Variable order explicit second derivative general linear methods

Abstract

Explicit second derivative general linear methods (ESDGLM) form an extension of the general linear methods (GLM). These special class of methods are designed for the numerical integration of non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs). Some ESDGLM are considered with numerical tests comparing the accuracy of the proposed algorithms with some other existing ones.

Appendix

The continuous coefficients of the scheme in (7), (8) and (10) for $k=1, s=3$,

$$\begin{aligned} E_1&= -20 t^6 c_1+30 t^5 c_1^2+10 t^6 c_2+30 t^5 c_1 c_2-75 t^4 c_1^2 c_2-24 t^5 c_2^2+15 t^4 c_1 c_2^2\\&+50 t^3 c_1^2 c_2^2+15 t^4 c_2^3-30 t^3 c_1 c_2^3,\\ E_2&= 10 t^6 c_3+30 t^5 c_1 c_3-75 t^4 c_1^2 c_3-42 t^5 c_2 c_3-30 t^4 c_1 c_2 c_3+200 t^3 c_1^2 c_2 c_3\\&+60 t^4 c_2^2 c_3-70 t^3 c_1 c_2^2 c_3,\\ E_3&= -150 t^2 c_1^2 c_2^2 c_3-30 t^3 c_2^3 c_3+90 t^2 c_1 c_2^3 c_3-24 t^5 c_3^2+15 t^4 c_1 c_3^2+50 t^3 c_1^2 c_3^2\\&+60 t^4 c_2 c_3^2-70 t^3 c_1 c_2 c_3^2,\\ E_4&= -150 t^2 c_1^2 c_2 c_3^2-40 t^3 c_2^2 c_3^2+120 t^2 c_1 c_2^2 c_3^2+150 t c_1^2 c_2^2 c_3^2-90 t c_1 c_2^3 c_3^2+\\&15 t^4 c_3^3-30 t^3 c_1 c_3^3,\\ E_5&= -30 t^3 c_2 c_3^3+90 t^2 c_1 c_2 c_3^3-90 t c_1 c_2^2 c_3^3+30 t c_2^3 c_3^3,\\ E_6&= t^3 c_1^3 (-10 t^2 c_1 (t-6 c_2)+3 t c_1^2 (8 t-5 c_2)-15 c_1^3 (t-2 c_2)+20 t^2 c_2 (-2 t+3 c_2)),\\ E_7&= 5 t^3 c_1^2 (-5 c_1^2 (3 t+2 c_1) c_2^2+(-8 t^3 c_1+12 t^2 c_1^2-3 t c_1^3+6 c_1^4-24 t^3 c_2) c_3),\\ E_8&= 5 t^2 c_1^2 c_2 (-15 t^2 c_1^2-4 t c_1^3-18 c_1^4+12 t^2 c_1 (4 t-5 c_2)+36 t^3 c_2) c_3,\\ E_9&= 5 t^2 c_1 c_3 (-10 (t-3 c_1) c_1^3 c_2^2+t (12 t^2 c_1^2-15 t c_1^3-10 c_1^4+12 t^3 c_2) c_3),\\ E_{10}&= 10 t c_1 c_2 (9 t^4 c_1-30 t^3 