,โ€ฆ,๐‘š.(2.13) Then (2.2) and (2.3) read ๐‘ˆ0(๐‘›+1)=๐‘ˆ0(๐‘›)+๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘๐‘—[๐‘ฃ]๐‘„๐‘—[๐‘ฃ](๐‘›),๐‘ˆ(2.14)๐‘–(๐‘›)=๐‘ˆ0(๐‘›)+๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๐‘„๐‘—[๐‘ฃ](๐‘›).(2.15)

Our main results about GDN-stability are contained in the following theorem.

Theorem 2.3. Assume ARK method (2.2) is strongly algebraically stable, and then the corresponding ARKLM (2.1) is GDN-stable, and satisfies โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ง๐‘›โ€–โ€–โ‰ค๐ถmax๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,๐‘›โ‰ฅ0,(2.16) where๎ƒฌ6๎€ท๐ถ=exp๐‘‡โˆ’๐‘ก0๎€ธ๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,๐ฟ๐‘ฃ=maxโˆ’๐‘‘โ‰ค๐‘๐‘ฃโ‰ค๐‘Ÿ๎€ฝ๐ฟ๐‘๐œˆ๎€พ,(2.17)

Proof. From (2.14) and (2.15) we get โ€–โ€–๐‘ˆ0(๐‘›+1)โ€–โ€–2=๎„”๐‘ˆ0(๐‘›)+๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘–=1๐‘๐‘–[๐‘ฃ]๐‘„๐‘–[๐‘ฃ](๐‘›),๐‘ˆ0(๐‘›)+๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘–=1๐‘๐‘–[๐‘ฃ]๐‘„๐‘–[๐‘ฃ](๐‘›)๎„•=โ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+2๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘–=1๐‘๐‘–[๐‘ฃ]๎‚ฌ๐‘„Re๐‘–[๐‘ฃ](๐‘›),๐‘ˆ0(๐‘›)๎‚ญ+๐‘š๎“๐‘ ๐‘ข,๐‘ฃ=1๎“๐‘–,๐‘—=1๐‘๐‘–[๐‘ข]๐‘๐‘—[๐‘ฃ]๎‚ฌ๐‘„๐‘–[๐‘ข](๐‘›),๐‘„๐‘—[๐‘ฃ](๐‘›)๎‚ญ=โ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+2๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘–=1๐‘๐‘–[๐‘ฃ]๎„”๐‘„Re๐‘–[๐‘ฃ](๐‘›),๐‘ˆ๐‘–(๐‘›)โˆ’๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๐‘„๐‘—[๐‘ฃ](๐‘›)๎„•+๐‘š๎“๐‘ ๐‘ข,๐‘ฃ=1๎“๐‘–,๐‘—=1๐‘๐‘–[๐‘ข]๐‘๐‘—[๐‘ฃ]๎‚ฌ๐‘„๐‘–[๐‘ฃ](๐‘›),๐‘„๐‘—[๐‘ข](๐‘›)๎‚ญ=โ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+2๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘–=1๐‘๐‘–[๐‘ฃ]๎‚ฌ๐‘„Re๐‘–[๐‘ฃ](๐‘›),๐‘ˆ๐‘–(๐‘›)๎‚ญโˆ’๐‘š๎“๐‘ ๐‘ข,๐‘ฃ=1๎“๐‘–,๐‘—=1๎‚€๐‘๐‘–[๐‘ข]๐‘Ž[๐‘ฃ]๐‘–๐‘—+๐‘๐‘—[๐‘ฃ]๐‘Ž[๐‘ข]๐‘–๐‘—โˆ’๐‘๐‘–[๐‘ข]๐‘๐‘—[๐‘ฃ]๐‘„๎‚๎‚ฌ๐‘–[๐‘ฃ](๐‘›),๐‘„๐‘—[๐‘ข](๐‘›)๎‚ญ.(2.18) If the matrices ๐‘€๐›พ๐œ‡ are nonnegative definite, then โ€–โ€–๐‘ˆ0(๐‘›+1)โ€–โ€–2โ‰คโ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+2๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘๐‘—[๐‘ฃ]๎‚ฌ๐‘„Re๐‘—[๐‘ฃ](๐‘›),๐‘ˆ๐‘—(๐‘›)๎‚ญ.(2.19) Furthermore, by conditions (1.2)~(1.4) and Schwartz inequality we have ๎‚ฌ๐‘„Re๐‘—[๐‘ฃ](๐‘›),๐‘ˆ๐‘—(๐‘›)๎‚ญ๎‚ฌ๐‘“=โ„ŽRe[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ง๐‘—(๐‘›),ฬƒ๐‘ง๐‘—[๐‘ฃ](๐‘›),๎๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚,๐‘ˆ๐‘—(๐‘›)๎‚ญโ‰คโ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+โ„Ž๐›ฝ๐‘ฃโ€–โ€–๎‚๐‘ˆ๐‘—[๐‘ฃ](๐‘›)โ€–โ€–+โ„Ž๐œŽ๐‘ฃโ€–โ€–๐‘ค๐‘—[๐‘ฃ](๐‘›)โˆ’๎๐‘ค๐‘—[๐‘ฃ](๐‘›)โ€–โ€–2โ‰คโ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+โ„Ž๐›ฝ๐‘ฃโ€–โ€–๎‚๐‘ˆ๐‘—[๐‘ฃ](๐‘›)โ€–โ€–+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2(2.20)=โ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค๐‘ก0=โ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+โ„Ž๐›ฝ๐‘ฃโ€–โ€–โ€–โ€–๐‘Ÿ๎“๐‘๐‘ฃ=โˆ’๐‘‘๐ฟ๐‘๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธ๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ>๐‘ก0(2.21)โ‰คโ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค๐‘ก0(2.22)โ‰คโ„Ž๐›ผ๐‘ฃโ€–โ€–๐‘ˆ๐‘—(๐‘›)โ€–โ€–2+2โ„Ž๐›ฝ๐‘ฃ๐ฟ2๐‘ฃโ€–โ€–โ€–โ€–๐‘Ÿ๎“๐‘๐‘ฃ=โˆ’๐‘‘๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ>๐‘ก0.