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Jacobian-free explicit multiderivative general linear methods for hyperbolic conservation laws

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Abstract

We study explicit strong stability preserving (SSP) multiderivative general linear methods (MDGLMs) for the numerical solution of hyperbolic conservation laws. Sufficient conditions for MDGLMs up to four derivatives to be SSP are determined. In this work, we describe the construction of two external stage explicit SSP MDGLMs based on Taylor series conditions, and present examples of constructed methods up to order nine and three internal stages along with their SSP coefficients. It is difficult to apply these methods directly to the discretization of partial differential equations, as higher-order flux derivatives must be calculated analytically. We hence use a Jacobian-free approach based on the recent development of explicit Jacobian-free multistage multiderivative solvers (Chouchoulis et al. J. Sci. Comput. 90, 96, 2022) that provides a practical application of MDGLMs. To show the capability of our novel methods in achieving the predicted order of convergence and preserving required stability properties, several numerical test cases for scalar and systems of equations are provided.

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Data Availability

No datasets were generated or analyzed during the current study.

Notes

  1. The dot \((\cdot )\) stands for the time derivative d/dt, whereas the prime \((')\) stands for the Jacobian of the vector-valued \(\varPhi \) w.r.t. y.

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Funding

R. D’Ambrosio is supported by GNCS-INDAM project and PRIN2017-MIUR project 2017JYCLSF “Structure preserving approximation of evolutionary problems.”

Author information

Authors and Affiliations

  1. Department of Information Engineering and Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    Afsaneh Moradi & Raffaele D’Ambrosio

  2. Faculty of Sciences & Data Science Institute, Hasselt University, Agoralaan Gebouw D, 3590 Diepenbeek, Hasselt, Belgium

    Jeremy Chouchoulis & Jochen Schütz

Authors
  1. Afsaneh Moradi
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  2. Jeremy Chouchoulis
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  3. Raffaele D’Ambrosio
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  4. Jochen Schütz
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Contributions

Afsaneh Moradi: conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—orginal draft, writing—review and editing. Jeremy Choichoulis: conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—review and editing. Jochen Schutz: conceptualization, formal analysis, investigation, methodology, software, supervision, writing—review and editing. Rafeale D’Ambrosio: conceptualization, formal analysis, investigation, methodology, software, supervision, writing—review and editing.

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Correspondence to Afsaneh Moradi.

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Appendices

Appendices

A   Coefficients of third-derivative GLMs

Here, we give the coefficient matrices of the constructed third-derivative GLMs for \(s=2\) and \(s=3\). The coefficient matrices of two stage methods take the form

and three stage methods take the following form

1. SSP 3DGLM4-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.481524563110426, &{} a^{\{2\}}_{21} = 0.107666492481270, &{} a^{\{3\}}_{21} = 0.021992397120078,&{} b^{\{1\}}_{11} = 0.498435904450369, \\ b^{\{1\}}_{12} = 0.522530753430964, &{} b^{\{1\}}_{21} = 0.410124141823035, &{} b^{\{1\}}_{22} = 0.463432449474462, &{} b^{\{2\}}_{11} = 0.095786801825143, \\ b^{\{2\}}_{12} = 0.152075022730067, &{} b^{\{2\}}_{21} = 0.137529315089621, &{} b^{\{2\}}_{22} = 0.088062569968302, &{} b^{\{3\}}_{11} = 0.018694337702556,\\ b^{\{3\}}_{12} = 0.027732589352203, &{} b^{\{3\}}_{21} = 0.015749712492286, &{} b^{\{3\}}_{22} = 0.005659319478742, &{} u_{11} = 0.935650184213693, \\ u_{12} = 0.064349815786307, &{} u_{21} = 0.886315321544333,&{} u_{22} = 0.113684678455667, &{} v = 0.142233555463516, \\ c_1 = 0.525747892280569, &{} w_{10} = 1.0, &{} w_{11} = 0.535233702910286,&{} w_{12} = 0.136943388092356, \\ w_{13} = 0.027743278545692, &{} w_{14} = 0.002690489400088, &{}w_{20} = 1.0, &{} w_{21} = 0.387823636326451, \\ w_{22} = 0.156555488157499, &{} w_{23} = -0.027002414204259, &{} w_{24} = 0.010351227670424. &{} \end{array}$$

