[petsc-users] ARKIMEX produces incorrect values

[Ed Bueler]

Dear PETSc --

I tried twice to make this an issue at the gitlab.com host site, but both
times got "something went wrong (500)".  So this is a bug report by
old-fashioned means.

I created a TS example,
https://github.com/bueler/p4pdes-next/blob/master/c/fix-arkimex/ex54.c at
my github, also attached.  It solves a 2D linear ODE
```
   x' + y' = 6 y
        y' = x
```
Pretty basic; the known exact solution is just exponentials.  The code
writes it as F(t,u,u')=G(t,u) and supplies all the pieces, namely
IFunction,IJacobian,RHSFunction,RHSJacobian.  Note both F and G must be
seen by TS to get the correct solution.  In summary, a boring (and
valgrind-clean ;-)) example.

For current master branch it runs fine for the fully-implicit methods (e.g.
BDF, CN, ROSW) which can use the IFunction F, including with
finite-differenced Jacobians.  With BDF2, BDF2+-snes_fd, BDF6+tight tol.,
CN, BEULER, ROSW:
$ ./ex54
error norm at tf = 1.000000 from 33 steps:  |u-u_exact| =  9.29170e-02
$ ./ex54 -snes_fd
error norm at tf = 1.000000 from 33 steps:  |u-u_exact| =  9.29170e-02
$ ./ex54 -ts_rtol 1.0e-14 -ts_atol 1.0e-14 -ts_bdf_order 6
error norm at tf = 1.000000 from 388 steps:  |u-u_exact| =  4.23624e-11
$ ./ex54 -ts_type beuler
error norm at tf = 1.000000 from 100 steps:  |u-u_exact| =  6.71676e-01
$ ./ex54 -ts_type cn
error norm at tf = 1.000000 from 100 steps:  |u-u_exact| =  2.22839e-03
$ ./ex54 -ts_type rosw
error norm at tf = 1.000000 from 21 steps:  |u-u_exact| =  5.64012e-03

But it produces wrong values with ARKIMEX:
$ ./ex54 -ts_type arkimex
error norm at tf = 1.000000 from 16 steps:  |u-u_exact| =  1.93229e+01

Neither tightening tolerance nor changing type (`-ts_arkimex_type`) helps
ARKIMEX.

Thanks!

Ed

PS  My book is at a late proofs stage, and out of my hands.  It should
appear SIAM Press in a couple of months.  In all the examples in my book,
only my diffusion-reaction system example using F(t,u,u') = G(t,u) is
broken.  Thus the motivation for a trivial ODE example as above.


-- 
Ed Bueler
Dept of Mathematics and Statistics
University of Alaska Fairbanks
Fairbanks, AK 99775-6660
306C Chapman
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