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Strong Stability Preserving Second Derivative General Linear Methods

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Abstract

In this paper, we find sufficient strong stability preserving (SSP) conditions for second derivative general linear methods (SGLMs). Then we construct some optimal SSP SGLMs of order p up to eight and stage order \(1\le q\le p\) with two external stages and \(2\le s\le 10\) internal stages, which have larger effective Courant–Friedrichs–Lewy coefficients than the other existing class of the methods such as two derivative Runge–Kutta methods investigated by Christlieb et al., the class of two-step Runge–Kutta methods introduced by Ketcheson et al., the class of multistep multistage methods studied by Constantinescu and Sandu, and general linear methods investigated by Izzo and Jackiewicz. Some numerical experiments for scalar and systems of equations are provided which demonstrate that the constructed SSP SGLMs achieve the predicted order of convergence and preserve monotonicity.

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Acknowledgements

The authors would like to thank the anonymous referees for their comments which have markedly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Sahand University of Technology, P.O. Box 51335-1996, Tabriz, Iran

    Afsaneh Moradi & Javad Farzi

  2. Faculty of Mathematical sciences, University of Tabriz, Tabriz, Iran

    Ali Abdi

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  1. Afsaneh Moradi
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  2. Javad Farzi
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Correspondence to Javad Farzi.

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Appendices

Appendices

Coefficients of Three Stage Third Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=1.401\), \({\mathscr {C}}_\text {eff}=0.467\). The coefficients matrices of the two external stages methods of order \(p=s=3\) and stage order \(q=1\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21}=0.367036327407744,&{} a_{31}=0.305289583777252,&{} a_{32}=0.594276924770852,\\ {\overline{a}}_{21}=0.087361998687504 ,&{} {\overline{a}}_{31}=0.072665036743429,&{} {\overline{a}}_{32}=0,\\ b_{11} = 0.378319518763646,&{} b_{12}=0.539413445719846,&{} b_{13}=0,\\ b_{21} = 0.224801836087141,&{} b_{22}=0.437599416854594,&{} b_{23}=0.526106658638147,\\ {\overline{b}}_{11} = 0.065956620927602,&{} {\overline{b}}_{12}= 0,&{} {\overline{b}}_{13}=0.156810544725088,\\ {\overline{b}}_{21} = 0.120798754439142,&{} {\overline{b}}_{22}= 0,&{} {\overline{b}}_{23}=0,\\ u_{11}= 0.883562994058718,&{} u_{12}= 0.116437005941282,&{} u_{21}= 0.756325839981176,\\ u_{22}= 0.243674160018824,&{} u_{31}= 0.629088685903634,&{} u_{32}= 0.370911314096366,\\ v = 0.303820705714043,&{}&{}\end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0.031528224123813 \\ 0.433017185195598\\ 1 \end{array}\right] ,\quad \alpha _{0}=\left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _{1}=\left[ \begin{array}{c} 0 \\ 0.270774947096390 \end{array}\right] ,\\ \alpha _{2}= & {} \left[ \begin{array}{c} 0 \\ 0.104435936229161 \end{array}\right] ,\quad \alpha _3=\left[ \begin{array}{c} 0 \\ -0.141473113475966 \end{array}\right] . \end{aligned}$$