c_1^2-5 t^2 c_1^3+15 t c_1^4+9 c_1^5-9 t^4 c_2) c_3^2,\\ E_{11}&= 10 c_1 c_3^2 (5 t c_1 (-3 t^3+8 t^2 c_1+12 t c_1^2-3 c_1^3) c_2^2+t (-2 t^5+3 t^4 c_1-3 c_1^5) c_3),\\ E_{12}&= -10 t c_2 (2 t^5+3 c_1 (2 t^4-5 t^3 c_1+6 c_1^4)-3 t^3 (t+5 c_1) c_2) c_3^3,\\ E_{13}&= 5 c_3^3 (-10 c_1^2 (8 t^3+15 t c_1^2) c_2^2+t (6 t^4 c_1-15 t^3 c_1^2+18 c_1^5+2 t^4 (t+3 c_2)) c_3),\\ E_{14}&= -25 t c_2 (-6 c_1^4+t^2 c_1 (3 t-2 c_2)+3 t^3 c_2+12 c_1^3 c_2-2 t c_1^2 (t+6 c_2)) c_3^4,\\ E_{15}&= t (60 c_1^4+50 t c_1^2 (t-3 c_2)+5 t^2 c_1 (3 t+4 c_2)+3 t^3 (-8 t+5 c_2)) c_3^5,\\ E_{16}&= 5 t c_3^5 (2 c_2 (24 c_1^3+5 (t^2-3 t c_1+6 c_1^2) c_2)+3 (t^3-2 t^2 c_1-4 c_1^3-2 (t^2-3 t c_1\\&+6 c_1^2) c_2) c_3),\\ E_{17}&= t^3 c_1^3 (10 t^2 c_1 (t-6 c_2)+15 c_1^3 (t-2 c_2)+20 t^2 (2 t-3 c_2) c_2+3 t c_1^2 (-8 t+5 c_2)),\\ E_{18}&= 5 t c_2^2 (5 t^2 c_1^4 (3 t+2 c_1)+2 c_1 (2 t^5-3 t^4 c_1+3 c_1^5) c_2-2 t^4 (t+3 c_1) c_2^2),\\ E_{19}&= t c_2^4 (75 t^3 c_1^2-90 c_1^5+(24 t^4-5 c_1 (3 t^3+10 t^2 c_1+12 c_1^3)) c_2-15 t^2 (t-2 c_1) c_2^2),\\ E_{20}&= 5 t c_1^2 (12 c_1 c_2^6+t^2 (8 t^3 c_1-12 t^2 c_1^2+3 t c_1^3-6 c_1^4+24 t^3 c_2) c_3),\\ E_{21}&= 5 t^2 c_1 c_2 (c_1^2 (-48 t^3+15 t^2 c_1+4 t c_1^2+18 c_1^3)-6 t^3 (2 t+3 c_1) c_2) c_3,\\ E_{22}&= 10 t c_2^2 (c_1^3 (30 t^3+c_1 (5 t^2-3 c_1 (5 t+3 c_1)))+2 t^4 (t+3 c_1) c_2) c_3,\\ E_{23}&= -5 t c_2^3 (6 c_1^2 (5 t^3-6 c_1^3)+(6 t^4+5 c_1 (-3 t^3+2 t^2 c_1+6 c_1^3)) c_2) c_3,\\ E_{24}&= -5 t c_2^5 (-30 t c_1^2+48 c_1^3+3 t^2 (t-2 c_2)+2 t c_1 (2 t+9 c_2)) c_3,\\ E_{25}&= 5 t c_1^2 c_3 (36 c_2^6+t^2 c_1 (-12 t^2+5 c_1 (3 t+2 c_1)) c_3+12 t^3 (-3 t+5 c_1) c_2 c_3),\\ E_{26}&= 10 t^2 c_1 c_2 (5 (t-3 c_1) c_1^3+(9 t^3-5 c_1 (-3 t^2+8 t c_1+12 c_1^2)) c_2) c_3^2,\\ E_{27}&= 5 t c_2^2 (30 c_1^5+2 (-3 t^4+5 c_1 (-3 t^3+8 t^2 c_1+15 c_1^3)) c_2+15 t^3 c_2^2) c_3^2,\\ E_{28}&= -50 t c_2^4 (-6 c_1^3+t c_1 (t-3 c_2)+t^2 c_2+6 c_1^2 (t+c_2)) c_3^2,\\ E_{29}&= t^3 c_1^3 (4 t^2 c_2 (-5 t+9 c_2)+5 c_1 (2 t^3-9 t c_2^2+8 c_2^3)-2 c_1^2 (6 t^2+5 c_2 (-3 t+2 c_2))),\\ E_{30}&= -2 t^3 c_1 (10 c_1^2 c_2^4+(2 t^2 c_1^2 (5 t-12 c_2)+30 t c_1^3 c_2+10 t^3 c_2^2+5 c_1^4 (-3 t+8 c_2)) c_3),\\ E_{31}&= 2 t^2 c_1 c_2^2 (30 c_1^4+20 c_1^3 (4 t-3 c_2)-40 t c_1^2 c_2+3 t^2 (8 t-5 c_2) c_2) c_3,\\ E_{32}&= t^2 c_1 c_3 (60 c_1^2 c_2^4+t (c_1^2 (36 t^2-5 c_1 (9 t+4 c_1))-20 (t^3-8 c_1^3) c_2) c_3),\\ E_{33}&= 2 t c_2 (30 t c_1^5+(5 t^5+42 t^4 c_1-100 t^2 c_1^3-105 t c_1^4-30 c_1^5) c_2-12 t^3 (t+5 c_1) c_2^2) c_3^2,\\ E_{34}&= t c_3^2 (15 c_2^3 (8 c_1^3 (2 t+c_1)+(t^3+4 t^2 c_1-4 c_1^3) c_2)+8 t^2 c_1 (5 c_1^3+6 t^2 c_2) c_3),\\ E_{35}&= -4 t c_2 (20 t c_1^3 (t-3 c_2)+30 c_1^4 (t-c_2)+3 t^3 (2 t-5 c_2) c_2+10 t^2 c_1 (3 t-2 c_2) c_2) c_3^3,\\ E_{36}&= -5 t c_3^3 (8 c_2^3 (6 c_1^3+t^2 c_2)+t (4 c_1^3 (t-3 c_2)+6 t^2 c_1 c_2-3 t^2 c_2^2) c_3),\\ E_{37}&= 10 t c_2^2 (-6 c_1^3+t c_2 (-4 t+3 c_2)+6 c_1 (t^2-c_2^2)) c_3^4,\\ E_{38}&= t^3 c_1^3 (t c_1^2 (10 t^2+3 c_1 (-8 t+5 c_1))-4 c_1 (-5 t^3+6 t^2 c_1+5 c_1^3) c_2-20 t^3 c_2^2),\\ E_{39}&= t^3 c_1^3 (2 c_2^2 (5 c_1^2 (3 t\!+\!4 c_1)\!+\!(12 t^2-5 c_1 (3 t+2 c_1)) c_2)\!+\!t c_1 (50 t^2-72 t c_1+15 c_1^2) c_3),\\ E_{40}&= 20 t^2 c_1^3 (-2 t c_1^4+(2 t^4+3 c_1 (-2 t^3+c_1 (t^2+c_1 (t+c_1)))) c_2) c_3,\\ E_{41}&= 6 t^2 c_1^2 c_2^2 (15 t^2 c_1^2-20 c_1^4+4 t^2 c_1 (2 t-5 c_2)+2 t^3 (-5 t+6 c_2)) c_3,\\ E_{42}&= 5 t^2 c_1^3 c_3 (-4 (t-3 c_1) c_1 c_2^3+t (4 t^3-24 t^2 c_1+21 t c_1^2+16 c_1^3) c_3),\\ E_{43}&= 2 t c_1^2 (15 t c_1^5+(-40 t^5+6 t^4 c_1+105 t^3 c_1^2-90 t c_1^4-30 