(2.23) For (2.23), we have (2.23)โ‰ค2โ„Ž๐›ฝ๐‘ฃ๐ฟ2๐‘ฃโ€–โ€–โ€–โ€–๐‘Ÿ๎“๐‘๐‘ฃ=โˆ’๐‘‘๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2+2โ„Ž3ฬƒ๐‘Ÿ2๐‘ฃ๐œ‚2๐‘ฃ๐œŽ๐‘ฃ๐‘š๐‘ฃ๎“๐‘ž=0โ€–โ€–๐‘ˆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–2โ‰ค3โ„Ž๐›ฝ๐‘ฃ๐ฟ2๐‘ฃโ€–โ€–โ€–โ€–๐‘š๐‘ฃ๎“๐‘๐‘ฃ=โˆ’๐‘‘๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2.(2.24) By the same way, we can also get (2.22)โ‰ค3โ„Ž๐›ฝ๐‘ฃ๐ฟ2๐‘ฃโ€–โ€–โ€–โ€–๐‘š๐‘ฃ๎“๐‘ƒ๐‘ฃ=โˆ’๐‘‘๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2.(2.25) Substituting (2.25) and (2.24) into (2.19), yields โ€–โ€–๐‘ˆ0(๐‘›+1)โ€–โ€–2โ‰คโ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+2โ„Ž๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=13๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๐‘๐‘—[๐‘ฃ]โ€–โ€–โ€–โ€–๐‘š๐‘ฃ๎“๐‘๐‘ฃ=โˆ’๐‘‘๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–โ€–โ€–2โ‰คโ€–โ€–๐‘ˆ(2.26)0(๐‘›)โ€–โ€–2+6โ„Ž๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๐‘๐‘—[๐‘ฃ]๎€ท๐‘š๐‘ฃ๎€ธ+๐‘‘+1maxโˆ’๐‘‘โ‰ค๐‘๐‘ฃโ‰ค๐‘š๐‘ฃโ€–โ€–๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–2โ‰คโ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ+๐‘‘+1max(๐‘—,๐‘๐‘ฃ)โˆˆ๐ธ๐‘ฃโ€–โ€–๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ๎‚ปโ€–โ€–๐‘ˆ+๐‘‘+1max0(๐‘›)โ€–โ€–2,max(๐‘—,๐‘๐‘ฃ)โˆˆ๐ธ๐‘ฃโ€–โ€–๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘—โ€–โ€–2๎‚ผ,(2.27) where ๐ธ๐‘ฃ={(๐‘—,๐‘ƒ๐‘ฃ)1โ‰ค๐‘—โ‰ค๐‘ ,โˆ’๐‘‘โ‰ค๐‘ƒ๐‘ฃโ‰ค๐‘Ÿ}.
Similar to (2.27), the inequalities: โ€–โ€–๐‘ˆ๐‘–(๐‘›)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ๎‚ปโ€–โ€–๐‘ˆ+๐‘‘+1max0(๐‘›)โ€–โ€–2,max(๐‘—,๐‘ƒ๐‘ฃ)โˆˆ๐ธโ€–โ€–๐‘ˆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘ƒ๐‘ฃ)๐‘—โ€–โ€–2๎‚ผ(2.28) follows for ๐‘–=1,2,โ€ฆ,๐‘ .
In the following, with the help of inequalities (2.27), (2.28), and induction we shall prove the inequalities: โ€–โ€–๐‘ˆ๐‘–(๐‘›)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1(๐‘›+1)max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,(2.29) for ๐‘›โ‰ฅ0, ๐‘–=1,2,โ€ฆ,๐‘ .
In fact, it is clear from (2.27), (2.28), and ๐‘š๐‘ฃโ‰ฅ๐‘Ÿ+1 such that โ€–โ€–๐‘ˆ๐‘–(0)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,๐‘–=0,1,2,โ€ฆ,๐‘ .(2.30) Suppose for ๐‘›โ‰ค๐‘˜(๐‘˜โ‰ฅ0)that โ€–โ€–๐‘ˆ๐‘–(๐‘›)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1(๐‘›+1)max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,๐‘–=0,1,2,โ€ฆ,๐‘ .(2.31) Then from (2.27) and (2.28), ๐‘š๐‘ฃโ‰ฅ๐‘Ÿ+1 and โˆ‘1+6โ„Ž๐‘š๐‘ ๐‘š๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ(๐‘š๐‘ฃ+๐‘‘+1)>1, we conclude that โ€–โ€–๐‘ˆ๐‘–(๐‘˜+1)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1(๐‘˜+2)max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2,๐‘–=0,1,2,โ€ฆ,๐‘ .(2.32) This completes the proof of inequalities (2.29). In view of (2.29), we get for ๐‘›โ‰ฅ0 that โ€–โ€–๐‘ˆ0(๐‘›)โ€–โ€–2โ‰ค๎ƒฌ1+6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1(๐‘›+1)max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)โ€–2๎ƒฌโ‰คexp(๐‘›+1)6โ„Ž๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)2๎ƒฌ6๎€ทโ‰คexp๐‘‡โˆ’๐‘ก0๎€ธ๐‘š๐‘ ๐‘š๎“๐‘ฃ=1๐›ฝ๐‘ฃ๐ฟ2๐‘ฃ๎€ท๐‘š๐‘ฃ๎€ธ๎ƒญ+๐‘‘+1max๐‘ก0โˆ’๐œโ‰ค๐‘กโ‰ค๐‘ก0โ€–โ€–๐œ‘(๐‘ก)โˆ’๐œ“(๐‘ก)2.(2.33)As a result, we know that method (2.1) is GDN-stable.

