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init_integrator.m
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function [A,b,c,s,type,order] = init_integrator(integrator)
% type:
% 1 - explicit
% 2 - diagonally implicit
% 3 - fully implicit
switch integrator
case{1}
% Forward Euler
type = 0; % explicit
s = 1; % number of stages
A = [0];
c = 0;
b = 1;
order = 1;
case{2}
% Midpoint
type = 0; % explicit
s = 2;
A = [0 0 ;
1/2 0];
c = [0; 0.5];
b = [0 1];
order = 2;
case{3}
% Trapezoid
type = 1; % implicit
s = 2;
A = [0 0;
1/2 1/2];
c = [0; 1];
b = [1/2 1/2];
order = 2;
case{4}
% RK3
type = 0; % explicit
s = 3;
A = [0 0 0;
0.5 0 0;
-1 2 0];
c = [0;0.5; 1];
b = [1/6 2/3 1/6];
order = 3;
case{5}
% RK4
type = 0; % explicit
s = 4;
A = [0 0 0 0;
0.5 0 0 0;
0 0.5 0 0;
0 0 1 0];
c = [0; 0.5; 0.5; 1];
b = [1/6 1/3 1/3 1/6];
order = 4;
case{6}
% Backward Euler
type = 1; % implicit
s = 1;
A = 1;
c = 1;
b = 1;
order = 1;
case{7}
% SDIRK 2
type = 1; % implicit
gamma = (2-sqrt(2))/2;
s = 2;
A = [gamma 0;
1-gamma gamma];
c = [gamma; 1];
b = [(1-gamma) gamma];
order = 2;
case{8}
% built in ode45 integrator
order = 4;
type = 2; % built in
A = [];
s = [];
c = [];
b = [];
case{9}
% RK5
order = 5;
type = 0;
s = 7;
c = [0;1/5;3/10;4/5;8/9;1;1];
A = zeros(s,s);
A(2,1) = 1/5;
A(3,1) = 3/40;
A(3,2) = 9/40;
A(4,1) = 44/45;
A(4,2) = -56/15;
A(4,3) = 32/9;
A(5,1) = 19372/6561;
A(5,2) = -25360/2187;
A(5,3) = 64448/6561;
A(5,4) = -212/729;
A(6,1) = 9017/3168;
A(6,2) = -355/33;
A(6,3) = 46732/5247;
A(6,4) = 49/176;
A(6,5) = -5103/18656;
A(7,1) = 35/384;
A(7,2) = 0;
A(7,3) = 500/1113;
A(7,4) = 125/192;
A(7,5) = -2187/6784;
A(7,6) = 11/84;
b = A(7,:);
case{10}
% RK6
order = 6;
type = 0;
s = 10;
A = zeros(s,s);
c = zeros(s,1);
c(1) = 0;
c(2) = .5e-2;
c(3) = .1088888888888888888888888888888888888889;
c(4) = .1633333333333333333333333333333333333333;
c(5) = .4555;
c(6) = .6095094489978381317087004421486024949638;
c(7) = .884;
c(8) = .925;
c(9) = 1.;
c(10) = 1.;
A(2,1) = 0.5e-2;
A(3,1) = -1.076790123456790123456790123456790123457;
A(3,2) = 1.185679012345679012345679012345679012346;
A(4,1) = .4083333333333333333333333333333333333333e-1;
A(4,2) = 0.;
A(4,3) = .1225;
A(5,1) = .6389139236255726780508121615993336109954;
A(5,2) = 0.;
A(5,3) = -2.455672638223656809662640566430653894211;
A(5,4) = 2.272258714598084131611828404831320283215;
A(6,1) = -2.661577375018757131119259297861818119279;
A(6,2) = 0.;
A(6,3) = 10.80451388645613769565396655365532838482;
A(6,4) = -8.353914657396199411968048547819291691541;
A(6,5) = .8204875949566569791420417341743839209619;
A(7,1) = 6.067741434696770992718360183877276714679;
A(7,2) = 0.;
A(7,3) = -24.71127363591108579734203485290746001803;
A(7,4) = 20.42751793078889394045773111748346612697;
A(7,5) = -1.906157978816647150624096784352757010879;
A(7,6) = 1.006172249242068014790040335899474187268;
A(8,1) = 12.05467007625320299509109452892778311648;
A(8,2) = 0.;
A(8,3) = -49.75478495046898932807257615331444758322;
A(8,4) = 41.14288863860467663259698416710157354209;
A(8,5) = -4.461760149974004185641911603484815375051;
