Efficient Long-Time Simulations of Multiscale Systems via High-Order Numerical Homogenization

By a high-order numerical homogenization method, a heterogeneous multiscale scheme was developed in Jin & Li (2022) for evolving differential equations containing two time scales. In this paper, we further explore the technique to propose an efficient algorithm able to carry out simulations up to a long time which was prohibitive before. The new algorithm is a multigrid-in-time method which combines coarse-grid high-order approximations with fine-grid low-order evaluations. The high efficiency is attained by minimizing the computational cost while the approximation accuracy is guaranteed. A priori error estimates are rigorously established and validated by numerical examples.