2. SSP 3DGLM5-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.444563089345968, &{} a^{\{2\}}_{21} = 0.220433797971554, &{} a^{\{3\}}_{21} = 0.047199086888770, &{}b^{\{1\}}_{11} = 0.677030364676214, \\ b^{\{1\}}_{12} = 0.109344513124094, &{} b^{\{1\}}_{21} = 0.600001026747633, &{} b^{\{1\}}_{22} = 0.643878289010242, &{} b^{\{2\}}_{11} = 0.187812461430404, \\ b^{\{2\}}_{12} = 0.167420842533560, &{} b^{\{2\}}_{21} = 0.335797629018218, &{} b^{\{2\}}_{22} = 0.027367894432842, &{} b^{\{3\}}_{11} = 0.019973582828875,\\ b^{\{3\}}_{12} = 0.014734064127359, &{} b^{\{3\}}_{21} = 0.076346090518484, &{} b^{\{3\}}_{22} = 0.007807990410220, &{} u_{11} = 0.691735860871995, \\ u_{12} = 0.308264139128005, &{} u_{21} = 0.362379132344019,&{}u_{22} = 0.637620867655981, &{} v = 0.466935628326092,\\ c_1 = 0.404754745681296, &{} w_{10} = 1.0, &{} w_{11} = 0.263722533967065,&{} w_{12} = 0.065333656992811, \\ w_{13} = 0.027169926386350, &{} w_{14} = 0.002545864356240, &{}w_{15} = -0.001260324593837, &{} w_{20} = 1.0, \\ w_{21} = 0.721226971924633, &{}w_{22} = 0.119117224325207, &{} w_{23} = -0.025117506961899, &{}w_{24} = -0.002085128124951, \\ w_{25} = 0.003121799371033.&{}&{}&{} \end{array}$$

3. SSP 3DGLM6-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.315517321907129, &{} a^{\{1\}}_{31} = 0.758864601103138, &{} a^{\{1\}}_{32} = 0.067336415726727,&{} a^{\{2\}}_{21} = 0.076494847917531, \\ a^{\{2\}}_{31} = 0.165996492414755, &{} a^{\{2\}}_{32} = 0.131818841412823, &{} a^{\{3\}}_{21} = 0.026098214000471,&{} a^{\{3\}}_{31} = 0.010407236225120, \\ a^{\{3\}}_{32} = 0.011694043676070, &{} b^{\{1\}}_{11} = 0.466407769409968, &{} b^{\{1\}}_{12} = 0.588029782942878,&{} b^{\{1\}}_{13} = 0.090753993408768,\\ b^{\{1\}}_{21} = 0.714352536776548, &{} b^{\{1\}}_{22} = 0.004057520548612, &{} b^{\{1\}}_{23} = 0.043314290009643,&{} b^{\{2\}}_{11} = 0.101470387228004, \\ b^{\{2\}}_{12} = 0.309600095382871, &{} b^{\{2\}}_{13} = 0.003137222126554, &{} b^{\{2\}}_{21} = 0.208922227866803,&{} b^{\{2\}}_{22} = 0.036586254679526,\\ b^{\{2\}}_{23} = 0.023218417766703, &{} b^{\{3\}}_{11} = 0.016855407713879, &{} b^{\{3\}}_{12} = 0.051517368037844,&{} b^{\{3\}}_{13} = 0.001065531053630,\\ b^{\{3\}}_{21} = 0.035888517238621, &{} b^{\{3\}}_{22} = 0.000525893188072, &{} b^{\{3\}}_{23} = 0.007526287669863,&{} u_{11} = 0.429393552342717, \\ u_{12} = 0.570606447657283, &{} u_{21} = 0.714766096918914, &{} u_{22} = 0.285233903081086, &{} u_{31} = 0.373124486427176, \\ u_{32} = 0.626875513572824, &{} v = 0.378628332116197, &{} c_1 = 0.195376324234861, &{} c_2 = 0.620324656318555,\\ w_{10} = 1.0, &{} w_{11} = 0.414185180122275,&{} w_{12} = 0.089420710803121, &{} w_{13} = -0.015802200386162, \\ w_{14} = -0.001623142712040,&{} w_{15} = 0.000303750873378, &{} w_{16} = 0.218129334015903, &{} w_{20} = 1.0, \\ w_{21} = 0.030717981695463, &{} w_{22} = -0.033842454301531,&{} w_{23} = 0.014069844764195, &{}w_{24} = 0.001327849153771, \\ w_{25} = -0.000224421436006, &{}w_{26} = 0.217923761039387.&{} &{} \end{array}$$