3. D-Convergence

In order to study the convergence of numerical methods for MDIDEs, we have to mention the concept of the convergence for stiff ODEs.

In 1981, Frank et al. [26] introduced the important concept of B-convergence for numerical methods applied to nonlinear stiff initial value problems of ordinary differential equations. Later, there have been rapid developments in the study of B-convergence, and a significant number of important results have already been found for Runge-Kutta methods. In fact, B-convergence result is nothing but a realistic global error estimate based on one-sided Lipschitz constant [27]. In this section, we start discussing the convergence of ARKLM (2.1) for MDIDEs (1.1) with conditions (1.2)โ€“(1.4). The approach to the derivation of these estimates is similar to that used in [25]. We assume the analytic solution ๐‘ฆ(๐‘ก) of (1.1) is smooth enough, and its derivatives used later are bounded byโ€–โ€–๐ท(๐‘–)โ€–โ€–โ‰ค๎‚‹๐‘€๐‘ฆ(๐‘ก)๐‘–๎€บ๐‘ก,๐‘กโˆˆ0๎€ปโˆ’๐œ,๐‘‡,(3.1)

where ๐ท(๐‘–)๎‚ป๐‘ฆ๐‘ฆ(๐‘ก)=(๐‘–)๎€ท๐‘ก(๐‘ก),๐‘กโˆˆ0+(๐‘—โˆ’1)โ„Ž,๐‘ก0๎€ธ,๐‘ฆ+๐‘—โ„Ž(๐‘–)๎€ท๐‘ก0๎€ธ+๐‘—โ„Žโˆ’0,๐‘ก=๐‘ก0+๐‘—โ„Ž.(3.2) If we introduce some notations๐‘Œ(๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘1โ„Ž๎€ธ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘2โ„Ž๎€ธโ‹ฎ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘๐‘ โ„Ž๎€ธโŽคโŽฅโŽฅโŽฅโŽฆ,๎‚๐‘Œ[๐‘ฃ](๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘1โ„Žโˆ’๐œ๐‘ฃ๎€ธ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘2โ„Žโˆ’๐œ๐‘ฃ๎€ธโ‹ฎ๐‘ฆ๎€ท๐‘ก๐‘›+๐‘๐‘ โ„Žโˆ’๐œ๐‘ฃ๎€ธโŽคโŽฅโŽฅโŽฅโŽฆ,๎‚๐‘ค[๐‘ฃ](๐‘›)=โŽกโŽขโŽขโŽขโŽฃ๐‘ค๎€ท๐‘ก๐‘›+๐‘1โ„Žโˆ’๐œ๐‘ฃ๎€ธ๐‘ค๎€ท๐‘ก๐‘›+๐‘2โ„Žโˆ’๐œ๐‘ฃ๎€ธโ‹ฎ๐‘ค๎€ท๐‘ก๐‘›+๐‘๐‘ โ„Žโˆ’๐œ๐‘ฃ๎€ธโŽคโŽฅโŽฅโŽฅโŽฆ.(3.3) With the above notations, the local errors in (2.9) can be defined as๐‘ฆ๎€ท๐‘ก๐‘›+1๎€ธ๎€ท๐‘ก=๐‘ฆ๐‘›๎€ธ+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),๐‘Œ(๐‘›),๎‚๐‘Œ[๐‘ฃ](๐‘›),๎‚๐‘ค[๐‘ฃ](๐‘›)๎‚+๐‘„๐‘›,๐‘Œ(3.4)(๐‘›)๎€ท๐‘ก=ฬƒ๐‘’๐‘ฆ๐‘›๎€ธ+โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),๐‘Œ(๐‘›),๎‚๐‘Œ[๐‘ฃ](๐‘›),๎‚๐‘ค[๐‘ฃ](๐‘›)๎‚+๐‘Ÿ๐‘›๎‚๐‘Œ,(3.5)[๐‘ฃ](๐‘›)=๎‚€๎‚๐‘Œ1[๐‘ฃ](๐‘›),๎‚๐‘Œ2[๐‘ฃ](๐‘›)๎‚๐‘Œ,โ€ฆ,๐‘ [๐‘ฃ](๐‘›)๎‚๐‘‡,(3.6) with๎‚๐‘Œ๐‘—[๐‘ฃ](๐‘›)=โŽงโŽชโŽจโŽชโŽฉ๐œ‘๎€ท๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ๎€ธ,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค๐‘ก0,๐‘Ÿ๎“๐‘ƒ๐‘ฃ=โˆ’๐‘‘๐ฟ๐‘ƒ๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธ๐‘ฆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘ƒ๐‘ฃ)๐‘—+๐œŒ๐‘—[๐‘ฃ](๐‘›),๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ>๐‘ก0,๐‘ค(3.7)๐‘—[๐‘ฃ](๐‘›)=โ„Ž๐‘š๐‘ฃ๎“๐‘ž=0๐‘‘๐‘ž๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›+๐‘๐‘—โ„Ž,๐‘ก๐‘›โˆ’๐‘ž+๐‘๐‘—โ„Ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚+๐‘…๐‘—[๐‘ฃ](๐‘›).(3.8) If we take ฬ†๐‘ฆ๐‘›=๐‘ฆ(๐‘ก๐‘›), ฬ†๐‘ฆ(๐‘›)=๐‘Œ(๐‘›), ฬ†๐‘ฆ[๐‘ฃ](๐‘›)=๎‚๐‘Œ[๐‘ฃ](๐‘›), and ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)=๎‚๐‘ค[๐‘ฃ](๐‘›)

