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A MultiOrder Discontinuous Galerkin Monte Carlo Method for Hyperbolic Problems with Stochastic Parameters

Research paper by Mohammad Motamed, Daniel Appelö

Indexed on: 01 Jun '18Published on: 01 Feb '18Published in: SIAM journal on numerical analysis



Abstract

SIAM Journal on Numerical Analysis, Volume 56, Issue 1, Page 448-468, January 2018. We present a new multiorder Monte Carlo algorithm for computing the statistics of stochastic quantities of interest described by linear hyperbolic problems with stochastic parameters. The method is a nonintrusive technique based on a recently proposed high-order energy-based discontinuous Galerkin method for the second-order acoustic and elastic wave equations. The algorithm is built upon a hierarchy of degrees of polynomial basis functions rather than a mesh hierarchy used in multilevel Monte Carlo. Through complexity theorems and numerical experiments, we show that the proposed multiorder method is a valid alternative to the current multilevel Monte Carlo method for hyperbolic problems. Moreover, in addition to the convenience of working with a fixed mesh, which is desirable in many real applications with complex geometries, the multiorder method is particularly beneficial in reducing errors due to numerical dispersion in long-distance propagation of waves. The numerical examples verify that the multiorder approach is faster than the mesh-based multilevel approach for waves that traverse long distances.