A(8,6) = 2.042334822239174959821717077708608543738;
A(8,7) = -0.9834843665406107379530801693870224403537e-1;
A(9,1) = 10.13814652288180787641845141981689030769;
A(9,2) = 0.;
A(9,3) = -42.64113603171750214622846006736635730625;
A(9,4) = 35.76384003992257007135021178023160054034;
A(9,5) = -4.348022840392907653340370296908245943710;
A(9,6) = 2.009862268377035895441943593011827554771;;
A(9,7) = .3487490460338272405953822853053145879140;
A(9,8) = -.2714390051048312842371587140910297407572;
A(10,1) = -45.03007203429867712435322405073769635151;
A(10,2) = 0.;
A(10,3) = 187.3272437654588840752418206154201997384;
A(10,4) = -154.0288236935018690596728621034510402582;
A(10,5) = 18.56465306347536233859492332958439136765;
A(10,6) = -7.141809679295078854925420496823551192821;
A(10,7) = 1.308808578161378625114762706007696696508;
A(10,8) = 0.;
A(10,9) = 0.;
% sixth order weights
b(1) = .4460860660634117628731817597479197781432e-1;
b(2) = 0.;
b(3) = 0.;
b(4) = .2671640378571372680509102260943837899738;
b(5) = .2201018300177293019979715776650753096323;
b(6) = .2188431703143156830983120833512893824578;
b(7) = .2289871705411202883378173889763552365362;
b(8) = 0.;
b(9) = 0.;
b(10) = .2029518466335628222767054793810430358554e-1;
case{11}
% RK 7
order = 7;
type = 0;
s = 10;
A = zeros(s,s);
c = zeros(s,1);
c(1) = 0;
c(2) = .5e-2;
c(3) = .1088888888888888888888888888888888888889;
c(4) = .1633333333333333333333333333333333333333;
c(5) = .4555;
c(6) = .6095094489978381317087004421486024949638;
c(7) = .884;
c(8) = .925;
c(9) = 1.;
c(10) = 1.;
A(2,1) = 0.5e-2;
A(3,1) = -1.076790123456790123456790123456790123457;
A(3,2) = 1.185679012345679012345679012345679012346;
A(4,1) = .4083333333333333333333333333333333333333e-1;
A(4,2) = 0.;
A(4,3) = .1225;
A(5,1) = .6389139236255726780508121615993336109954;
A(5,2) = 0.;
A(5,3) = -2.455672638223656809662640566430653894211;
A(5,4) = 2.272258714598084131611828404831320283215;
A(6,1) = -2.661577375018757131119259297861818119279;
A(6,2) = 0.;
A(6,3) = 10.80451388645613769565396655365532838482;
A(6,4) = -8.353914657396199411968048547819291691541;
A(6,5) = .8204875949566569791420417341743839209619;
A(7,1) = 6.067741434696770992718360183877276714679;
A(7,2) = 0.;
A(7,3) = -24.71127363591108579734203485290746001803;
A(7,4) = 20.42751793078889394045773111748346612697;
A(7,5) = -1.906157978816647150624096784352757010879;
A(7,6) = 1.006172249242068014790040335899474187268;
A(8,1) = 12.05467007625320299509109452892778311648;
A(8,2) = 0.;
A(8,3) = -49.75478495046898932807257615331444758322;
A(8,4) = 41.14288863860467663259698416710157354209;
A(8,5) = -4.461760149974004185641911603484815375051;
A(8,6) = 2.042334822239174959821717077708608543738;
A(8,7) = -0.9834843665406107379530801693870224403537e-1;
A(9,1) = 10.13814652288180787641845141981689030769;
A(9,2) = 0.;
A(9,3) = -42.64113603171750214622846006736635730625;
A(9,4) = 35.76384003992257007135021178023160054034;
A(9,5) = -4.348022840392907653340370296908245943710;
A(9,6) = 2.009862268377035895441943593011827554771;;
A(9,7) = .3487490460338272405953822853053145879140;