Then we can get the perturbed scheme of (2.9),ฬ†๐‘ฆ๐‘›+1=ฬ†๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)๎‚+๐‘„๐‘›,(3.9)ฬ†๐‘ฆ(๐‘›)=ฬƒ๐‘’ฬ†๐‘ฆ๐‘›+โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)๎‚+๐‘Ÿ๐‘›ฬ†,(3.10)ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)=โŽงโŽชโŽจโŽชโŽฉ๎€ท๐‘กฬƒ๐‘’๐œ‘๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ๎€ธ,๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃโ‰ค0,๐‘Ÿ๎“๐‘ƒ๐‘ฃ=โˆ’๐‘‘๐ฟ๐‘ƒ๐‘ฃ๎€ท๐›ฟ๐‘ฃ๎€ธฬ†๐‘ฆ(๐‘›โˆ’๐‘š๐‘ฃ+๐‘ƒ๐‘ฃ)๐‘—+๐œŒ๐‘—[๐‘ฃ](๐‘›),๐‘ก๐‘›+๐‘๐‘—โ„Žโˆ’๐œ๐‘ฃ๐‘ค>0,(3.11)๐‘—[๐‘ฃ](๐‘›)=โ„Ž๐‘š๐‘ฃ๎“๐‘ž=0๐‘‘๐‘ž๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›+๐‘๐‘—โ„Ž,๐‘ก๐‘›โˆ’๐‘ž+๐‘๐‘—โ„Ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚+๐‘…๐‘—[๐‘ฃ](๐‘›).(3.12) With perturbations, ๐‘„๐‘›โˆˆ๐ถ๐‘, ๐‘Ÿ๐‘›=(๐‘Ÿ(๐‘›)๐‘‡1,๐‘Ÿ(๐‘›)๐‘‡2,โ€ฆ,๐‘Ÿ(๐‘›)๐‘‡s)๐‘‡,๐‘…[๐‘ฃ](๐‘›)=(๐‘…1[๐‘ฃ](๐‘›),๐‘…2[๐‘ฃ](๐‘›),โ€ฆ,๐‘…๐‘ [๐‘ฃ](๐‘›))๐‘‡, ๐œŒ(๐‘›)=(๐œŒ(๐‘›)๐‘‡1,๐œŒ(๐‘›)๐‘‡2,โ€ฆ,๐œŒ(๐‘›)๐‘‡s)โˆˆ๐ถ๐‘๐‘†, according to Taylor formula and the formula in [28, pages 205โ€“212], ๐‘„๐‘›,๐‘Ÿ๐‘› and ๐œŒ๐‘› can be determined respectively, as follows:๐‘„๐‘›=๐‘ƒ๎“๐‘™=1โ„Ž๐‘™๎ƒฉ1(๐‘™โˆ’1)!๐‘™โˆ’๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘๐‘—[๐‘ฃ]๐‘๐‘—๐‘™โˆ’1๎ƒช๐ท(๐‘™)๐‘ฆ๎€ท๐‘ก๐‘›๎€ธ+๐‘…0(๐‘›),๐‘Ÿ(3.13)๐‘–(๐‘›)=๐‘ƒ๎“๐‘™=1โ„Ž๐‘™๎ƒฉ1(๐‘™โˆ’1)!๐‘™๐‘๐‘™๐‘–โˆ’๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๐‘๐‘—๐‘™โˆ’1๎ƒช๐ท(๐‘™)๐‘ฆ๎€ท๐‘ก๐‘›๎€ธ+๐‘…๐‘–(๐‘›)๐œŒ,(3.14)๐‘–(๐‘›)=โ„Ž๐‘ž+1(๐‘ž+1)!๐‘š๎“๐‘Ÿ๐‘ฃ=1๎‘๐‘ƒ๐‘ฃ=โˆ’๐‘‘๎€ท๐›ฟ๐‘ฃโˆ’๐‘ƒ๐‘ฃ๎€ธ๐ท(๐‘ž+1)๐‘ฆ๎‚€๐œ‰๐‘–(๐‘›)๎‚,๐œ‰๐‘–(๐‘›)โˆˆ๎€ท๐‘ก๐‘›โˆ’๐‘š๐‘ฃโˆ’๐‘‘+๐‘๐‘–โ„Ž,๐‘ก๐‘›โˆ’๐‘š๐‘ฃ+๐‘Ÿ+๐‘๐‘–โ„Ž๎€ธ,(3.15) where ๐‘ž=๐‘‘+๐‘Ÿ,๐‘…๐‘–(๐‘›), and ๐œ‰๐‘–(๐‘›) satisfy ||๐‘…๐‘–(๐‘›)๎‚Š๐‘€||โ‰ค๐‘–โ„Ž๐‘–+1,๐‘–=0,1,2,โ€ฆ,๐‘ , โ„Žโˆˆ(0,โ„Ž0], โ„Ž0 depends only on the method, and ๎‚Š๐‘€๐‘–(๐‘–=0,1,2,โ€ฆ,๐‘ ) depends only on the method and some ๎‚‹๐‘€๐‘– in (3.2).