4. SSP 3DGLM7-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.379665884421877, &{} a^{\{1\}}_{31} = 0.515488221507126, &{}a^{\{1\}}_{32} = 0.008239732993902,&{} a^{\{2\}}_{21} = 0.060210073470162, \\ a^{\{2\}}_{31} = 0.114667967793747, &{} a^{\{2\}}_{32} = 0.010050929935034, &{} a^{\{3\}}_{21} = 0.027175182796611,&{} a^{\{3\}}_{31} = 0.093175561916561, \\ a^{\{3\}}_{32} = 0.008173511911974, &{} b^{\{1\}}_{11} = 1.107792953001359, &{} b^{\{1\}}_{12} = 0.090330329859403,&{} b^{\{1\}}_{13} = 0.000000066635348,\\ b^{\{1\}}_{21} = 0.664653168823024, &{} b^{\{1\}}_{22} = 0.309711371128129, &{} b^{\{1\}}_{23} = 0.015546721568938,&{} b^{\{2\}}_{11} = 0.189649927487829, \\ b^{\{2\}}_{12} = 0.600387177342980, &{} b^{\{2\}}_{13} = 0.000000000094460, &{} b^{\{2\}}_{21} = 0.107129004361048,&{} b^{\{2\}}_{22} = 0.214103657229089, \\ b^{\{2\}}_{23} = 0.018964113794848, &{} b^{\{3\}}_{11} = 0.003171262661914, &{} b^{\{3\}}_{12} = 0.067436297519191,&{} b^{\{3\}}_{13} = 0.000000000076816,\\ b^{\{3\}}_{21} = 0.006076703572795, &{} b^{\{3\}}_{22} = 0.000052086287613, &{} b^{\{3\}}_{23} = 0.015421809057899,&{} u_{11} = 0.050799054224709, \\ u_{12} = 0.949200945775291, &{} u_{21} = 0.271811785253415, &{} u_{22} = 0.728188214746585, &{} u_{31} = 0.848358688706512, \\ u_{32} = 0.151641311293488, &{} v = 0.951545856064458, &{} c_1 = 0.310210488718125, &{} c_2 = 0.735893895336771,\\ w_{10} = 1.0, &{} w_{11} = 0.507845599546811,&{} w_{12} = 0.239955454479258, &{} w_{13} = -0.006520488569236, \\ w_{14} = -0.005551486351131, &{} w_{15} = 0.000113698479916, &{} w_{16} = 0.000223765806822, &{} w_{17} = 1.999940921849224, \\ w_{20} = 1.0,&{}w_{21} = 0.299633511570792, &{} w_{22} = 0.037848427850458,&{} w_{23} = 0.005590515047944, \\ w_{24} = 0.000703599013788,&{} w_{25} = 0.000019134996526, &{} w_{26} = -0.000010671520193, &{} w_{27} = 2.0. \end{array}$$

B   Coefficients of Fourth-derivative GLMs

Here, we give the coefficients matrices of the constructed fourth-derivative GLMs for \(s=2\) and \(s=3\). The coefficient matrices of two stage methods take the form

and three stage methods take the following form

1. SSP 4DGLM4-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.492502401255857, &{} a^{\{2\}}_{21} = 0.121549058540707, &{} a^{\{3\}}_{21} = 0.019860332137553, &{} a^{\{4\}}_{21} = 0.002469334621926,\\ b^{\{1\}}_{11} = 0.493035280152582, &{} b^{\{1\}}_{12} = 0.519129890196680, &{} b^{\{1\}}_{21} = 0.492159016664885, &{} b^{\{1\}}_{22} = 0.507134722289626,\\ b^{\{2\}}_{11} = 0.120822362679484, &{} b^{\{2\}}_{12} = 0.118796068553098, &{} b^{\{2\}}_{21} = 0.121514758761671, &{} b^{\{2\}}_{22} = 0.129069521144118,\\ b^{\{3\}}_{11} = 0.019733454185988, &{} b^{\{3\}}_{12} = 0.020130886150193, &{} b^{\{3\}}_{21} = 0.019811609177391, &{} b^{\{3\}}_{22} = 0.021899025203918,\\ b^{\{4\}}_{11} = 0.002452844681870, &{} b^{\{4\}}_{12} = 0.002512408254589, &{} b^{\{4\}}_{21} = 0.002463116135828, &{} b^{\{4\}}_{22} = 0.002754839201374,\\ u_{11} = 0.043647614240132, &{} u_{12} = 0.956352385759868, &{} u_{21} = 0.054763091495361, &{} u_{22} = 0.945236908504639,\\ v = 0.945129564550485, &{} c_1 = 0.507354526641232, &{} w_{10}= 1.0, &{} w_{11} = 0.519664150763746, \\ w_{12} = 0.117804247188642, &{} w_{13} = 0.020428440495935, &{} w_{14} = 0.002458895114308,&{}w_{20} = 1.0, \\ w_{21} = 0.506792719368996, &{} w_{22} = 0.129201783102532, &{} w_{23} = 0.021827294362888, &{} w_{24} = 0.002774578642803. \end{array}$$