A(9,8) = -.2714390051048312842371587140910297407572;
A(10,1) = -45.03007203429867712435322405073769635151;
A(10,2) = 0.;
A(10,3) = 187.3272437654588840752418206154201997384;
A(10,4) = -154.0288236935018690596728621034510402582;
A(10,5) = 18.56465306347536233859492332958439136765;
A(10,6) = -7.141809679295078854925420496823551192821;
A(10,7) = 1.308808578161378625114762706007696696508;
A(10,8) = 0.;
A(10,9) = 0.;
% seventh order weights
b(1) = .4715561848627222170431765108838175679569e-1;
b(2) = 0.;
b(3) = 0.;
b(4) = .2575056429843415189596436101037687580986;
b(5) = .2621665397741262047713863095764527711129;
b(6) = .1521609265673855740323133199165117535523;
b(7) = .4939969170032484246907175893227876844296;
b(8) = -.2943031171403250441557244744092703429139;
b(9) = .8131747232495109999734599440136761892478e-1;
b(10) = 0.;
case{12}
% RK 8
order = 8;
type = 0;
s = 16;
A = zeros(s,s);
c = zeros(s,1);
c(1) = 0.;
c(2) = .3462e-1;
c(3) = .9702435063878044594828361677100617517633e-1;
c(4) = .1455365259581706689224254251565092627645;
c(5) = .561;
c(6) = .2290079115904850126662751771814700052182;
c(7) = .5449920884095149873337248228185299947818;
c(8) = .645;
c(9) = .4837500000000000000000000000000000000000;
c(10) = .6757e-1;
c(11) = .2500;
c(12) = .6590650618730998549405331618649220295334;
c(13) = .8206;
c(14) = .9012;
c(15) = 1.;
c(16) = 1.;
A(2,1) = .3462e-1;
A(3,1) = -.389335438857287327017042687229284478532e-1;
A(3,2) = .1359578945245091786499878854939346230295;
A(4,1) = .3638413148954266723060635628912731569111e-1;
A(4,2) = 0.;
A(4,3) = .1091523944686280016918190688673819470733;
A(5,1) = 2.025763914393969636805657604282571047511;
A(5,2) = 0.;
A(5,3) = -7.638023836496292020387602153091964592952;
A(5,4) = 6.173259922102322383581944548809393545442;
A(6,1) = .5112275589406060872792270881648288397197e-1;
A(6,2) = 0.;
A(6,3) = 0.;
A(6,4) = .1770823794555021537929910813839068684087;
A(6,5) = .80277624092225014536138698108025283759e-3;
A(7,1) = .1316006357975216279279871693164256985334;
A(7,2) = 0.;
A(7,3) = 0.;
A(7,4) = -.2957276252669636417685183174672273730699;
A(7,5) = .878137803564295237421124704053886667082e-1;
A(7,6) = .6213052975225274774321435005639430026100;
A(8,1) = .7166666666666666666666666666666666666667e-1;
A(8,2) = 0.;
A(8,3) = 0.;
A(8,4) = 0.;
A(8,5) = 0.;
A(8,6) = .3305533578915319409260346730051472207728;
A(8,7) = .2427799754418013924072986603281861125606;
A(9,1) = .7180664062500000000000000000000000000000e-1;
A(9,2) = 0.;
A(9,3) = 0.;
A(9,4) = 0.;
A(9,5) = 0.;
A(9,6) = .3294380283228177160744825466257672816401;
A(9,7) = .1165190029271822839255174533742327183599;
A(9,8) = -.3401367187500000000000000000000000000000e-1;
A(10,1) = .4836757646340646986611287718844085773549e-1;
A(10,2) = 0.;
A(10,3) = 0.;
A(10,4) = 0.;
A(10,5) = 0.;
A(10,6) = .3928989925676163974333190042057047002852e-1;
A(10,7) = .1054740945890344608263649267140088017604;
A(10,8) = -.2143865284648312665982642293830533996214e-1;
A(10,9) = -.1041229174627194437759832813847147895623;
A(11,1) = -.2664561487201478635337289243849737340534e-1;
A(11,2) = 0.;