c_1^5) c_2) c_3^2,\\ E_{44}&= 3 t c_1 (10 t^5+c_1 (48 t^4+5 c_1 (-11 t^3+4 c_1 (-4 t^2-3 t c_1+2 c_1^2)))) c_2^2 c_3^2,\\ E_{45}&= -4 t c_1 c_3^2 ((9 t^4+5 c_1 (3 t^3+c_1 (-8 t^2+3 c_1 (-4 t+c_1)))) c_2^3+t^4 c_1 (5 t+6 c_1) c_3),\\ E_{46}&= t c_1 (15 t c_1^3 (7 t^2-6 c_1^2)+2 (10 t^5+c_1 (78 t^4+5 c_1 (-21 t^3-2 (4 t-3 c_1)\\&c_1 (2 t+3 c_1)))) c_2) c_3^3,\\ E_{47}&= t c_2^2 (-72 t^4 c_1-45 t^3 c_1^2+320 t^2 c_1^3+180 t c_1^4+240 c_1^5+2 t^4 (-5 t+6 c_2)) c_3^3,\\ E_{48}&= 2 t c_1 c_3^3 (10 (3 t^3\!\!-\!8 t^2 c_1-15 c_1^3) c_2^3\!+\!t c_1 (24 t^3-5 c_1 (3 t^2\!+\!8 t c_1\!+\!6 c_1^2)) c_3-24 t^4 c_2 c_3),\\ E_{49}&= 3 t c_2 (10 c_1^2 (-t^3\!+\!2 c_1 (3 t^2+2 c_1 (t+c_1)))\!+\!t (8 t^3-5 c_1 (-3 t^2\!+\!8 t c_1+10 c_1^2)) c_2) c_3^4,\\ E_{50}&= 10 t c_3^4 ((-3 t^3+2 c_1 (t^2+6 (t-c_1) c_1)) c_2^3+t c_1 (-3 t^2 c_1+4 t c_1^2+6 c_1^3+3 t^2 c_2) c_3),\\ E_{51}&= -5 t c_2 (12 c_1^2 (t^2+2 c_1^2)+3 (t^3-6 t c_1^2+8 c_1^3) c_2-4 (t^2-3 t c_1+6 c_1^2) c_2^2) c_3^5,\\ E_{52}&= t^3 c_1^3 (t c_1^2 (10 t^2+3 c_1 (-8 t+5 c_1))+c_1 (50 t^3+c_1 (-72 t^2+5 (3 t-8 c_1) c_1))\\&c_2+20 t^2 (t-6 c_1) c_2^2),\\ E_{53}&= t^2 c_1^2 c_2^2 (5 c_1^3 (21 t^2+16 t c_1+6 c_1^2)-(20 t^4+3 c_1 (8 t^3-35 t^2 c_1+30 c_1^3)) c_2),\\ E_{54}&= 2 t^2 c_1^2 (c_2^4 (24 t^3-5 c_1 (3 t^2+8 t c_1+6 c_1^2)+5 (-3 t^2+4 t c_1+6 c_1^2) c_2)+10 t^4 c_1^2 c_3),\\ E_{55}&= 4 t^2 c_1^3 (-6 t^3 c_1^2-5 t c_1^4+5 (2 t^4+3 c_1 (-2 t^3+c_1 (t^2+c_1 (t+c_1)))) c_2) c_3,\\ E_{56}&= 2 t c_1 c_2^2 (c_1 (-40 t^5+6 t^4 c_1+105 t^3 c_1^2-90 t c_1^4-30 c_1^5)+2 t^4 (5 t+39 c_1) c_2) c_3,\\ E_{57}&= -2 t c_1 c_2^3 (5 c_1^2 (21 t^3+2 (4 t-3 c_1) c_1 (2 t+3 c_1))+3 t^2 (8 t^2+5 (t-6 c_1) c_1) c_2) c_3,\\ E_{58}&= 10 t c_1 c_3 (3 c_2^4 (4 c_1^3 (t+c_1)+(t^3-2 t^2 c_1-4 