2. SSP 4DGLM5-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.500492666826293, &{} a^{\{2\}}_{21} = 0.114756600078513, &{} a^{\{3\}}_{21} = 0.018148178751572, &{} a^{\{4\}}_{21} = 0.002714589878061,\\ b^{\{1\}}_{11} = 0.431186537508394, &{} b^{\{1\}}_{12} = 0.404171392100681, &{} b^{\{1\}}_{21} = 0.483662234321112, &{} b^{\{1\}}_{22} = 0.560179604710392,\\ b^{\{2\}}_{11} = 0.081227126476428, &{} b^{\{2\}}_{12} = 0.047086826677170, &{} b^{\{2\}}_{21} = 0.109480320931676, &{} b^{\{2\}}_{22} = 0.135274055565348,\\ b^{\{3\}}_{11} = 0.013178225742208, &{} b^{\{3\}}_{12} = 0.008882729874764, &{} b^{\{3\}}_{21} = 0.017492912408121, &{} b^{\{3\}}_{22} = 0.025272005673387,\\ b^{\{4\}}_{11} = 0.001589026148517, &{} b^{\{4\}}_{12} = 0.000129548838296, &{} b^{\{4\}}_{21} = 0.002446465182670, &{} b^{\{4\}}_{22} = 0.003740837321860,\\ u_{11} = 0.088185135295424, &{} u_{12} = 0.911814864704576, &{} u_{21} = 0.176145583277641, &{} u_{22} = 0.823854416722359, \\ v = 0.789711162108572, &{} c_1 = 0.517845671243588, &{} w_{10}= 1.0,&{} w_{11} = 0.327746943580494,\\ w_{12} = 0.050980139124519, &{} w_{13} = 0.011125839350588, &{} w_{14} = -0.001282009647015, &{} w_{15} = 0.000603507965945,\\ w_{20} = 1.0,&{} w_{21} = 0.536230853002924, &{} w_{22} = 0.142119178095184, &{} w_{23} = 0.024306987806757, \\ w_{24} = 0.003410108652328, &{} w_{25} = 0.000281973075456. &{}&{} \end{array}$$

3. SSP 4DGLM6-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.517293370447585, &{} a^{\{2\}}_{21} = 0.133925352925481, &{} a^{\{3\}}_{21} = 0.027371115505543, &{} a^{\{4\}}_{21} = 0.004940436426576,\\ b^{\{1\}}_{11} = 0.562577084182751, &{} b^{\{1\}}_{12} = 0.421763934840916, &{} b^{\{1\}}_{21} = 0.572954574408108, &{} b^{\{1\}}_{22} = 0.478728558134980,\\ b^{\{2\}}_{11} = 0.163935914431203, &{} b^{\{2\}}_{12} = 0.100241069539339, &{} b^{\{2\}}_{21} = 0.170412942747712, &{} b^{\{2\}}_{22} = 0.165570428503403,\\ b^{\{3\}}_{11} = 0.035658092350845, &{} b^{\{3\}}_{12} = 0.016327465256022, &{} b^{\{3\}}_{21} = 0.037215832658113, &{} b^{\{3\}}_{22} = 0.035849937970705,\\ b^{\{4\}}_{11} = 0.005287177412273, &{} b^{\{4\}}_{12} = 0.002944889939538, &{} b^{\{4\}}_{21} = 0.006694878863016, &{} b^{\{4\}}_{22} = 0.000000025053928,\\ u_{11} = 0.991894960507822, &{} u_{12} = 0.008105039492178, &{} u_{21} = 0.677452760618707, &{} u_{22} = 0.322547239381293, \\ v = 0.232528801933385, &{} c_1 = 0.461531427232185, &{} w_{10}= 1.0, &{} w_{11} = 0.460985616742624, \\ w_{12} = 0.105968926131271, &{} w_{13} = 0.016230449697124, &{} w_{14} = 0.001904410077557,&{} w_{15} = 0.000179896607685,\\ w_{16} = 0.000011946083550,&{}w_{20} = 1.0, &{} w_{21} = 0.528327730262044, &{} w_{22} = 0.172187361061285, \\ w_{23} = 0.035327454889158, &{} w_{24} = 0.000197418142938, &{} w_{25} = 0.000484463733041, &{} w_{26} = 0.000194265652679. \end{array}$$