A(11,3) = 0.;
A(11,4) = 0.;
A(11,5) = 0.;
A(11,6) = .3333333333333333333333333333333333333333e-1;
A(11,7) = -.1631072244872467239162704487554706387141;
A(11,8) = .3396081684127761199487954930015522928244e-1;
A(11,9) = .1572319413814626097110769806810024118077;
A(11,10) = .2152267478031879552303534778794770376960;
A(12,1) = .3689009248708622334786359863227633989718e-1;
A(12,2) = 0.;
A(12,3) = 0.;
A(12,4) = 0.;
A(12,5) = 0.;
A(12,6) = -.1465181576725542928653609891758501156785;
A(12,7) = .2242577768172024345345469822625833796001;
A(12,8) = .2294405717066072637090897902753790803034e-1;
A(12,9) = -.35850052905728761357394424889330334334e-2;
A(12,10) = .8669223316444385506869203619044453906053e-1;
A(12,11) = .4383840651968337846196219974168630120572;
A(13,1) = -.4866012215113340846662212357570395295088;
A(13,2) = 0.;
A(13,3) = 0.;
A(13,4) = 0.;;
A(13,5) = 0.;
A(13,6) = -6.304602650282852990657772792012007122988;
A(13,7) = -.281245618289472564778284183790118418111;
A(13,8) = -2.679019236219849057687906597489223155566;
A(13,9) = .518815663924157511565311164615012522024;
A(13,10) = 1.365353187603341710683633635235238678626;
A(13,11) = 5.885091088503946585721274891680604830712;
A(13,12) = 2.802808786272062889819965117517532194812;
A(14,1) = .4185367457753471441471025246471931649633;
A(14,2) = 0.;
A(14,3) = 0.;
A(14,4) = 0.;
A(14,5) = 0.;
A(14,6) = 6.724547581906459363100870806514855026676;
A(14,7) = -.425444280164611790606983409697113064616;
A(14,8) = 3.343279153001265577811816947557982637749;
A(14,9) = .617081663117537759528421117507709784737;
A(14,10) = -.929966123939932833937749523988800852013;
A(14,11) = -6.099948804751010722472962837945508844846;
A(14,12) = -3.002206187889399044804158084895173690015;
A(14,13) = .2553202529443445472336424602988558373637;
A(15,1) = -.779374086122884664644623040843840506343;
A(15,2) = 0.;
A(15,3) = 0.;
A(15,4) = 0.;
A(15,5) = 0.;
A(15,6) = -13.93734253810777678786523664804936051203;
A(15,7) = 1.252048853379357320949735183924200895136;
A(15,8) = -14.69150040801686878191527989293072091588;
A(15,9) = -.494705058533141685655191992136962873577;
A(15,10) = 2.242974909146236657906984549543692874755;
A(15,11) = 13.36789380382864375813864978592679139881;
A(15,12) = 14.39665048665068644512236935340272139005;
A(15,13) = -.7975813331776800379127866056663258667437;
A(15,14) = .4409353709534277758753793068298041158235;
A(16,1) = 2.058051337466886442151242368989994043993;
A(16,2) = 0.;
A(16,3) = 0.;
A(16,4) = 0.;
A(16,5) = 0.;
A(16,6) = 22.35793772796803295519317565842520212899;
A(16,7) = .90949810997556332745009198137971890783;
A(16,8) = 35.89110098240264104710550686568482456493;
A(16,9) = -3.442515027624453437985000403608480262211;
A(16,10) = -4.865481358036368826566013387928704014496;
A(16,11) = -18.90980381354342625688427480879773032857;
A(16,12) = -34.26354448030451782929251177395134170515;
A(16,13) = 1.264756521695642578827783499806516664686;
A(16,14) = 0.;
A(16,15) = 0.;
b(1) = .1996996514886773085518508418098868756464e-1;
b(2) = 0.;
b(3) = 0.;
b(4) = 0.;
b(5) = 0.;
b(6) = 0.;;
b(7) = 0.;