Coefficients of Four Stage Fourth Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=1.644\), \({\mathscr {C}}_\text {eff}=0.411\). The coefficients matrices of the two external stages methods of order \(p=s=4\) and stage order \(q=1\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21} =0.264016422097677,&{} a_{31}=0.208728827122131,&{} a_{32}=0.280067373408790,\\ a_{41} =0.156642291124758,&{} a_{42}=0.210178898836894,&{} a_{43}=0.292626140134642,\\ {\overline{a}}_{21} = 0.069927142101271,&{}{\overline{a}}_{31}= 0.055054745190716,&{} {\overline{a}}_{32}= 0.060229350201948,\\ {\overline{a}}_{41} = 0.041316293215779,&{}{\overline{a}}_{42}= 0.045199618752556,&{} {\overline{a}}_{43}= 0.049611867625336,\\ b_{11}= 0.149050523837038,&{} b_{12} = 0.203464159352573,&{} b_{13}= 0.239948323340150,\\ b_{14}= 0.081851422369239,&{} b_{21} = 0.148523308458399,&{} b_{22} = 0.199285041091082,\\ b_{23}= 0.277458929910524,&{} b_{24} = 0.576062630073757,&{}{\overline{b}}_{11}= 0.050710732728729,\\ {\overline{b}}_{12}= 0.042125352335649,&{}{\overline{b}}_{13}= 0.006683865350595,&{}{\overline{b}}_{14}= 0,\\ {\overline{b}}_{21}= 0.042847486964744,&{}{\overline{b}}_{22}= 0.042856861132357,&{}{\overline{b}}_{23}= 0.047040417154307,\\ {\overline{b}}_{24}= 0,&{}u_{11} = 0.734518464168064,&{} u_{12} = 0.265481535831936,\\ u_{21} = 0.597488310192903,&{} u_{22} = 0.402511689807097,&{} u_{31} = 0.471457186431044,\\ u_{32} = 0.528542813568956,&{} u_{41} = 0.353808982055841,&{} u_{42} = 0.646191017944159,\\ v = 0.617981032945615,&{}&{} \end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0.139912879206122\\ 0.476146313762474\\ 0.767346445460013\\ 1 \end{array}\right] ,\quad \alpha _0 = \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _1=\left[ \begin{array}{c} 0 \\ 0.527015480634761 \end{array}\right] ,\\ \alpha _2= & {} \left[ \begin{array}{c} 0 \\ 0.027140536548443 \end{array}\right] ,\quad \alpha _3=\left[ \begin{array}{c} 0 \\ -0.002762467585087 \end{array}\right] \\ \alpha _4= & {} \left[ \begin{array}{c} 0 \\ -0.001638534898442 \end{array}\right] . \end{aligned}$$