4. SSP 4DGLM7-2

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.499507160885063, &{} a^{\{2\}}_{21} = 0.119968933288827, &{} a^{\{3\}}_{21} = 0.017081896566529, &{} a^{\{4\}}_{21} = 0.004031320198658,\\ b^{\{1\}}_{11} = 0.666614451332563, &{} b^{\{1\}}_{12} = 0.303306256194171, &{} b^{\{1\}}_{21} = 0.576409644843040, &{} b^{\{1\}}_{22} = 0.462190555937428,\\ b^{\{2\}}_{11} = 0.183560532895898, &{} b^{\{2\}}_{12} = 0.155207228786631, &{} b^{\{2\}}_{21} = 0.259780581201447 , &{} b^{\{2\}}_{22} = 0.034723177133322,\\ b^{\{3\}}_{11} = 0.031183451078632, &{} b^{\{3\}}_{12} = 0.041000165610189, &{} b^{\{3\}}_{21} = 0.057101861689900 , &{} b^{\{3\}}_{22} = 0.010915027907463, \\ b^{\{4\}}_{11} = 0.001433739889497, &{} b^{\{4\}}_{12} = 0.004322242467946, &{} b^{\{4\}}_{21} = 0.001529153305801, &{} b^{\{4\}}_{22} = 0.002575963845641,\\ u_{11} = 0.618158814106136, &{} u_{12} = 0.381841185893864, &{} u_{21} = 0.688026611571673 , &{} u_{22} = 0.311973388428327, \\ v = 0.437966138773607, &{} c_1 = 0.505291324039624, &{} w_{10}= 1.0, &{} w_{11} = 0.479066664889029,\\ w_{12} = 0.127521779354434, &{} w_{13} = 0.041705623460671, &{} w_{14} = 0.003986082383866,&{}w_{15} = -0.000245284111217, \\ w_{16} = 0.000137223973872 , &{} w_{17} = 0.210228581679632, &{}w_{20} = 1.0,&{} w_{21} = 0.547746158142762, \\ w_{22} = 0.127882876390368 ,&{} w_{23} = -0.011206034972369, &{} w_{24} = 0.000660308249256, &{} w_{25} = 0.001115949397675 \\ w_{26} = -0.000161611421203, &{} w_{27} = 0.210270682891451. &{}&{} \end{array}$$

