Fast integrators for dynamical systems with several temporal scales
This work addresses the computational bottleneck of simulating multiscale dynamical systems with more than two scales, offering a scalable solution for scientists and engineers.
The paper proposes a fast integrator for dynamical systems with multiple temporal scales, achieving computational complexity linear in the number of scales, unlike exponential growth with iterated two-scale methods. Numerical tests on dissipative and oscillatory problems, including multiscale PDEs, demonstrate its efficiency.
We propose a fast integrator to a class of dynamical systems with several temporal scales. The proposed method is developed as an extension of the variable step size Heterogeneous Multiscale Method (VSHMM), which is a two-scale integrator developed by the authors. While iterated applications of multiscale integrators for two different scales increase the computational complexity exponentially as the number of different scales increases, the proposed method, on the other hand, has computational complexity linearly proportional to the number of different scales. This efficiency is achieved by solving different scale components of the vector fields with variable time steps. It is shown that variable time stepping of different force components has an effect of fast integration for the effective force of the slow dynamics. The proposed fast integrator is numerically tested on problems with several different scales which are dissipative and highly oscillatory including multiscale partial differential equations with sparsity in the solution space.