Combining (2.2), (2.3), (2.5), and (2.6) with (3.9), (3.10), (3.11), and (3.12) yields the following recursion scheme for the ๐œ€0(๐‘›+1)=ฬ†๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›+1:๐œ€0(๐‘›+1)=๐œ€0(๐‘›)+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]T๎‚†๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ+๐‘”๐‘›[๐‘ฃ]๐œ€๐‘›+๐ป[๐‘ฃ](๐‘›)๎‚€ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)โˆ’๐‘ค[๐‘ฃ](๐‘›)๐œ€๎‚๎‚‡+๐‘„๐‘›,๐‘›=ฬƒ๐‘’๐œ€0(๐‘›)+โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎‚†๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ+๐‘”๐‘›[๐‘ฃ]๐œ€๐‘›+๐ป[๐‘ฃ](๐‘›)๎‚€ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)โˆ’๐‘ค[๐‘ฃ](๐‘›)๎‚๎‚‡+๐‘Ÿ๐‘›,(3.16) where ๐œ€0(๐‘›+1)=ฬ†๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›+1, ๐œ€๐‘›=(๐œ€1๐‘‡(๐‘›),๐œ€2๐‘‡(๐‘›),โ€ฆ,๐œ€๐‘ ๐‘‡(๐‘›))๐‘‡=ฬ†๐‘ฆ(๐‘›)โˆ’๐‘ฆ(๐‘›),๐‘”๐‘–[๐‘ฃ](๐‘›)=๎€œ10๐‘“2๎‚€๐‘ก๐‘›+๐‘๐‘–โ„Ž,๐‘ฆ๐‘–(๐‘›)๎‚€+๐œƒฬ†๐‘ฆ๐‘–(๐‘›)โˆ’๐‘ฆ๐‘–(๐‘›)๎‚,ฬ†ฬƒ๐‘ฆ๐‘–[๐‘ฃ](๐‘›),ฬ†๎‚๐‘ค๐‘–[๐‘ฃ](๐‘›)๎‚๐ป๐‘‘๐œƒ,๐‘–=1,2,โ€ฆ,๐‘ ,๐‘–[๐‘ฃ](๐‘›)=๎€œ10๐‘“4๎‚€๐‘ก๐‘›+๐‘๐‘–โ„Ž,๐‘ฆ๐‘–(๐‘›),ฬ†ฬƒ๐‘ฆ๐‘–[๐‘ฃ](๐‘›),ฬ†๎‚๐‘ค๐‘–[๐‘ฃ](๐‘›)๎‚€ฬ†๎‚๐‘ค+๐œƒ๐‘–[๐‘ฃ](๐‘›)โˆ’๐‘ค๐‘–[๐‘ฃ](๐‘›)๎‚๎‚๐‘‘๐œƒ,(3.17) here, ๐‘“๐‘–(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,๐‘ฅ4) is the Jacobian matrix (๐œ•๐‘“(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3,๐‘ฅ4)/๐œ•๐‘ฅ๐‘–)๐‘–=1,2,3,4.๐ป[๐‘ฃ](๐‘›)๎‚€ฬ†๎‚๐‘ค[๐‘ฃ](๐‘›)โˆ’๐‘ค[๐‘ฃ](๐‘›)๎‚=โ„Ž๐ป๐‘š[๐‘ฃ](๐‘›)๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธ๎€ป+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›,ฬ†๐‘ฆ(๐‘›)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›,๐‘ฆ(๐‘›)๎€ธ๎€ป=โ„Ž๐ป๐‘š[๐‘ฃ](๐‘›)๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธ๎€ป+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๎€œ10๐‘”3[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›,๐‘ฆ(๐‘›)๎€ท+๐œƒฬ†๐‘ฆ(๐‘›)โˆ’๐‘ฆ(๐‘›)๎€ธ๎€ธ๐‘‘๐œƒโ‹…๐œ€๐‘›.(3.18) Assume that (๎‚๐ผ๐‘ โˆ‘โˆ’โ„Ž๐‘š๐‘ฃ=1๎‚๐ด[๐‘ฃ](๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›))) is regular, from (3.16) and (3.17), (3.18), we can get๐œ€0(๐‘›+1)=โŽงโŽชโŽจโŽชโŽฉ๐ผ๐‘+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๎ƒฌ๎‚๐ผ๐‘ โˆ’โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎‚€๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚๎ƒญโˆ’1๎‚€๐‘”ร—ฬƒ๐‘’[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚โŽซโŽชโŽฌโŽชโŽญ๐œ€0(๐‘›)+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๎‚€๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚๎ƒฌ๎‚๐ผ๐‘ โˆ’โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎‚€๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚๎ƒญโˆ’1ร—๎ƒฉ๐‘Ÿ๐‘›+โ„Ž2๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›)๎ƒช+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๎ƒฌ๎‚๐ผ๐‘ โˆ’โ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎‚€๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚๎ƒญโˆ’1ร—๎ƒฌโ„Ž๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎‚€๐‘”[๐‘ฃ](๐‘›)+โ„Ž๐ป[๐‘ฃ](๐‘›)๐‘‘0๐‘”3[๐‘ฃ](๐‘›)๎‚๎ƒญโ‹…๎ƒฏ๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ+โ„Ž๐ป๐‘š[๐‘ฃ](๐‘›)๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ(๐‘›โˆ’๐‘ž)๎ƒฐ๎€ธ๎€ป+๐‘„๐‘›+โ„Ž2๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›).(3.19) Now, we introduce the concept of D-convergence from [25].