5. SSP 4DGLM6-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.374778451641205, &{} a^{\{2\}}_{21} = 0.063406760818824, &{} a^{\{3\}}_{21} = 0.008463987299019, &{} a^{\{4\}}_{21} = 0.000850022276759,\\ a^{\{1\}}_{31} = 0.357597546030056, &{} a^{\{2\}}_{31} = 0.062908428776938, &{} a^{\{3\}}_{31} = 0.008164853981020, &{} a^{\{4\}}_{31} = 0.000653850758305,\\ a^{\{1\}}_{32} = 0.325477000217070, &{} a^{\{2\}}_{32} = 0.046441087898065, &{} a^{\{3\}}_{32} = 0.005297221144835, &{} a^{\{4\}}_{32} = 0.000532701706906,\\ b^{\{1\}}_{11} = 0.334959394295523, &{} b^{\{1\}}_{12} = 0.315171444182360, &{} b^{\{1\}}_{13} = 0.410190435750393, &{} b^{\{1\}}_{21} = 0.349958069221630,\\ b^{\{1\}}_{22} = 0.292876373365019, &{} b^{\{1\}}_{23} = 0.258165929344463, &{} b^{\{2\}}_{11} = 0.060633404000675, &{} b^{\{2\}}_{12} = 0.041104145868688,\\ b^{\{2\}}_{13} = 0.041008760892016, &{} b^{\{2\}}_{21} = 0.055477164332920, &{} b^{\{2\}}_{22} = 0.047166235937992, &{} b^{\{2\}}_{23} = 0.051817450556552,\\ b^{\{3\}}_{11} = 0.007643277336581, &{} b^{\{3\}}_{12} = 0.004674306560848, &{} b^{\{3\}}_{13} = 0.005497486923106, &{} b^{\{3\}}_{21} = 0.007256291024332,\\ b^{\{3\}}_{22} = 0.005093765887795, &{} b^{\{3\}}_{23} = 0.006947776401714, &{} b^{\{4\}}_{11} = 0.000621015210638, &{} b^{\{4\}}_{12} = 0.000470045478575,\\ b^{\{4\}}_{13} = 0.000552841163009, &{} b^{\{4\}}_{21} = 0.000622129834196, &{} b^{\{4\}}_{22} = 0.000341706070841, &{} b^{\{4\}}_{23} = 0.000698685934132,\\ u_{11} = 0.630474771711679, &{} u_{12} = 0.369525228288321, &{} u_{21} = 0.573012085130541, &{} u_{22} = 0.426987914869459, \\ u_{31} = 0.601083078559332, &{} u_{32} = 0.398916921440668,&{} v = 0.378614942286516, &{} c_1 = 0.321608164825949, \\ c_2 = 0.687231609392625, &{} w_{10}= 1.0, &{} w_{11} = 0.380481257618408, &{}w_{12} = 0.048570228152514, \end{array}$$
$$\begin{array}{llll} w_{13} = 0.003531837263071, &{} w_{14} = 0.000965597584732&{} w_{15} = -0.000093904905574,&{}w_{16} = 0.837871652069336,\\ w_{20} = 1.0,&{} w_{21} = 0.221160355321244, &{} w_{22} = 0.057082983028313,&{} w_{23} = 0.008977334456590, \\ w_{24} = -0.000441185296213,&{} w_{25} = 0.000237808975927,&{} w_{26} = 0.837823139921717.&{} \end{array}$$

6. SSP 4DGLM7-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.350457763059077, &{} a^{\{2\}}_{21} = 0.059030590724736, &{} a^{\{3\}}_{21} = 0.006167354392834, &{} a^{\{4\}}_{21} = 0.000820001027662,\\ a^{\{1\}}_{31} = 0.387500300301159, &{} a^{\{2\}}_{31} = 0.084838889640679, &{} a^{\{3\}}_{31} = 0.012031303235926, &{} a^{\{4\}}_{31} = 0.000492349845276,\\ a^{\{1\}}_{32} = 0.312881241973284, &{} a^{\{2\}}_{32} = 0.056243073304936, &{} a^{\{3\}}_{32} = 0.009017317429851, &{} a^{\{4\}}_{32} = 0.001125124849635,\\ b^{\{1\}}_{11} = 0.369607055094061, &{} b^{\{1\}}_{12} = 0.403154224734759, &{} b^{\{1\}}_{13} = 0.127967119709400, &{} b^{\{1\}}_{21} = 0.390879999306208,\\ b^{\{1\}}_{22} = 0.239443987884715 , &{} b^{\{1\}}_{23} = 0.494290470310205, &{} b^{\{2\}}_{11} = 0.058565784856034, &{} b^{\{2\}}_{12} = 0.104280749110758,\\ b^{\{2\}}_{13} = 0.008303419949253, &{} b^{\{2\}}_{21} = 0.087915116062722 , &{} b^{\{2\}}_{22} = 0.058148914188203, &{} b^{\{2\}}_{23} = 0.049797213094552,\\ b^{\{3\}}_{11} = 0.006572057700620, &{} b^{\{3\}}_{12} = 0.018705647077826, &{} b^{\{3\}}_{13} = 0.001466993217758, &{} b^{\{3\}}_{21} = 0.012839897311860,\\ b^{\{3\}}_{22} = 0.006947653665000, &{} b^{\{3\}}_{23} = 0.008829144278492, &{} b^{\{4\}}_{11} = 0.000610929864500, &{} b^{\{4\}}_{12} = 0.002527896988956,\\ b^{\{4\}}_{13} = 0.000199724984841, &{} b^{\{4\}}_{21} = 0.000420835098031, &{} b^{\{4\}}_{22} = 0.000858837048376, &{} b^{\{4\}}_{23} = 0.001202326356901,\\ u_{11} = 0.558164113618959, &{} u_{12} = 0.441835886381041, &{} u_{21} = 0.589995146907839, &{} u_{22} = 0.410004853092161, \\ u_{31} = 0.537204663265761, &{} u_{32} = 0.462795336734239, &{} v = 0.443402333155673, &{} c_1 = 0.294925929008912, \\ c_2 = 0.638257167504055, &{} w_{10}= 1.0, &{} w_{11} = 0.196005034140514, &{} w_{12} = 0.013033449258411,\\ w_{13} = 0.007614583616032, &{} w_{14} = -0.001144243875588,&{} w_{15} = -0.000104282181364, &{} w_{16} = 0.000081762590702,\\ w_{17} = 0.749648748323158, &{} w_{20} = 1.0, &{}w_{21} = 0.419891092103421, &{} w_{22} = 0.081966742096937,\\ w_{23} = 0.000057305547194,&{} w_{24} = 0.002158981114169, &{} w_{25} = 0.000173822525851,&{} w_{26} = -0.000101220718724,\\ w_{27} = 0.749692058749461.&{}&{}&{} \end{array}$$

