What are the implicit methods for solving a rigid system that has been discretized?

I have a continuous system where I can only solve it through the functions for rigid matlab systems (ode15s, ode23s, ode23t, ode23tb). So I discretized my model by the progressive finite difference method and I want to get the solution of it to compare with the solution of the continuous model. I obtained through the link https://www.mathworks.com/matlabcentral/answers/98293-is-there-a-fixed-step-ordinary-differential-equation-ode-solver-in-matlab-8-0-r2012b#answer_107643 informed by Walter Roberson https://www.mathworks.com/matlabcentral/answers/411195-implement-ode15s-function-in-a-discrete-model?s_tid=prof_contriblnk these functions. However they are explicit methods for non-rigid systems. I would like to get a method that solves my rigid discretized system. Can anyone help?

The problem codes are attached to the second link in the body of the text. Thank you very much for the help.