Coefficients of Five Stage Fifth Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=1.185\), \({\mathscr {C}}_\text {eff}=0.237\). The coefficients matrices of the two external stages methods of order \(p=s=5\) and stage order \(q=2\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21}= 0.061516399923018,&{} a_{31}= 0.043600570185111,&{} a_{32}= 0.444527820608227, \\ a_{41}= 0.165875098133232, &{} a_{42}= 0.383629364090464,&{} a_{43}= 0.136281919092130, \\ a_{51}= 0.242661894606699,&{} a_{52}= 0.229098986759035, &{}a_{53}= 0.081385974328652,\\ a_{54}= 0.424475494404930,&{}{\overline{a}}_{21}= 0,&{} {\overline{a}}_{31}= 0.073569636769899,\\ {\overline{a}}_{32}= 0.021058600222758,&{}{\overline{a}}_{41}= 0.041456427578800,&{} {\overline{a}}_{42}= 0.011866503266506,\\ {\overline{a}}_{43}= 0.093912282949104, &{} {\overline{a}}_{51}= 0.075476164472320,&{} {\overline{a}}_{52}= 0.007086537500003, \\ {\overline{a}}_{53}= 0.056083321251691,&{} {\overline{a}}_{54}= 0,&{} b_{11}= 0.187235442217099,\\ b_{12}= 0.322862243971992,&{} b_{13}= 0.248777699245909,&{} b_{14} = 0.220697796096535,\\ b_{15}= 0.002570885968929,&{} b_{21}= 0.188276056945535,&{} b_{22}= 0.322742421729971, \\ b_{23}= 0.252236989947234, &{} b_{24}= 0.239243854132961,&{} b_{25}= 0.012555275639019, \\ {\overline{b}}_{11}= 0.060195951745607,&{} {\overline{b}}_{12}= 0.011759266769746, &{} {\overline{b}}_{13}= 0.052645476899545,\\ {\overline{b}}_{14}= 0.063174241824732,&{} {\overline{b}}_{15}= 0.003372973897714,&{} {\overline{b}}_{21}= 0.059952119791420,\\ {\overline{b}}_{22}= 0.011801850994192,&{} {\overline{b}}_{23}= 0.051926886958876,&{} {\overline{b}}_{24}= 0.054650157061697, \\ {\overline{b}}_{25}= 0.021527446840733, &{} u_{11}=0,&{} u_{12}=1, \\ u_{21}= 0.913453478890676, &{} u_{22}= 0.086546521109324, &{} u_{31}= 0.647422355128817, \\ u_{32}= 0.352577644871183, &{} u_{41}= 0.535921231366958,&{} u_{42}= 0.464078768633042, \\ u_{51}= 0.320045915619410, &{} u_{52}= 0.679954084380590,&{} v=0.542559843744499,\end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0.032910530894256\\ 0.064364691879777\\ 0.499731908267495\\ 0.701059459968292\\ 1 \end{array}\right] ,\quad \alpha _0 = \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _1=\left[ \begin{array}{c} 0 \\ 0.032910530894256 \end{array}\right] ,\\ \alpha _2= & {} \left[ \begin{array}{c} 0 \\ 0.000541551521871 \end{array}\right] , \alpha _3=\left[ \begin{array}{c} 0 \\ 0.004799386403454 \end{array}\right] \\ \alpha _4= & {} \left[ \begin{array}{c} 0 \\ -0.000861456597941 \end{array}\right] ,\quad \alpha _5 = \left[ \begin{array}{c} 0 \\ 0.000133191758196 \end{array}\right] . \end{aligned}$$