7. SSP 4DGLM8-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.175790178224457, &{} a^{\{2\}}_{21} = 0.053976619001249, &{} a^{\{3\}}_{21} = 0.014560267078991, &{} a^{\{4\}}_{21} = 0.002914857159230,\\ a^{\{1\}}_{31} = 0.193854979888018, &{} a^{\{2\}}_{31} = 0.027727867380224, &{} a^{\{3\}}_{31} = 0.005382677196853, &{} a^{\{4\}}_{31} = 0.001097886442691,\\ a^{\{1\}}_{32} = 0.344954266207200, &{} a^{\{2\}}_{32} = 0.013247065452726, &{} a^{\{3\}}_{32} = 0.001182532004156, &{} a^{\{4\}}_{32} = 0.000163946507126,\\ b^{\{1\}}_{11} = 0.350470082796973, &{} b^{\{1\}}_{12} = 0.207739507894165, &{} b^{\{1\}}_{13} = 0.323978091749971, &{} b^{\{1\}}_{21} = 0.336851998589390,\\ b^{\{1\}}_{22} = 0.379664713420180, &{} b^{\{1\}}_{23} = 0.475765720663504, &{} b^{\{2\}}_{11} = 0.055651506187291, &{} b^{\{2\}}_{12} = 0.102852985251848,\\ b^{\{2\}}_{13} = 0.027042199636409, &{} b^{\{2\}}_{21} = 0.114564248168959, &{} b^{\{2\}}_{22} = 0.121907339183384, &{} b^{\{2\}}_{23} = 0.134966509007059,\\ b^{\{3\}}_{11} = 0.005747532077953, &{} b^{\{3\}}_{12} = 0.007546452267841, &{} b^{\{3\}}_{13} = 0.007094916636282, &{} b^{\{3\}}_{21} = 0.030041698640272,\\ b^{\{3\}}_{22} = 0.027573371342666, &{} b^{\{3\}}_{23} = 0.038269689121708, &{} b^{\{4\}}_{11} = 0.001039724633659, &{} b^{\{4\}}_{12} = 0.000079054390091,\\ b^{\{4\}}_{13} = 0.001572203392119, &{} b^{\{4\}}_{21} = 0.003491131979020, &{} b^{\{4\}}_{22} = 0.006063710347766, &{} b^{\{4\}}_{23} = 0.002985797295388,\\ u_{11} = 0.991567328795643, &{} u_{12} = 0.008432671204357, &{} u_{21} = 0.184632179631794 , &{} u_{22} = 0.815367820368206, \\ u_{31} = 0.678379155693658, &{} u_{32} = 0.321620844306342,&{} v = 0.379923611962994, &{} c_1 = 0.364072745591117, \\ c_2 = 0.790089277348930, &{} w_{10}= 1.0, &{} w_{11} = 0.361457818520213, &{} w_{12} = 0.064938219272829, \\ w_{13} = 0.007747210371254, &{} w_{14} = 0.000708041762913,&{} w_{15} = 0.000054105005786, &{} w_{16} = 0.000003491031849,\\ w_{17} = 0.000000163872239, &{} w_{18} = 0.982635662271492, &{} w_{20} = 1.0,&{}w_{21} = 0.671552568752178 ,\\ w_{22} = 0.223400791323222, &{} w_{23} = 0.042813264929839, &{} w_{24} = 0.003555229580558, &{} w_{25} = -0.000040886308458,\\ w_{26} = -0.000026939548375,&{} w_{27} = 0.000000679871749, &{} w_{28} = 0.982637254619139. &{} \end{array}$$

