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Efficient Stochastic Galerkin Methods for Maxwell’s Equations with Random Inputs

Abstract

In this paper, we are concerned with the stochastic Galerkin methods for time-dependent Maxwell’s equations with random input. The generalized polynomial chaos approach is first adopted to convert the original random Maxwell’s equation into a system of deterministic equations for the expansion coefficients (the Galerkin system). It is shown that the stochastic Galerkin approach preserves the energy conservation law. Then, we propose a finite element approach in the physical space to solve the Galerkin system, and error estimates is presented. For the time domain approach, we propose two discrete schemes, namely, the Crank–Nicolson scheme and the leap-frog type scheme. For the Crank–Nicolson scheme, we show the energy preserving property for the fully discrete scheme. While for the classic leap-frog scheme, we present a conditional energy stability property. It is well known that for the stochastic Galerkin approach, the main challenge is how to efficiently solve the coupled Galerkin system. To this end, we design a modified leap-frog type scheme in which one can solve the coupled system in a decouple way—yielding a very efficient numerical approach. Numerical examples are presented to support the theoretical finding.

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Correspondence to Jichun Li.

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Jichun Li: Work partially supported by NSF Grant DMS-1416742 and NSFC Project 11671340. Tao Tang: Work supported by the NSF of China (under the Grant No. 11731006) and the Science Challenge Project (No. TZ2018001). Tao Zhou: Work supported by the NSF of China (under Grant Nos. 11822111, 11688101, 91630203, 11571351, 11731006), the Science Challenge Project (No. TZ2018001), the National Key Basic Research Program (No. 2018YFB0704304), NCMIS, and the Youth Innovation Promotion Association (CAS).

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Fang, Z., Li, J., Tang, T. et al. Efficient Stochastic Galerkin Methods for Maxwell’s Equations with Random Inputs. J Sci Comput 80, 248–267 (2019). https://doi.org/10.1007/s10915-019-00936-z

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  • DOI: https://doi.org/10.1007/s10915-019-00936-z

Keywords

  • Maxwell’s equations
  • Finite element method
  • Random inputs
  • Polynomial chaos methods
  • Stochastic Galerkin

Mathematics Subject Classification

  • 65N30
  • 35L15
  • 78-08