b(8) = 2.191499304949330054530747099310837524864;
b(9) = .8857071848208438030833722031786358862953e-1;
b(10) = .1140560234865965622484956605091432032674;
b(11) = .2533163805345107065564577734569651977347;
b(12) = -2.056564386240941011158999594595981300493;
b(13) = .3408096799013119935160094894224543812830;
b(14) = 0.;
b(15) = 0.;
b(16) = .4834231373823958314376726739772871714902e-1;
case{13}
% RK 9
order = 9;
type = 0;
s = 16;
A = zeros(s,s);
c = zeros(s,1);
c(1) = 0.;
c(2) = .3462e-1;
c(3) = .9702435063878044594828361677100617517633e-1;
c(4) = .1455365259581706689224254251565092627645;
c(5) = .561;
c(6) = .2290079115904850126662751771814700052182;
c(7) = .5449920884095149873337248228185299947818;
c(8) = .645;
c(9) = .4837500000000000000000000000000000000000;
c(10) = .6757e-1;
c(11) = .2500;
c(12) = .6590650618730998549405331618649220295334;
c(13) = .8206;
c(14) = .9012;
c(15) = 1.;
c(16) = 1.;
A(2,1) = .3462e-1;
A(3,1) = -.389335438857287327017042687229284478532e-1;
A(3,2) = .1359578945245091786499878854939346230295;
A(4,1) = .3638413148954266723060635628912731569111e-1;
A(4,2) = 0.;
A(4,3) = .1091523944686280016918190688673819470733;
A(5,1) = 2.025763914393969636805657604282571047511;
A(5,2) = 0.;
A(5,3) = -7.638023836496292020387602153091964592952;
A(5,4) = 6.173259922102322383581944548809393545442;
A(6,1) = .5112275589406060872792270881648288397197e-1;
A(6,2) = 0.;
A(6,3) = 0.;
A(6,4) = .1770823794555021537929910813839068684087;
A(6,5) = .80277624092225014536138698108025283759e-3;
A(7,1) = .1316006357975216279279871693164256985334;
A(7,2) = 0.;
A(7,3) = 0.;
A(7,4) = -.2957276252669636417685183174672273730699;
A(7,5) = .878137803564295237421124704053886667082e-1;
A(7,6) = .6213052975225274774321435005639430026100;
A(8,1) = .7166666666666666666666666666666666666667e-1;
A(8,2) = 0.;
A(8,3) = 0.;
A(8,4) = 0.;
A(8,5) = 0.;
A(8,6) = .3305533578915319409260346730051472207728;
A(8,7) = .2427799754418013924072986603281861125606;
A(9,1) = .7180664062500000000000000000000000000000e-1;
A(9,2) = 0.;
A(9,3) = 0.;
A(9,4) = 0.;
A(9,5) = 0.;
A(9,6) = .3294380283228177160744825466257672816401;
A(9,7) = .1165190029271822839255174533742327183599;
A(9,8) = -.3401367187500000000000000000000000000000e-1;
A(10,1) = .4836757646340646986611287718844085773549e-1;
A(10,2) = 0.;
A(10,3) = 0.;
A(10,4) = 0.;
A(10,5) = 0.;
A(10,6) = .3928989925676163974333190042057047002852e-1;
A(10,7) = .1054740945890344608263649267140088017604;
A(10,8) = -.2143865284648312665982642293830533996214e-1;
A(10,9) = -.1041229174627194437759832813847147895623;
A(11,1) = -.2664561487201478635337289243849737340534e-1;
A(11,2) = 0.;
A(11,3) = 0.;
A(11,4) = 0.;
A(11,5) = 0.;
A(11,6) = .3333333333333333333333333333333333333333e-1;
A(11,7) = -.1631072244872467239162704487554706387141;
A(11,8) = .3396081684127761199487954930015522928244e-1;
A(11,9) = .1572319413814626097110769806810024118077;
A(11,10) = .2152267478031879552303534778794770376960;
A(12,1) = .3689009248708622334786359863227633989718e-1;
A(12,2) = 0.;
A(12,3) = 0.;
A(12,4) = 0.;
A(12,5) = 0.;
A(12,6) = -.1465181576725542928653609891758501156785;