8. SSP 4DGLM9-3

$$\begin{array}{llll} a^{\{1\}}_{21} = 0.568808081207977, &{} a^{\{2\}}_{21} = 0.048114396995161, &{} a^{\{3\}}_{21} = 0.010788380615035, &{} a^{\{4\}}_{21} = 0.006008124562918,\\ a^{\{1\}}_{31} = 0.344187432821678, &{} a^{\{2\}}_{31} = 0.042382496857406, &{} a^{\{3\}}_{31} = 0.003582825049991, &{} a^{\{4\}}_{31} = 0.000689205385252,\\ a^{\{1\}}_{32} = 0.340398899955966, &{} a^{\{2\}}_{32} = 0.014335262109803, &{} a^{\{3\}}_{32} = 0.000313906335632, &{} a^{\{4\}}_{32} = 0.000145276246639,\\ b^{\{1\}}_{11} = 0.982954717825614, &{} b^{\{1\}}_{12} = 0.014490770169346, &{} b^{\{1\}}_{13} = 0.013584892713300, &{} b^{\{1\}}_{21} = 0.443809258398672,\\ b^{\{1\}}_{22} = 0.135975551664887, &{} b^{\{1\}}_{23} = 0.118857631207925, &{} b^{\{2\}}_{11} = 0.212259343381131, &{} b^{\{2\}}_{12} = 0.273158542191652,\\ b^{\{2\}}_{13} = 0.002916848832074, &{} b^{\{2\}}_{21} = 0.142520928035879, &{} b^{\{2\}}_{22} = 0.136853198913849, &{} b^{\{2\}}_{23} = 0.018625818520990,\\ b^{\{3\}}_{11} = 0.018327446754578, &{} b^{\{3\}}_{12} = 0.002170604033030, &{} b^{\{3\}}_{13} = 0.002168545523328, &{} b^{\{3\}}_{21} = 0.056096865472845,\\ b^{\{3\}}_{22} = 0.096389327446308, &{} b^{\{3\}}_{23} = 0.012468767290534, &{} b^{\{4\}}_{11} = 0.000413806824034, &{} b^{\{4\}}_{12} = 0.000000761021627,\\ b^{\{4\}}_{13} = 0.001208578960785, &{} b^{\{4\}}_{21} = 0.006136911418370, &{} b^{\{4\}}_{22} = 0.033053608345553, &{} b^{\{4\}}_{23} = 0.004291265251185,\\ u_{11} = 0.957045804986185, &{} u_{12} = 0.042954195013815, &{} u_{21} = 0.684034016720173, &{} u_{22} = 0.315965983279827, \\ u_{31} = 0.929009329287935, &{} u_{32} = 0.070990670712065,&{} v = 0.035309880170621, &{} c_1 = 0.324171924094799, \\ c_2 = 0.807694415324405, &{} w_{10}= 1.0, &{} w_{11} = 0.337590296565330, &{} w_{12} = 0.046071695897029,\\ w_{13} = 0.001607713367915, &{} w_{14} = 0.000143963069728,&{} w_{15} = 0.000063618469522, &{}w_{16} = 0.000006654666131,\\ w_{17} = -0.000000395816761, &{} w_{18} = -0.000000198789037, &{} w_{19} = 0.986560378780447,&{}w_{20} = 1.0,\\ w_{21} = 0.025202357128554, &{} w_{22} = 0.196744343505741,&{} w_{23} = 0.096360260165478, &{} w_{24} = 0.007504764895165,\\ w_{25} = -0.000722929866516,&{} w_{26} = -0.000110745594193,&{} w_{27} = 0.000010556807649,&{} w_{28} = 0.000004499558562,\\ w_{29} = 0.986558896598726. &{}&{}&{} \end{array}$$

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Moradi, A., Chouchoulis, J., D’Ambrosio, R. et al. Jacobian-free explicit multiderivative general linear methods for hyperbolic conservation laws. Numer Algor 97, 1823–1858 (2024). https://doi.org/10.1007/s11075-024-01771-6

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Keywords

Mathematics Subject Classification (2010)