EPODE

Test problems

Tests were done with success for approx. 100 problems, including the stiff ODEs from the stiff classification provided by Enright, W.H., Hull T.E., Lindberg B., BIT, 1975,cite from Hairer E., Wanner G., Solving ODEs I. and II.; and other sources;

The most popular classification of the stiff systems is due to Enright:

Class A: the linear systems with real matrix eigenvalues;
Class B: the linear systems with complex matrix eigenvalues;
Class C: the systems with nonlinear coupling between the smooth and stiff solution components;
Class D: the nonlinear systems with a real \sc Jacobian's matrix eigenvalues;
Class E: the nonlinear systems with a complex \sc Jacobian's matrix eigenevalues.

For proving the efficiency of a new method designed for stiff problems, the method must integrate with success all the stiff systems from STIFF-DETEST package.

In the next Tables we present the classical stiff systems. Note S the ratio between the maximum and minimum absolute eigenvalues evaluated at the starting point. The second notation correspond to that of STIFF-DETEST package.

Class A:

`No.`	`System`	`T`	`y(0)`	`S`
S1 (A1)	`y₁'=-0.5y₁, y₂'=-y₂, y₃'=-100y₃, y₄'=-90y₄`	`20`	`1, 1, 1, 1`	`2` x `10²`
S2	`y'₁=998y₁+1998y₂, y₂'=-999y₁-1999y₂`	`100`	`1, 0`	`10³`
S3	`y₁'=-10⁶y₁+0.075y₂, y₂'=7500y₁-0.075y₂`	`10`	`1, -1`	`7.4` x `10⁴`
S4 (A2)	`y₁'=-1800y₁+900y₂, y_i'=y_i-1-2y_i+y_i+1, i=2, ..., 8, y₉'=1000y₈-2000y₉+10000`	`120`	`0, ... , 0`	`2` x `10⁴`
S5 (A3)	`y₁'=-10⁴y₁+100y₂-10y₃+y₄, y₂'=-10³y₂+10y₃-10y₄, y₃'=-y₃+10y₄, y₄'=-0.1y₄`	`20`	`1, 1, 1, 1`	`10⁵`
S6 (A4)	`y_i'=-i⁵y_i, i=1, ..., 10`	`1`	`1, ..., 1`	`10⁵`
S7	`y₁'=-k₁y₁+k₂y₃, y₂'=-k₄y₂+k₃y₃, y₃'=k₁y₁+k₄y₂-(k₂+k₃)y₃, k₁=8.43` x `10^-10, k₂=2.9` x `10¹¹, k₃=2.46` x `10¹⁰, k₄=8.76` x `10^-6`	`8` x `10⁵`	`0, 1, 0`	`3.146` x `10¹¹`

Class B:

`No.`	`System`	`T`	`y(0)`	`S`
S8 (B1)	`y₁'=-y₁+y₂, y₂'=-100y₁-y₂, y₃'=-100y₃+y₄, y₄'=-10000y₃-100y₄`	`20`	`1, 0, 1, 0`	`10²`
S9, (B2-B5)	`y₁'=-10y₁+a y₂, y₂'=-a y₁-10y₂, y₃'=-4y₃, y₄'=-y₄, y₅'=-0.5y₅, y₆'=-0.1y₆a=3, 8, 25, 100`	`20`	`0, 1, 1, 1, 1, 1`	`10²`

Stiff perturbated linear systems from Classes A and B.

`No.`	`System`	`T`	`y(0)`	`S`
S10	`y'=-200(y-E)+E', E(t)=10-(10+t)e^-t`	`100`	`10`	`2` x `10²`
S11	`y'=ay+ade^dt, a=10^-4, d=i` in `C-R`	`100`	`y₀`	`10³`
S12	`y₁'=-6y₁+5y₂+2 sin t, y₂'=94y₁-95y₂`	`100`	`0, 0`	`10³`
S13	`y₁'=-4498y₁-5996y₂+0.006-t, y₂'=2248.5y₁+2997y₂-0.503+3t`	`25`	`25498/ 1500, -16499/ 1500`
`1.5` x `10³`
S14	`y₁'=-2000y₁+1000y₂+1, y₂'=y₁-y₂`	`100`	`0, 0`	`4` x `10³`
S15	`y'=UBUy+UD, u_ii=-1/ 3 , u_ij=2/ 3, i# j, B=[(-100, 1000, 0), (-1000, -100, 0), (0, 0, -0.1)], D=(0, 0, 0, 1)^T`	`100`	`-1/ 3, 2/ 3, 2/ 3`	`10³`
S16	`y'=UBU^Ty+Uf(t), u_ii=-1/ 2, u_ij=1/ 2, i# j, B=[(0, 1, 0, 0), (-1, 0, 0, 0), (0, 0, -100, -900), (0, 0, 900, -100)], f(t)=(t²+2t, t²-2t, -800t+1, -1000t-1)^T`	`1`	`1, 0, 0, 1`	`10²`

Stiff linear systems with variable coefficients from Class A.

