Third-order Symplectic Integration Method with Inverse Time Dispersion Transform for Long-term Simulation

Long-term problems are popular in various numerical simulations for understanding physical phenomena. However, conventional symplectic integration method can only handle a relatively small travel time and would exhibit very strong numerical artifacts in the presence of a long duration, which greatly pollute the modeling results.

We employ the inverse time dispersion transform to reduce these artifacts and the accuracy is significantly improved, especially for those long travel times. We use the inverse time dispersion transform as a post processing method, which seems to aggravate the computational cost; however, our numerical experiments show that the total computational efficiency after applying the new method is similar or even much higher than those of traditional schemes, since it allows us to use much larger temporal interval that is close to the upper limit given by stability conditions.

Our scheme is a general tool to improve the numerical simulations results by reducing the time-dispersion error, which allows us to obtain accurate simulations results at much longer travel times, thus is necessary and would be popular for long-term problems.