Definition 3.1. An ARKLM (2.1) with๐‘ฆ๐‘›=๐‘ฆ(๐‘ก๐‘›)(๐‘›โ‰ค0), ๐‘ฆ๐‘–(๐‘›)=๐‘ฆ(๐‘ก๐‘›+๐‘๐‘–โ„Ž)(๐‘›<0) and ฬƒ๐‘ฆ๐‘–[๐‘ฃ](๐‘›)=๐‘ฆ(๐‘ก๐‘›+๐‘๐‘–โ„Žโˆ’๐œ๐‘ฃ)(๐‘›<0) is called D-convergence of order ๐‘ if this method, when applied to any given DIDEs (1.1) subject to (1.2)โ€“(1.4); produce an approximation sequence ๐‘ฆ๐‘› and the global error satisfies a bound of the form: โ€–โ€–๐‘ฆ๎€ท๐‘ก๐‘›๎€ธโˆ’๐‘ฆ๐‘›โ€–โ€–๎€ท๐‘กโ‰ค๐ถ๐‘›๎€ธโ„Ž๐‘ƒ๎€ท,โ„Žโˆˆ0,โ„Ž0๎€ป,(3.20) where the maximum stepsize โ„Ž0 depends on characteristic parameter ๐›ผ๐‘ฃ,๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,๎‚๐‘Ÿ๐‘ฃ and the method, the function ๐ถ(๐‘ก) depends only on some ๎‚‹๐‘€๐‘– in (3.2), delay ๐œ๐‘ฃ, characteristic parameters ๐›ผ๐‘ฃ,๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,ฬƒ๐‘Ÿ๐‘ฃ, ๐‘ฃ=1,2,โ€ฆ,๐‘š, and the method.

Definition 3.2. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be DA-stable if the matrix(๐ผ๐‘ โˆ’โˆ‘๐‘š๐œˆ=1๐ด[๐œˆ]๐œ‰) is regular for ๐œ‰โˆˆ๐ถโˆ’โˆถ={๐œ‰โˆˆ๐ถโˆฃRe๐œ‰โ‰ค0}, and |๐‘…๐‘–(๐œ‰)|โ‰ค1forall๐œ‰โˆˆ๐ถโˆ’,๐‘–=0,1,โ€ฆ,๐‘ .
Where ๐‘…๐‘–๎€ท๐œ€1๎€ธ=1+๐‘š๎“๐‘ฃ=1๐ด๐‘–[๐‘ฃ]๐œ€1๎ƒฉ๐ผ๐‘ โˆ’๐‘š๎“๐œˆ=1๐ด[๐œˆ]๐œ‰๎ƒชโˆ’1๐ด๐‘’,0[๐‘ฃ]=๐‘[๐‘ฃ],๐ด๐‘–[๐‘ฃ]=๎€ท๐‘Ž[๐‘ฃ]๐‘–1,๐‘Ž[๐‘ฃ]๐‘–2,โ€ฆ,๐‘Ž[๐‘ฃ]๐‘–๐‘ ๎€ธ๐‘‡,๐‘–=0,1,โ€ฆ,๐‘ .(3.21)

Definition 3.3. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be ASI-stable if the matrix (๐ผ๐‘ โˆ’โˆ‘๐‘€๐œˆ=1๐ด[๐œˆ]๐œ‰) is regular for๐œ‰โˆˆ๐ถโˆ’, and (๐ผ๐‘ โˆ’โˆ‘๐‘€๐œˆ=1๐ด[๐œˆ]๐œ‰)โˆ’1 is uniformly bounded for ๐œ‰โˆˆ๐ถโˆ’.

Definition 3.4. The ARKLM (2.2), (2.3), (2.5), and (2.6) is said to be DAS-stable if the matrix(๐ผ๐‘ โˆ’โˆ‘๐‘€๐œˆ=1๐ด[๐œˆ]๐œ‰) is regular for ๐œ‰โˆˆ๐ถโˆ’, and โˆ‘๐‘š๐œˆ=1๐ด[๐œˆ]๐‘‡๐‘–๐œ‰(๐ผ๐‘ โˆ’โˆ‘๐‘€๐œˆ=1๐ด[๐œˆ]๐œ‰)โˆ’1(๐‘–=0,1,โ€ฆ,s) is uniformly bounded for ๐œ‰โˆˆ๐ถโˆ’.

Lemma 3.5. Suppose the ARKLM (2.2), (2.3), (2.5), and (2.6) is DA- DAS- and ASI-stable, then there exist positive constants โ„Ž0,๐›พ1,๐›พ2,๐›พ3, which depend only on the method and the parameter ๐›ผ๐‘ฃ,๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,๎‚๐‘Ÿ๐‘ฃ such that โ€–โ€–โ€–โ€–๎‚๐ผ๐‘ โˆ’๐‘€๎“๐œˆ=1๎‚๐ด[๐œˆ]๐œ‰โ€–โ€–โ€–โ€–โ‰ค๐›พ1,โ€–โ€–โ€–โ€–๐ผ๐‘+๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐‘‡๐‘–๐œ‰๎ƒฉ๎‚๐ผ๐‘ โˆ’๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐œ‰๎ƒชโˆ’1โ€–โ€–โ€–โ€–ฬƒ๐‘’โ‰ค1+๐›พ2โ€–โ€–โ€–โ€–โ„Ž,๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐‘‡๐‘–๐œ‰๎ƒฉ๎‚๐ผ๐‘ โˆ’๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐œ‰๎ƒชโˆ’1๐‘ฃโ€–โ€–โ€–โ€–โ‰ค๐›พ3โ€–๐‘ฃโ€–,๐‘ฃโˆˆ๐ถ๐‘๐‘†,๎€ทโ„Žโˆˆ0,โ„Ž0๎€ป,๐‘–=0,1,2โ€ฆ,๐‘ .(3.22)

Proof. This Lemma can be proved in the similar way as that of in [29, Lemmas 3.5โ€“3.7].

