Along with exponential integrators, another innovative category of time integration methods for multiphysics problems characterized by multiple time scales are multirate time integrators. These methods use multiple time steps for integrating the system when the right-hand-side $F(t,U)$ can be partitioned into fast and slow components, i.e.,
$U'(t)=F(t,U)=F_{\text{fast}}(t,U)+F_{\text{slow}}(t,U)$.
The idea (originally due to Gear et al. (1984) is to use micro time steps to integrate the fast components and macro time steps to integrate slow components. So far, several types of multirate integrators have been proposed in the literature (both one-step and multistep methods), e.g., multirate Runge-Kutta methods, multirate Richardson extrapolation methods, multirate infinitesimal step (MIS) methods, multirate general-structure additive Runge-Kutta (GARK) methods (see Sandu and Günther, SINUM 2015). Constructing multirate methods usually requires fulfilling a set of coupling and decoupling order conditions. As the desired order of accuracy increases, the number of order conditions in this set increases dramatically and thus is much more complicated to solve for coefficients. Very recently, several fourth-order MIS and GARK schemes have been derived. The extension to higher-order methods, however, is not straightforward.
Contributions:
1. Derivation of a new class of multirate integrators for multiphysics problems:
I have developed a new class of multirate methods based on the newly constructed exponential Runge–Kutta schemes applied to semilinear differential equations with fast and slow dynamics. As a trade-off, this approach does not need to deal with a set of complicated coupling and decoupling order conditions like competing approaches. Consequently, new multirate methods of orders up to 5 have been derived in an elegant and systematic way. In collaboration with Daniel R. Reynolds (SMU) and Rujeko Chinomona (graduate student), we have shown that the new methods allow for both fast and slow components to be integrated in explicit-explicit and implicit-explicit manners. Moreover, in contrast to many previous works, we have investigated both situations where the fast (stiff) process does and does not contribute significantly to overall temporal error. For more details, see Luan, Chinomona & Reynolds: SIAM Journal on Scientific Computing, 42(2), A1245–A1268 (2020).
2. Construction of multirate exponential Rosenbrock (MERB) methods:
In another collaboration with Daniel R. Reynolds (SMU) and Rujeko Chinomona (Temple Uni.), I constructed multirate schemes by approximating the action of matrix φ-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified “fast” initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to providing how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods. For more details, see Luan, Chinomona & Reynolds: SIAM Journal on Scientific Computing 44 (5), A3265–A3289 (2022); link to arXiv version.
Numerical experiments:
a) Reaction-diffusion systems
b) One-way system with fast variables coupled into the slow
c) Two-way system with fast-slow coupling
