Multi-Order Monte Carlo IMEX hierarchies for uncertainty quantification in multiscale hyperbolic systems
This provides a more efficient approach for uncertainty quantification in multiscale hyperbolic systems like stiff relaxation flows, addressing computational challenges of traditional methods.
The paper tackles uncertainty quantification in multiscale hyperbolic systems by introducing a Multi-Order Monte Carlo method that uses Implicit-Explicit Runge-Kutta integrators to vary spatial and temporal discretization orders, achieving substantial error and variance reduction while maintaining asymptotic consistency.
We introduce a novel Multi-Order Monte Carlo approach for uncertainty quantification in the context of multiscale time-dependent partial differential equations. The new framework leverages Implicit-Explicit Runge-Kutta time integrators to satisfy the asymptotic-preserving property across different discretization orders of accuracy. In contrast to traditional Multi-Level Monte Carlo methods, which require costly hierarchical re-meshing, our method constructs a multi-order hierarchy by varying both spatial and temporal discretization orders within the Monte Carlo framework. This enables efficient variance reduction while naturally adapting to the multiple scales inherent in the problem ensuring asymptotic consistency. The proposed method is particularly well-suited for hyperbolic systems with stiff relaxation, kinetic equations, and low Mach number flows, where standard Multi-Level Monte Carlo techniques often encounter computational challenges. Numerical experiments demonstrate that the novel Multi-Order Monte Carlo approach achieves substantial reduction of both error and variance while maintaining asymptotic consistency in the asymptotic limit.