Theorem 3.6. Suppose the ARKLM (2.2), (2.3), (2.5), and (2.6) is DA- DAS- and ASI-stable, then there exist positive constants โ„Ž0,๐›พ3,๐›พ4,๐›พ5, which depend only on the method and the parameters๐›ผ๐‘ฃ,๐›ฝ๐‘ฃ,๐œŽ๐‘ฃ,๐‘Ÿ๐‘ฃ,ฬƒ๐‘Ÿ๐‘ฃ, such that for โ„Žโˆˆ(๐‘œ,โ„Ž0], โ€–โ€–๐œ€๐‘–(๐‘›)โ€–โ€–โ‰คโŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๎€ท1+โ„Ž๐›พ4๎€ธ๎‚ปโ€–โ€–๐œ€max0(๐‘›+1)โ€–โ€–,max(๐‘–,๐‘๐‘ฃ)โˆˆ๐ธโ€–โ€–๐œ€(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘–โ€–โ€–,max(๐‘–,๐‘ž)โˆˆ๐ธ๐‘žโ€–โ€–๐œ€๐‘–(๐‘›โˆ’๐‘ž)โ€–โ€–๎‚ผ+โ„Ž๐›พ5max1โ‰ค๐‘–โ‰ค๐‘ โ€–โ€–๐œŒ๐‘–(๐‘›โˆ’1)โ€–โ€–+โ€–โ€–๎‚๐‘„๐‘›โˆ’1โ€–โ€–+๐›พ3โ€–โ€–ฬƒ๐›พ๐‘›โˆ’1โ€–โ€–๎€ท,๐‘–=0,1+โ„Ž๐›พ4๎€ธ๎‚ปโ€–โ€–๐œ€max0(๐‘›+1)โ€–โ€–,max(๐‘–,๐‘๐‘ฃ)โˆˆ๐ธโ€–โ€–๐œ€(๐‘›โˆ’๐‘š๐‘ฃ+๐‘๐‘ฃ)๐‘–โ€–โ€–,max(๐‘–,๐‘ž)โˆˆ๐ธ๐‘žโ€–โ€–๐œ€๐‘–(๐‘›โˆ’๐‘ž)โ€–โ€–๎‚ผ+โ„Ž๐›พ5max1โ‰ค๐‘–โ‰ค๐‘ โ€–โ€–๐œŒ๐‘–(๐‘›)โ€–โ€–+โ€–โ€–๎‚๐‘„๐‘›โ€–โ€–+๐›พ3โ€–โ€–ฬƒ๐›พ๐‘›โ€–โ€–,๐‘–=1,2,โ€ฆ,๐‘ ,(3.23) where ๐œ€0(๐‘›)=ฬ†๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›,๐œ€๐‘–(๐‘›)=ฬ†๐‘ฆ๐‘–(๐‘›)โˆ’๐‘ฆ๐‘–(๐‘›),๐ธ={(๐‘–,๐‘๐‘ฃ)โˆฃ1โ‰ค๐‘–โ‰ค๐‘ ,โˆ’๐‘‘โ‰ค๐‘๐‘ฃโ‰ค๐›พ}, ๐ธ๐‘ž={(๐‘–,๐‘ž)โˆฃ1โ‰ค๐‘–โ‰ค๐‘ ,1โ‰ค๐‘žโ‰ค๐‘š},๎‚๐‘„๐‘›=๐‘„๐‘›+โ„Ž2โˆ‘๐‘š๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›), ฬƒ๐‘Ÿ๐‘›=๐‘Ÿ๐‘›+โ„Ž2โˆ‘๐‘š๐‘ฃ=1๎‚๐ด[๐‘ฃ]๐ป[๐‘ฃ](๐‘›)๐‘…[๐‘ฃ](๐‘›).