Coefficients of Six Stage Sixth Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=2.556\), \({\mathscr {C}}_\text {eff}=0.426\). The coefficients matrices of the two external stages methods of order \(p=s=6\) and stage order \(q=3\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21}= 0.195519007044922,&{} a_{31}= 0.182698834641781,&{} a_{32}= 0.224355689307196,\\ a_{41}= 0.200798240963995,&{} a_{42}= 0.202087232337342,&{} a_{43}= 0.214237056626004,\\ a_{51}= 0.180742417657866,&{} a_{52}= 0.206337715995125, &{} a_{53}= 0.311032694918027,\\ a_{54}=0,&{} a_{61}= 0.179400758954363,&{} a_{62}= 0.231484391689903,\\ a_{63}= 0.258646300437170,&{} a_{64}=0,&{} a_{65}= 0.209570365940087,\\ {\overline{a}}_{21}= 0.028978386926393, &{} {\overline{a}}_{31}= 0.027078275413061,&{} {\overline{a}}_{32}= 0.027503486292418,\\ {\overline{a}}_{41}= 0.024390617200713,&{} {\overline{a}}_{42}= 0.024773623711643, &{} {\overline{a}}_{43}= 0.026908793160004,\\ {\overline{a}}_{51}= 0.028375394881790,&{} {\overline{a}}_{52}= 0.025294685242906,&{} {\overline{a}}_{53}= 0.009443683013244,\\ {\overline{a}}_{54}= 0.026792613259658,&{} {\overline{a}}_{61}= 0.027844686173205,&{} {\overline{a}}_{62}= 0.020895996142354,\\ {\overline{a}}_{63}= 0.007469018252919, &{} {\overline{a}}_{64}= 0.021190304375858,&{} {\overline{a}}_{65}= 0.019434763451396,\\ b_{11}= 0.162799979155626,&{} b_{12}= 0.210064073093223, &{} b_{13}= 0.234712565126676,\\ b_{14}= 0,&{} b_{15}= 0.190177853235071,&{} b_{16}= 0.043897073679799,\\ b_{21}= 0.186288536307596,&{}b_{22}= 0.201414279006715,&{} b_{23}= 0.299976348757000,\\ b_{24}= 0, &{} b_{25}= 0.372215734846917,&{} b_{26}= 0.015729879093722,\\ {\overline{b}}_{11}= 0.028681836912939,&{} {\overline{b}}_{12}= 0.018962393226422, &{} {\overline{b}}_{13}= 0.006777875539520,\\ {\overline{b}}_{14}= 0.019229467761441,&{} {\overline{b}}_{15}= 0.017636375137000,&{} {\overline{b}}_{16}= 0.060702028317952,\\ {\overline{b}}_{21}= 0.027537753865251,&{} {\overline{b}}_{22}= 0.024389959300741,&{} {\overline{b}}_{23}= 0.009092524569629,\\ {\overline{b}}_{24}= 0.025796343863547, &{} {\overline{b}}_{25}= 0.000782358307294,&{} {\overline{b}}_{26}= 0,\\ u_{11}= 0.382332042901846,&{} u_{12}= 0.617667957098154, &{} u_{21}= 0.336433540760703, \\ u_{22}= 0.663566459239297, &{} u_{31}= 0.314373608788154,&{} u_{32}= 0.685626391211846, \\ u_{41}= 0.325150848279669, &{} u_{42}= 0.674849151720331,&{}u_{51}= 0.321246382913682,\\ u_{52}= 0.678753617086318, &{} u_{61}= 0.337338926920657,&{} u_{62}= 0.662661073079343,\\ v=0.676780216227852,\end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0.110370998137150\\ 0.316629026151011\\ 0.533325976674361\\ 0.740872397078304\\ 0.822776236109616\\ 1 \end{array}\right] , \quad \alpha _0 = \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _1=\left[ \begin{array}{c} 0 \\ 0.093858080521929 \end{array}\right] ,\\ \alpha _2= & {} \left[ \begin{array}{c} 0 \\ -0.008889300639912 \end{array}\right] ,\\ \\ \alpha _3= & {} \left[ \begin{array}{c} 0 \\ 0.001888057837179 \end{array}\right] \quad \alpha _4 = \left[ \begin{array}{c} 0 \\ -0.000344064832613 \end{array}\right] ,\\ \\ \alpha _5= & {} \left[ \begin{array}{c} 0 \\ -0.369325560056622 \end{array}\right] ,\quad \alpha _6 = \left[ \begin{array}{c} 0 \\ 0.199826462570264 \end{array}\right] . \end{aligned}$$

