I’m using an ODE Matlab integrator (Bulirsch-Stoer) that is (supposedly, I never used it before) quite good for solving ODEs with high accuracy. I’m using it to solve highly non-linear diff. eqs. of which one does not know the true solution and which are highly senstivie to initial conditions and numerical accuracy. To be sure that it does the right thing I integrate the curves back and forth and see whether the back-integration returns to the intial conditions. It does. Of course, if the time interval (the ODE’s time-parameter "t") is large enough one sees it diverging (i.e., it doesn’t return to the intital conditions.) This is normal, and expected. However, since the integrator’s accuracy also depends on three independent parameters (the error tolerance, and two internal loops, midpoint and nr. of segmentation points) and it is difficult to find the optimal setting in a large 3D parameter space, I’m wondering whether the divergence from a true solution at some time, say t_max, beyond which the integrator no longer funrishes the correct values, is due to a non-optimal setting of the accuracy parameters (that is, I could still increase the accuracy), or wheteher it has reached the precision that is limited by the internal 15 digits number representation (that is, I reached the max. accuracy, and can’t do anything about it as long I work with double precision.)
So, my question is: Is there a method to know when one has effectively reached the best numerical evaluation in solving an ODE inside the limitation of a double precision integratioon? A method that essentially tells me: "Yes, that’s the best integration, beyond which you can’t go with a 15 digits internal representation, no matter how you fine-tune your solver."
I hope that I expressed clearly what my issue is. Feel free to ask for more information.I’m using an ODE Matlab integrator (Bulirsch-Stoer) that is (supposedly, I never used it before) quite good for solving ODEs with high accuracy. I’m using it to solve highly non-linear diff. eqs. of which one does not know the true solution and which are highly senstivie to initial conditions and numerical accuracy. To be sure that it does the right thing I integrate the curves back and forth and see whether the back-integration returns to the intial conditions. It does. Of course, if the time interval (the ODE’s time-parameter "t") is large enough one sees it diverging (i.e., it doesn’t return to the intital conditions.) This is normal, and expected. However, since the integrator’s accuracy also depends on three independent parameters (the error tolerance, and two internal loops, midpoint and nr. of segmentation points) and it is difficult to find the optimal setting in a large 3D parameter space, I’m wondering whether the divergence from a true solution at some time, say t_max, beyond which the integrator no longer funrishes the correct values, is due to a non-optimal setting of the accuracy parameters (that is, I could still increase the accuracy), or wheteher it has reached the precision that is limited by the internal 15 digits number representation (that is, I reached the max. accuracy, and can’t do anything about it as long I work with double precision.)
So, my question is: Is there a method to know when one has effectively reached the best numerical evaluation in solving an ODE inside the limitation of a double precision integratioon? A method that essentially tells me: "Yes, that’s the best integration, beyond which you can’t go with a 15 digits internal representation, no matter how you fine-tune your solver."
I hope that I expressed clearly what my issue is. Feel free to ask for more information. I’m using an ODE Matlab integrator (Bulirsch-Stoer) that is (supposedly, I never used it before) quite good for solving ODEs with high accuracy. I’m using it to solve highly non-linear diff. eqs. of which one does not know the true solution and which are highly senstivie to initial conditions and numerical accuracy. To be sure that it does the right thing I integrate the curves back and forth and see whether the back-integration returns to the intial conditions. It does. Of course, if the time interval (the ODE’s time-parameter "t") is large enough one sees it diverging (i.e., it doesn’t return to the intital conditions.) This is normal, and expected. However, since the integrator’s accuracy also depends on three independent parameters (the error tolerance, and two internal loops, midpoint and nr. of segmentation points) and it is difficult to find the optimal setting in a large 3D parameter space, I’m wondering whether the divergence from a true solution at some time, say t_max, beyond which the integrator no longer funrishes the correct values, is due to a non-optimal setting of the accuracy parameters (that is, I could still increase the accuracy), or wheteher it has reached the precision that is limited by the internal 15 digits number representation (that is, I reached the max. accuracy, and can’t do anything about it as long I work with double precision.)
So, my question is: Is there a method to know when one has effectively reached the best numerical evaluation in solving an ODE inside the limitation of a double precision integratioon? A method that essentially tells me: "Yes, that’s the best integration, beyond which you can’t go with a 15 digits internal representation, no matter how you fine-tune your solver."
I hope that I expressed clearly what my issue is. Feel free to ask for more information. ode, numerical integration MATLAB Answers — New Questions

​