Practical improvements to SIMPR codes for stiff ODEs (Q1124287)

The paper discusses codes using extrapolation of the semi-implicit mid- point rule for the solution of stiff ordinary differential equations (ODEs). Some comparisons are made between applying such a code to systems of equations in autonomous form or non-autonomous form. Arguments in favour of using the autonomous form are given and numerical evidence is also presented that prior conversion of the ODEs to autonomous form may produce significantly better results. A way of writing the code so that this conversion is not required of the user is derived. It is argued that methods of extrapolation type can have problems with a stepsize which is out of scale. A stability monitor is derived for the purpose of rejecting such a step as early as possible in the interests of efficiency and of avoiding more serious difficulties. Some tests using the new monitor are described.