c_1^3) c_2)+t^2 c_1^2 (-2 t^3+3 t c_1^2+4 c_1^3) c_3),\\ E_{59}&= -3 t^2 c_1 c_2 (40 c_1^5+4 t^3 c_1 (5 t-12 c_2)-10 t^4 c_2+10 t c_1^3 (-3 t+8 c_2)\\&+t^2 c_1^2 (-16 t+55 c_2)) c_3^2,\\ E_{60}&= t c_2^2 (60 c_1^5 (-3 t+2 c_1)+t (-10 t^4+c_1 (-72 t^3+5 c_1 (-9 t^2+64 t c_1+36 c_1^2))) c_2) c_3^2,\\ E_{61}&= 3 t c_2^3 (80 c_1^5+t (8 t^3-5 c_1 (-3 t^2+8 t c_1+10 c_1^2)) c_2-5 (t^3-6 t c_1^2+8 c_1^3) c_2^2) c_3^2,\\ E_{62}&= 2 t^2 c_1 (12 t^2 c_1^2 (t-5 c_2)-10 c_1^4 (t-3 c_2)+36 t^3 c_1 c_2-18 t^3 c_2^2-5 t c_1^3 (3 t+2 c_2)) c_3^3,\\ E_{63}&= 4 t c_2^2 (40 t^2 c_1^3-15 c_1^5+15 c_1^4 (4 t-5 c_2)+3 t^4 c_2+15 t^3 c_1 c_2-5 t^2 c_1^2 (3 t+8 c_2)) c_3^3,\\ E_{64}&= -10 t c_2^4 (12 c_1^3-2 t c_1 (t-3 c_2)+t^2 (3 t-2 c_2)-12 c_1^2 (t+c_2)) c_3^3,\\ T_{1}&= 30 c_1^6-90 c_1^5 c_2-60 c_1^4 c_2^2+60 c_1^3 c_2^3-90 c_1^5 c_3-330 c_1^4 c_2 c_3-60 c_1^3 c_2^2 c_3+180 c_1^2 c_2^3 c_3,\\ T_{2}&= -60 c_1^4 c_3^2-60 c_1^3 c_2 c_3^2+390 c_1^2 c_2^2 c_3^2-90 c_1 c_2^3 c_3^2+60 c_1^3 c_3^3+180 c_1^2 c_2 c_3^3-90 c_1 c_2^2 c_3^3\\&+30 c_2^3 c_3^3,\\ T_{3}&= 30 c_1^3 c_2^2 (c_2 (c_1+c_2) (c_1^2-4 c_1 c_2+2 c_2^2)-(3 c_1^3-6 c_1^2 c_2+5 c_1 c_2^2+8 c_2^3) c_3),\\ T_4&= 30 c_1 c_3 (6 c_1 c_2^6+c_2 (3 c_1^5+25 c_1^3 c_2^2+10 c_1^2 c_2^3-5 c_1 c_2^4-3 c_2^5) c_3-c_1^5 c_3^2),\\ T_5&= 30 c_3^3 (-15 c_1^2 c_2^4+6 c_1 c_2^5+c_2^6+5 c_1^4 c_2 (-5 c_2+c_3)+3 c_1^5 (-2 c_2+c_3)),\\ T_6&= 30 c_3^4 (-3 c_2^5+2 c_1^4 c_3-6 c_1 c_2^3 c_3+5 c_1^2 c_2^2 (3 c_2+c_3)+2 c_1^3 c_2 (-5 c_2+4 c_3)),\\ T_7&= 30 c_3^5 (3 c_2^4-(2 c_1^3+6 c_1^2 c_2-3 c_1 c_2^2+c_2^3) c_3),\\ T_8&= 30 c_1^2 c_2 (c_1 c_2^2 (c_1+c_2) (c_1^2-4 c_1 c_2+2 c_2^2)+c_2 (c_1+c_2) (-3 c_1^3+9 c_1^2 c_2-14 c_1 c_2^2\\&+6 c_2^3) c_3+3 c_1^4 c_3^2),\\ T_9&= -30 c_1 c_3^2 (-10 c_1^2 c_2^4+3 