Proof. Using (3.19) and Lemma 3.5, for โ„Žโˆˆ(0,โ„Ž0], we obtain that ๐œ€0(๐‘›+1)โ‰ค๎€ท1+๐›พ2โ„Ž๎€ธโ€–โ€–๐œ€0(๐‘›)โ€–โ€–+๐›พ3โ€–โ€–ฬƒ๐‘Ÿ๐‘›โ€–โ€–+โ€–โ€–๎‚๐‘„๐‘›โ€–โ€–+โ„Ž๐›พ3โ€–โ€–โ€–โ€–๐‘š๎“๐‘ฃ=1๎‚๐ด[๐‘ฃ]๎ƒฏ๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ+โ„Ž๐ป๐‘š[๐‘ฃ](๐‘›)๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ(๐‘›โˆ’๐‘ž)๎ƒฐโ€–โ€–โ€–โ€–โ€–โ€–โ€–โ€–๎€ธ๎€ป+โ„Ž๐‘š๎“๐‘ฃ=1ฬƒ๐‘[๐‘ฃ]๐‘‡๎ƒฏ๐‘“[๐‘ฃ]๎‚€๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬ†ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎€ท๐‘‡(๐‘›),ฬ†๐‘ฆ(๐‘›),ฬƒ๐‘ฆ[๐‘ฃ](๐‘›),๐‘ค[๐‘ฃ](๐‘›)๎€ธ+โ„Ž๐ป[๐‘ฃ](๐‘›)โ‹…๐‘š๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž๎€บ๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ(๐‘›โˆ’๐‘ž)๎€ธโˆ’๐‘”[๐‘ฃ]๎€ท๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ(๐‘›โˆ’๐‘ž)๎ƒฐโ€–โ€–โ€–โ€–โ‰ค๎€ท๎€ธ๎€ป(3.24)1+๐›พ2โ„Ž๎€ธโ€–โ€–๐œ€0(๐‘›)โ€–โ€–+๐›พ3โ€–โ€–ฬƒ๐‘Ÿ๐‘›โ€–โ€–+โ€–โ€–๎‚๐‘„๐‘›โ€–โ€–(3.25a)+โ„Ž๐›พ3๐‘š๎“๐‘ฃ=1๎ƒฏ๐‘ ๎“๐‘–=1โ€–โ€–โ€–โ€–๐‘ ๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๎‚ƒ๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚๎‚„+โ„Ž๐ป๐‘—๐‘š[๐‘ฃ](๐‘›)๐‘‰๎“๐‘ž=1๐‘‘๐‘ž๎‚ƒ๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–โ€–โ€–๎‚๎‚„2โŽซโŽชโŽฌโŽชโŽญ1/2(3.25b)+โ„Ž๐‘š๎“๐‘ฃ=1โ€–โ€–โ€–โ€–๐‘ ๎“๐‘—=1๐‘๐‘—[๐‘ฃ]๎ƒฏ๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚+โ„Ž๐ป๐‘—๐‘š[๐‘ฃ](๐‘›)๐‘‰๎“๐‘ž=1๐‘‘๐‘ž๎‚ƒ๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎ƒฐโ€–โ€–โ€–โ€–.๎‚๎‚„(3.25c)For โ„Ž๐›พ3๐‘š๎“๐‘ฃ=1๎ƒฏ๐‘ ๎“๐‘–=1โ€–โ€–โ€–โ€–๐‘ ๎“๐‘—=1๐‘Ž[๐‘ฃ]๐‘–๐‘—๎‚ƒ๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚๎‚„+โ„Ž๐ป๐‘—๐‘š[๐‘ฃ](๐‘›)๐‘‰๎“๐‘ž=1๐‘‘๐‘ž๎‚ƒ๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–โ€–โ€–๎‚๎‚„2โŽซโŽชโŽฌโŽชโŽญ1/2=(3.25b).(3.26) Then (3.25b)โ‰คโ„Ž๐›พ3๐‘š๎“๐‘ฃ=1๎ƒฏ๐‘ ๎“๐‘–=12๐‘ ๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||2๎ƒฏโ€–โ€–๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โˆ’๐‘“[๐‘ฃ]๎‚€๐‘ก๐‘—(๐‘›),๐‘ฆ๐‘—(๐‘›),ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›),๐‘ค๐‘—[๐‘ฃ](๐‘›)๎‚โ€–โ€–2+โ€–โ€–โ€–โ€–โ„Ž๐ป๐‘—๐‘š[๐‘ฃ](๐‘›)๐‘‰๎“๐‘ž=1๐‘‘๐‘ž๎‚ƒ๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)โ€–โ€–โ€–โ€–๎‚๎‚„2๎ƒฐ๎ƒฐ1/2โ‰คโ„Ž๐›พ3๐‘š๎“๐‘ฃ=1๎ƒฏ๐‘ ๎“๐‘–=1๎ƒฏ2๐‘ ๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||2๐‘Ÿ2๐‘ฃโ€–โ€–ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โˆ’ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โ€–โ€–2+2๐‘ ๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||2โ„Ž2๐ป๐‘—2[๐‘ฃ](๐‘›)2๐‘š๐‘ฃ๎“๐‘ž=1๐‘‘๐‘žโ‹…โ€–โ€–๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โ€–โ€–2๎ƒฐ๎ƒฐ1/2โ‰คโ„Ž๐›พ3๐‘š๎“๐‘ฃ=1๎„ถ๎„ต๎„ตโŽท๐‘ ๎“๐‘–=12๐‘ ๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||2๐‘Ÿ2๐‘ฃโ€–โ€–ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โˆ’ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โ€–โ€–2+2โ„Ž2๐›พ3๐‘š๎“๐‘ฃ=1๎„ถ๎„ต๎„ตโŽท๐‘ ๎“๐‘ ๐‘–=1๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||2๐ป๐‘—2๐‘š[๐‘ฃ](๐‘›)๐‘ฃ๎“๐‘ž=1๐‘‘2๐‘žโ‹…โ€–โ€–๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,ฬ†๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โˆ’๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž,๐‘ฆ๐‘—(๐‘›โˆ’๐‘ž)๎‚โ€–โ€–2โ‰ค2โ„Ž๐›พ3๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘ ๐‘–=1๎“๐‘—=1||๐‘Ž[๐‘ฃ]๐‘–๐‘—||๐‘Ÿ๐‘ฃโ€–โ€–ฬ†ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โˆ’ฬƒ๐‘ฆ๐‘—[๐‘ฃ](๐‘›)โ€–โ€–+2โ„Ž2๐›พ3๐‘š๎“๐‘ ๐‘ฃ=1๎“๐‘ ๐‘–=1๎“๐‘š๐‘—=1๐‘ฃ๎“๐‘ž=1๐‘‘๐‘ž||๐‘Ž[๐‘ฃ]๐‘–๐‘—||||๐ป๐‘—[๐‘ฃ](๐‘›)||โ‹…โ€–โ€–๐‘”[๐‘ฃ]๎‚€๐‘ก๐‘›,๐‘ก๐‘›โˆ’๐‘ž