(* Runge - Kutta pair of orders 8(6) for linear inho problems *)

T86bvec={314527/4021920,0,0,5727/1232,-87349/5670,45545/1764,-1227/49,
         93395/6048,-2543/378,1730048/829521,1/10,0};

T86amat={{1/5},{3/40,9/40},
{-40355761601083472/266140230441105939,379205628299487986/443567050735176565,
 -80711523202166944/266140230441105939},
{-695463111469361764/1500196446724802203,1260442490511067231/788875102592301206,
-270448114268444353/457441976633771789,-27251633927536895/634169504640478649},
{-324985794948140570/279362987511647517,2067789770618503777/539709130919242078,
-1024281445080204601/271760704640117271,1267426191530190207/414089865092880655,
-312635769063330587/229931829445938805},
{-1991868306773221465/988006971090061434,2066883310365527951/280309913359190828,
-1149915980509893214/105627821090836979,5877046745400870627/551178302290848343,
-2307633641349644207/534928889341982614,-19336393482604757/161379954985036024},
{3421988792443320320/409958906743348487,-7719526057460011553/583879427237260871,
-14327944929325885917/974705522708371870,5169381944138214741/193127113008949648,
-3485110191323692511/454226838873771266,954500654845243233/243523757961617086,
-1194811460443290403/452590789241887404},
{17264226447133602112/272996897031641451,-32295720487824629515/241904388356288873,
2953206026652849252/303413816706367747,12956688961776592125/220733668141103896,
-52264594687490789/8300814588386032,8242534511359177399/334193487031316324,
-3550153210913076029/227155312778390915,-1532175456666191/388107427043540147},
{26688207385003289504/264585390926238097,-56507224747685848649/253583197096431440,
11428471378372210538/207643458100451045,22065085407467690258/509695469676088365,
1766268407864809339/156440651140379502,12163744429792102954/310450202476834919,
-22159013041367309573/796071732134508657,1197252865130107127/509462498080681282,
-63611354765744053/159614734793971724},
{114537892779893654389/192922971090262140,-510740282904871030564/415586341949265143,
-47597666620490009567/897862996765222138,188835790411128503725/232069536271070424,
-51119528850220842269/182287831866373472,168104550605285163532/542064106458782789,
-35470180775173364810/256387766512111747,-3139869671811831263/170707935556822437,
206571767992602104/130392041890225475,865024/829521},
{314527/4021920,0,0,5727/1232,-87349/5670,45545/1764,-1227/49,93395/6048,-2543/378,
1730048/829521,1/10}};

T86cvec={1/5,3/10,2/5,1/2,3/5,7/10,4/5,9/10,19/20,1,1};

T86evec= {-2193001/205922304,0,0,27567979/14192640,-6553007/725760,8277295/451584,
-11701679/564480,21853871/1548288,-6452/945,35430481/16590420,1/8,-1/20};

T86Coefficients[8, p_] :=  N[{T86amat, T86bvec, T86cvec, T86evec}, p];

(*-----------------------------*)
(* run y'=-10*y+cos(t), y(0)=1 for tol=10^-22 *)
Needs["DifferentialEquations`NDSolveProblems`"];
Needs["DifferentialEquations`NDSolveUtilities`"];
T86={"ExplicitRungeKutta","Coefficients"->T86Coefficients,"DifferenceOrder"->8,"StiffnessTest"->False};
system=NDSolveProblem[{{y'[t]==-10*y[t]+Cos[t]},{y[0]==1},{y[t]},{t,0,10*Pi},{},{},{}}];
refsol={10/101};
CompareMethods[system,refsol,{T86},WorkingPrecision->33,AccuracyGoal->22,PrecisionGoal->22]