Coefficients of Seven Stage Seventh Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=2.134\), \({\mathscr {C}}_\text {eff}=0.305\). The coefficients matrices of the two external stages methods of order \(p=s=7\) and stage order \(q=4\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21}= 0.121020696167578,&{} a_{31}= 0.281444806288868,&{} a_{32}= 0.037608568428153, \\ a_{41}= 0.200813428997174, &{} a_{42}= 0.232400061029153,&{} a_{43}= 0.062293761908094,\\ a_{51}= 0.216857168039921,&{} a_{52}= 0.062163125667391, &{} a_{53}= 0.287091624201198, \\ a_{54}= 0 ,&{} a_{61}= 0.239675395084010,&{} a_{62}= 0.046284663476529,\\ a_{63}= 0.280968628857240,&{} a_{64}= 0 ,&{} a_{65}= 0.181270548883475,\\ a_{71}= 0.235733878768210, &{} a_{72}= 0.048169668125128,&{} a_{73}= 0.293087501905743,\\ a_{74}= 0 ,&{} a_{75}= 0.220909177318860, &{} a_{76}= 0.167399084919805,\\ {\overline{a}}_{21}= 0.033622871277955,&{} {\overline{a}}_{31}= 0.026467610174312,&{} {\overline{a}}_{32}= 0.031079337949210,\\ {\overline{a}}_{41}= 0.032008230975419,&{} {\overline{a}}_{42}= 0.016102719248223, &{}{\overline{a}}_{43}= 0.042281281346407,\\ {\overline{a}}_{51}= 0.025647568937754,&{} {\overline{a}}_{52}= 0.035890029157894,&{} {\overline{a}}_{53}= 0.007273692588794,\\ {\overline{a}}_{54}= 0.018884429077112,&{} {\overline{a}}_{61}= 0.025961504128808,&{} {\overline{a}}_{62}= 0.036146387300470,\\ {\overline{a}}_{63}= 0.004412734061480, &{} {\overline{a}}_{64}= 0.011456624321538,&{} {\overline{a}}_{65}= 0.024128365116758,\\ {\overline{a}}_{71}= 0.026879006637649,&{} {\overline{a}}_{72}= 0.035311062754025, &{} {\overline{a}}_{73}= 0.004582359260718,\\ {\overline{a}}_{74}= 0.011897016186551, &{} {\overline{a}}_{75}= 0.017401819177701, &{} {\overline{a}}_{76}= 0.039952472018132,\\ b_{11}= 0.233258330804281,&{} b_{12}= 0.050729114746177,&{} b_{13}= 0.299319442929729,\\ b_{14}= 0, &{} b_{15}= 0.199509841349740,&{} b_{16}= 0.150164018941345,\\ b_{17}= 0.030540153122196,&{} b_{21}= 0.230532619342581, &{} b_{22}= 0.065827883118964,\\ b_{23}= 0.268468793805008,&{} b_{24}= 0,&{} b_{25}= 0.149621931436028,\\ b_{26}= 0.386751078492987,&{} b_{27}= 0,&{} {\overline{b}}_{11}= 0.026041050133540,\\ {\overline{b}}_{12}= 0.036370124880282, &{} {\overline{b}}_{13}= 0.004965241452746, &{} {\overline{b}}_{14}= 0.012891079588596,\\ {\overline{b}}_{15}= 0.028193631918574,&{} {\overline{b}}_{16}= 0.018659341701326, &{} {\overline{b}}_{17}= 0.002789442108399,\\ {\overline{b}}_{21}= 0.026679627338015,&{} {\overline{b}}_{22}= 0.034405351273111,&{} {\overline{b}}_{23}= 0.004568460172558,\\ {\overline{b}}_{24}= 0.011860930478858,&{} {\overline{b}}_{25}= 0.033893167589117,&{} {\overline{b}}_{26}= 0,\\ {\overline{b}}_{27}= 0, &{} u_{11}= 1,&{} u_{12}= 0, \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{lll} u_{21}= 0.564576649034832, &{} u_{22}= 0.435423350965168, &{} u_{31}= 0.841772743229203,\\ u_{32}= 0.158227256770797, &{} u_{41}= 0.720161981469503,&{} u_{42}= 0.279838018530497, \\ u_{51}= 0.696459064529954, &{} u_{52}= 0.303540935470046,&{} u_{61}= 0.748017724899665,\\ u_{62}= 0.251982275100335, &{} u_{71}= 0.747963865286320,&{} u_{72}= 0.252036134713680,\\ v= 0.264952978155935,\end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0\\ 0.180970394594390\\ 0.340838325628094\\ 0.534035743302817\\ 0.607903860167200\\ 0.782892509796306\\ 1.0 \end{array}\right] , \quad \alpha _0 = \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _1=\left[ \begin{array}{c} 0 \\ 0.137681404302102 \end{array}\right] ,\\ \alpha _2= & {} \left[ \begin{array}{c} 0 \\ -0.039611402052506 \end{array}\right] , \alpha _3=\left[ \begin{array}{c} 0 \\ 0.002268608912332 \end{array}\right] \quad \\ \alpha _4= & {} \left[ \begin{array}{c} 0 \\ 0.000102637762511 \end{array}\right] , \quad \alpha _5 = \left[ \begin{array}{c} 0 \\ -0.000016024680216 \end{array}\right] ,\\ \\ \alpha _6= & {} \left[ \begin{array}{c} 0 \\ -0.000000253926561 \end{array}\right] ,\quad \alpha _7 = \left[ \begin{array}{c} 0 \\ -0.000000239969313 \end{array}\right] . \end{aligned}$$