`No.`	`System`	`T`	`y(0)`	`S`
S17	`y₁'=-(80+{1 / 3}(1+t)^-1)y₁ -(40-{2 / 5}(1+t)^-1)y₂, y₂'=-(40-{2 / 5}(1+t)^-1)y₁-(20+{4 / 5}(1+t)^-1)y₂`	100	0, 1	`10²`
S18, (D1)	`y₁'=0.2(y₂-y₁), y₂'=10y₁-(60-0.125t)y₂+0.125t`	`400`	`0, 0`	`3.53` x `10²`

Stiff nonlinear systems from Class C.

`No.`	`System`	`T`	`y(0)`	`S`
S19, (C1)	`y₁'=-y₁+y₂²+y₃²+y₄², y₂'=-10y₂+10(y₃²+y₄²), y₃'=-40y₃+40y₄², y₄'=-100y₄+2`	`20`	`1, 1, 1, 1`	`10²`
S20, (C2)	`y₁'=-y₁+2, y₂'=-10y₂+0.1y₁², y₃'=-40y₃+4(y₁²+y₂²), y₄'=-100y₄+y₁²+y₂²+y₃²`	`20`	`1, 1, 1, 1`	`10²`

Stiff nonlinear systems from Class D.

`No.`	`System`	`T`	`y(0)`	`S`
S21,	`y_i'=-b_iy_i+y_i², (b_i)_i=1,...,4=(1000, 800, -10, 0.001)^T`	`20`	`-1, -1, -1, -1`	`10⁶`
S22, (D2)	`y₁'=-0.04y₁+0.01y₂y₃, y₂'=400y₁-100y₂y₃-3000y₂², y₃'=30y₂²`	`40`	`1, 0, 0`	`10⁵`
S23,	`y₁'=-0.04y₁+10⁴y₂y₃, y₂'=0.04y₁-10⁴y₂y₃-3` x `10⁷y₂², y₃'=3` x `10⁷y₂²`	`40`	`1, 0, 0`	`10⁵`
S24, (D3)	`y₁'=y₃-100y₁y₂, y₂'=y₃+2y₄-100y₁y₂-2` x `10⁴y₂², y₃'=-y₃+100y₁y₂, y₄'=-y₄+10⁴y₂²`	`20`	`1, 1, 0, 0`	`4` x `10²`
S25, (D4)	`y₁'=-0.013y₁-1000y₁y₃, y₂'=-2500y₂y₃, y₃'=-0.013y₁-1000y₁y₃-2500y₂y₃`	`50`	`1, 1, 0`	`3` x `10⁵`
S26, (D5)	`y₁'=0.01-(0.01+y₁+y₂)` x `,` x `(y₁²+1001y₁+1001), y₂'=0.01-(0.01+y₁+y₂)(1+y₂²)`	`100`	`0, 0`	`10⁶`
S27, (D6)	`y₁'=-y₁+10⁸y₃(1-y₁), y₂'=-10y₂+3` x `10⁷y₃(1-y₂), y₃'=-y₁'-y₂'`	`1`	`1, 0, 0`	`3` x `10⁷`

Stiff nonlinear systems from Class E.

`No.`	`System`	`T`	`y(0)`	`S`
S28, (E2)	`y₁'=y₂, y₂'=5(1-y₁²)y₂-y₁`	`1`	`2, 0`	`5.2` x `10⁶`
S29, (E3)	`y₁'=-(55+y₃)y₁+65y₂, y₂'=0.0785(y₁-y₂), y₃'=0.1y₁`	`500`	`1, 1, 0`	`10⁴`
S30, (E4)	`y'=-By+U^T(z₁²/2-z₂²/2, z₁z₂, z₃², z₄²]^T , z=Uy, , B=U[(b₁, -b₂, 0, 0), (b₂, b₁, 0, 0), (0, 0, b₃, 0), (0, 0, 0, b₄)]U, u_ij=1/ 2, i# j, u_ii=-1/ 2, , b₁=-b₂=-10, , b₃=1000, b₄=0.01`	`1`	`0, -2, -1, -1`	`10⁶`
S31, (E5)	`y₁'=-7.89` x `10^-10y₁-1.1` x `10⁷ y₁y₃, y₂'=7.89` x `10^-10y₁-1.13` x `10⁹y₂y₃, y₃'=7.89` x `10^-10y₁-1.1` x `10⁷y₁y₃+1.13` x `10³ y₄-1.13` x `10⁵ y₂y₃, y₄'=1.1` x `10⁷y₁y₃-1.13` x `10³y₄`	`1000`	`0.002, 0, 0, 0`	`2.7` x `10⁷`
S32,	`y₁'=10⁴y₁y₃+10⁴y₂y₄, y₂'=-10⁴y₁y₄+10⁴y₂y₃, y₃'=1-y₃, y₄'=-y₄-0.5y₃+0.5`	`10`	`1, 1, -1, 0`	`10⁴`