c_2^5 (c_2-2 c_3)+c_1^5 c_3+6 c_1^4 c_2 c_3+25 c_1^3 c_2^2 (-c_2+c_3)\\&+5 c_1 c_2^4 (c_2+3 c_3)),\\ T_{10}&= 30 c_3^3 (c_2^6+(3 c_1^5+5 c_1^4 c_2-10 c_1^3 c_2^2+15 c_1^2 c_2^3-3 c_2^5) c_3+2 c_1^3 (c_1+4 c_2) c_3^2),\\ T_{11}&= -30 c_3^5 (3 c_1 c_2^2 (2 c_2-c_3)+2 c_1^3 c_3+c_2^3 (-3 c_2+c_3)+c_1^2 c_2 (-5 c_2+6 c_3)),\\ T_{12}&= 60 c_1^4 (c_1-c_2) (c_1^3+2 c_2^3-2 c_1 c_2 (c_2+4 c_3)-c_1^2 (3 c_2+4 c_3)),\\ T_{13}&= 60 c_1^3 c_3 (4 (c_1-c_2) c_2^3+(c_1^3+8 c_1^2 c_2+6 c_1 c_2^2-24 c_2^3) c_3),\\ T_{14}&= 60 c_1 c_3^2 (9 c_1 c_2^4+4 (c_1-c_2)^{2} (c_1^2+3 c_1 c_2-c_2^2) c_3),\\ T_{15}&= -60 (c_1-c_2) (2 c_1^3+6 c_1^2 c_2-3 c_1 c_2^2+c_2^3) c_3^4,\\ T_{16}&= 60 c_1^3 c_2 ((c_1-c_2) c_2 (c_1+c_2) (c_1^2-4 c_1 c_2+2 c_2^2)+c_1^3 (-2 c_1+5 c_2) c_3),\\ T_{17}&= 60 c_1^2 c_3 (c_2^3 (-10 c_1^3+c_1^2 c_2+12 c_1 c_2^2-6 c_2^3)+c_1^4 (c_1+2 c_2) c_3),\\ T_{18}&= -60 c_1 c_3^2 (4 c_1 c_2^5-3 c_2^6+c_1^3 c_2^2 (14 c_2-11 c_3)+3 c_1^5 c_3+c_1^4 c_2 (-15 c_2+4 c_3)\\&+3 c_1^2 c_2^3 (c_2+6 c_3)),\\ T_{19}&= -60 c_3^3 (c_2^4 (-17 c_1^2+2 c_1 c_2+c_2^2)+c_1 (2 c_1^4+4 c_1^3 c_2-7 c_1^2 c_2^2-2 c_1 c_2^3+5 c_2^4) c_3),\\ T_{20}&= 60 c_3^4 (2 c_2^5+(c_1-c_2) (2 c_1^3+6 c_1^2 c_2-3 c_1 c_2^2+c_2^3) c_3),\\ T_{21}&= 60 c_1^3 (2 c_1 c_2^4 (c_2-2 c_3)+c_1^4 (c_2-c_3)^{2}+4 c_2^5 c_3-c_1^3 c_2 (3 c_2-5 c_3) (c_2+c_3)\\&-2 c_1^2 c_2^3 (c_2+2 c_3)),\\ T_{22}&= -60 c_1 c_3^2 (c_1 c_2^4 (9 c_2-2 c_3)+4 c_1^5 c_3-4 c_2^5 c_3+5 c_1^4 c_2 (-3 c_2+2 c_3)+c_1^3 c_2^2 (-11 c_2\\&+14 c_3)+c_1^2 c_2^3 (-7 c_2+18 c_3)),\\ T_{23}&= 60 c_3^4 (c_1^5-c_2^5-3 c_1^3 c_2 (c_2-4 c_3)+c_1^2 c_2^2 (17 c_2-4 c_3)-c_1 c_2^3 (5 c_2+2 c_3)\\&+c_1^4 (c_2+4 c_3)),\\ T_{24}&= 60 c_3^5 (2 c_2^4-(2 c_1^3+6 c_1^2 c_2-3 c_1 c_2^2+c_2^3) c_3). \end{aligned}$$