Coefficients of Six Stage Eighth Order Methods

For \(K=\frac{1}{\sqrt{2}}\) we obtain an SSP SGLM with SSP coefficient \({\mathscr {C}}=0.378\), \({\mathscr {C}}_\text {eff}=0.063\). The coefficients matrices of the two external stages methods of order \(p=8\) and stage order \(q=5\) with \(s=6\) take the form

with

$$\begin{aligned} \begin{array}{lll} a_{21}= 0.194941336033860,&{} a_{31}= 0.258869155652665,&{} a_{32}= 0,\\ a_{41}= 0.375879165544731,&{}a_{42}= 0,&{} a_{43}= 0,\\ a_{51}= 0.176133472453509,&{} a_{52}= 0.398391061034415,&{}a_{53}= 0,\\ a_{54}=0,&{} a_{61}= 0.070686734394561,&{} a_{62}= 0.815297398839893,\\ a_{63}= 0,&{} a_{64}=0,&{} a_{65}= 0,\\ {\overline{a}}_{21}= 0.033799601974334, &{} {\overline{a}}_{31}= 0.052105299549334,&{} {\overline{a}}_{32}= 0.005225668603390,\\ {\overline{a}}_{41}= 0.099091721590713,&{} {\overline{a}}_{42}= 0.000031326729575,&{} {\overline{a}}_{43}= 0.021231212047832,\\ {\overline{a}}_{51}= 0.080948456912720,&{} {\overline{a}}_{52}= 0.000000493810882,&{} {\overline{a}}_{53}= 0.000334672776228,\\ {\overline{a}}_{54}= 0.055827373072186,&{} {\overline{a}}_{61}= 0.045434215470162,&{} {\overline{a}}_{62}= 0.095476339616561, \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{lll} {\overline{a}}_{63}= 0.000013012202321, &{}{\overline{a}}_{64}= 0.002170589080106,&{} {\overline{a}}_{65}= 0.137699389678188,\\ b_{11}= 0.963639559043213,&{} b_{12}= 0.048191855456677,&{} b_{13}= 0,\\ b_{14}= 0,&{} b_{15}= 0.004365945043872,&{} b_{16}= 0,\\ b_{21}= 0.927829178576764,&{} b_{22}= 0.069185537428087,&{} b_{23}= 0,\\ b_{24}= 0, &{} b_{25}= 0,&{} b_{26}= 0.000727414372750,\\ {\overline{b}}_{11}= 0.090918490440034,&{} {\overline{b}}_{12}= 0.082575044411426, &{} {\overline{b}}_{13}= 0.001654982790583,\\ {\overline{b}}_{14}= 0.276070682292649,&{} {\overline{b}}_{15}= 0,&{} {\overline{b}}_{16}= 0,\\ {\overline{b}}_{21}= 0.051336508261314,&{} {\overline{b}}_{22}= 0.246328650630155,&{} {\overline{b}}_{23}= 0.000015640267445,\\ {\overline{b}}_{24}= 0.002608981392075, &{}{\overline{b}}_{25}= 0.165510436159075,&{} {\overline{b}}_{26}= 0.025121245889126,\\ u_{11}= 0.110314127130126,&{} u_{12}= 0.889685872869874, &{} u_{21}= 0.341312840349621,\\ u_{22}= 0.658687159650379,&{}u_{31}= 0.475349130736242,&{} u_{32}= 0.524650869263758,\\ u_{41}= 0.846907416948989,&{} u_{42}= 0.153092583051011,&{} u_{51}= 0.833172220938727,\\ u_{52}= 0.166827779061273, &{} u_{61}= 0.769967140635720,&{} u_{62}= 0.230032859364280,\\ v=0.877656917610160,\end{array} \end{aligned}$$

and

$$\begin{aligned} c= & {} \left[ \begin{array}{c} 0.101841819231152\\ 0.301046289454566\\ 0.367447779529038\\ 0.491314982741747\\ 0.689706864494928\\ 1 \end{array}\right] , \quad \alpha _0 = \left[ \begin{array}{c} 1 \\ 1 \end{array}\right] ,\quad \alpha _1=\left[ \begin{array}{c} 0.118261175900861 \\ 0.099805946734701 \end{array}\right] ,\\ \alpha _2= & {} \left[ \begin{array}{c} -0.046902346937077\\ 0.011644412764986 \end{array}\right] , \alpha _3=\left[ \begin{array}{c} -0.000139597202405 \\ 0.000215183772645 \end{array}\right] \\ \alpha _4= & {} \left[ \begin{array}{c} 0.000498054309704 \\ -0.000056716877362 \end{array}\right] , \alpha _5 = \left[ \begin{array}{c} 0.000052818879022\\ -0.000006446514645 \end{array}\right] ,\\ \alpha _6= & {} \left[ \begin{array}{c} 0.000008284070774 \\ -0.000001103052897 \end{array}\right] , \alpha _7 = \left[ \begin{array}{c} -0.000015242323561 \\ 0.000002223023071 \end{array}\right] ,\\ \alpha _8= & {} \left[ \begin{array}{c} -0.330455420500514 \\ -0.330461271414987 \end{array}\right] . \end{aligned}$$

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Moradi, A., Farzi, J. & Abdi, A. Strong Stability Preserving Second Derivative General Linear Methods. J Sci Comput 81, 392–435 (2019). https://doi.org/10.1007/s10915-019-01021-1

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