A(12,7) = .2242577768172024345345469822625833796001;
A(12,8) = .2294405717066072637090897902753790803034e-1;
A(12,9) = -.35850052905728761357394424889330334334e-2;
A(12,10) = .8669223316444385506869203619044453906053e-1;
A(12,11) = .4383840651968337846196219974168630120572;
A(13,1) = -.4866012215113340846662212357570395295088;
A(13,2) = 0.;
A(13,3) = 0.;
A(13,4) = 0.;;
A(13,5) = 0.;
A(13,6) = -6.304602650282852990657772792012007122988;
A(13,7) = -.281245618289472564778284183790118418111;
A(13,8) = -2.679019236219849057687906597489223155566;
A(13,9) = .518815663924157511565311164615012522024;
A(13,10) = 1.365353187603341710683633635235238678626;
A(13,11) = 5.885091088503946585721274891680604830712;
A(13,12) = 2.802808786272062889819965117517532194812;
A(14,1) = .4185367457753471441471025246471931649633;
A(14,2) = 0.;
A(14,3) = 0.;
A(14,4) = 0.;
A(14,5) = 0.;
A(14,6) = 6.724547581906459363100870806514855026676;
A(14,7) = -.425444280164611790606983409697113064616;
A(14,8) = 3.343279153001265577811816947557982637749;
A(14,9) = .617081663117537759528421117507709784737;
A(14,10) = -.929966123939932833937749523988800852013;
A(14,11) = -6.099948804751010722472962837945508844846;
A(14,12) = -3.002206187889399044804158084895173690015;
A(14,13) = .2553202529443445472336424602988558373637;
A(15,1) = -.779374086122884664644623040843840506343;
A(15,2) = 0.;
A(15,3) = 0.;
A(15,4) = 0.;
A(15,5) = 0.;
A(15,6) = -13.93734253810777678786523664804936051203;
A(15,7) = 1.252048853379357320949735183924200895136;
A(15,8) = -14.69150040801686878191527989293072091588;
A(15,9) = -.494705058533141685655191992136962873577;
A(15,10) = 2.242974909146236657906984549543692874755;
A(15,11) = 13.36789380382864375813864978592679139881;
A(15,12) = 14.39665048665068644512236935340272139005;
A(15,13) = -.7975813331776800379127866056663258667437;
A(15,14) = .4409353709534277758753793068298041158235;
A(16,1) = 2.058051337466886442151242368989994043993;
A(16,2) = 0.;
A(16,3) = 0.;
A(16,4) = 0.;
A(16,5) = 0.;
A(16,6) = 22.35793772796803295519317565842520212899;
A(16,7) = .90949810997556332745009198137971890783;
A(16,8) = 35.89110098240264104710550686568482456493;
A(16,9) = -3.442515027624453437985000403608480262211;
A(16,10) = -4.865481358036368826566013387928704014496;
A(16,11) = -18.90980381354342625688427480879773032857;
A(16,12) = -34.26354448030451782929251177395134170515;
A(16,13) = 1.264756521695642578827783499806516664686;
A(16,14) = 0.;
A(16,15) = 0.;
b(1) = .1461197685842315252051541915018784713459e-1;
b(2) = 0.;
b(3) = 0.;
b(4) = 0.;
b(5) = 0.;
b(6) = 0.;
b(7) = 0.;
b(8) = -.3915211862331339089410228267288242030810;
b(9) = .2310932500289506415909675644868993669908;
b(10) = .1274766769992852382560589467488989175618;
b(11) = .2246434176204157731566981937082069688984;
b(12) = .5684352689748512932705226972873692126743;
b(13) = .5825871557215827200814768021863420902155e-1;
b(14) = .1364317403482215641609022744494239843327;
b(15) = .3057013983082797397721005067920369646664e-1;
b(16) = 0.;
otherwise
error('invalid integrator option');
end