Some stiff nonlinear systems which model real systems.

`No.`	`System`	`T`	`y(0)`
S33	`y₁'=-10⁴y₁+y₂⁴-2y₃²-y₄²-y₅, y₂'=-1/ 2y₂+y₁-y₃², y₃'=-0.01y₂², y₄'=-y₃+y₁³-y₅³, y₅'=-y₁-y₃y₄`	`1`	`1, 10, 1, 1, 1`
S34	`y₁'=-y₁-y₁y₂+e ky₂, e y₂'=y₁-y₁y₂-e ky₂, e=98, k=3`	`100`	`1, 1`
S35	`y₁'=y₂, y₂'=((1-y₁²)\|1-y₁²\|^1/2y₂-y₁)/e, e=10^-6`	`11`	`2, 0`
S36	`y₁'=-(b+ai)y₁+by₂ⁱ, y₂'=y₁-ay₂-y₂ⁱ, , a=0.1, c=1, b=10², i=4`	`10`	`cⁱ, c`
S37	`y₁'=250[(-0.0048(y₃-660.2)-, -0.032(y₅-273.9)-1)y₁+y₂], y₂'=0.1(y₁-y₂), y₃'=93y₁-0.26(y₃-y₄), y₄'=0.87(y₃-y₄)-11(y₄-y₅), y₅'=1.8(y₄-y₅)-13(y₅-270)`	`1`	`1, 1, 660.2, 302.2, 273.9`
S38	`y₁'=100y₁/y₂(y₃-y₁), y₂'=-100(y₃-y₁), y₃'=1/y₄ [0.9-1000(y₃-y₅)-100y₃(y₃-y₁)], y₄'=100(y₃-y₁), y₅'=-100(y₅-y₃), , a=0.99026, 0.99, 0.9`	`1`	`1, 1, 1, -10, a`

Other stiff nonlinear systems which model real systems.

`No.`	`System`	`T`	`y(0)`
S39	`y₁'=77.27[y₂+y₁(1-8.375` x `10^-6y₁-y₂)], y₂'=[y₃-(1+y₁)y₂]/77.27, y₃'=0.161(y₁-y₃)`	`300`	`3, 1, 2`
S40	`y₁'=-1.71y₁+0.43y₂+8.32y₃+7` x `10^-4, y₂'=1.71y₁-8.75y₂, y₃'=-10.03y₃+0.43y₄+0.035y₅, y₄'=8.32y₂+1.71y₃-1.12y₄, y₅'=-1.745y₅+0.43y₆+0.43y₇, y₆'=-280y₆y₈+0.69y₄+1.71y₅-0.43y₆+0.69y₇, y₇'=280y₆y₈-1.81y₇, y₈'=-y₇'`	`400`	`1, 0, 0, 0, 0, 0, 0, 0.0057`
S41	`y'₁=-(2+e^-1)y₁+e^-1y₂², y₂'=y₁-y₂(1+y₂), e=10^-8`	`4`	`1, 1`
S42	`y₁'=-k₁y₁-k₂y₁y₂, y₂'=k₁y₁+k₃y₃-k₄y₂y₄-2k₅y₂², y₃'=k₂y₁y₂-k₃y₃, y₄'=-k₄y₂y₄, , k₁=10^-4, k₂=2.9` x `10⁴, k₃=5` x `10³, k₄=10⁴, k₅=6.7` x `10¹⁰`	`20`	`0.6, 0, 0, 0.4`
S43	`y₁'=1.30(y₃-y₁)+2.13` x `10⁶k(y₁)y₂, y₂'=1.80` x `10³ [y₄-y₂(1+k(y₁))], y₃'=1752-269y₃+267y₁, y₄'=0.1+320y₂-321y₄, k(y₁)=0.006` x `e^{20.7-15000/y₁}`	`1`	`761, 0, 600, 0.1`