Multi-Order Monte Carlo IMEX hierarchies for uncertainty quantification in multiscale hyperbolic systems

TL;DR

This paper presents a new Multi-Order Monte Carlo method using IMEX Runge-Kutta integrators for efficient uncertainty quantification in multiscale hyperbolic PDEs, avoiding costly re-meshing.

Contribution

It introduces a multi-order hierarchy varying spatial and temporal discretizations within Monte Carlo, ensuring asymptotic-preserving properties and improved efficiency.

Findings

Significant error and variance reduction demonstrated in numerical experiments.

Method maintains asymptotic consistency across multiple scales.

Effective for hyperbolic systems with stiff relaxation and kinetic equations.

Abstract

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.