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Journal of Scientific Computing

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07-03-2019

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

Authors: Zhiwei Fang, Jichun Li, Tao Tang, Tao Zhou

Published in: Journal of Scientific Computing | Issue 1/2019

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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.

previous article Numerical Approximation of a Phase-Field Surfactant Model with Fluid Flow

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Title: Efficient Stochastic Galerkin Methods for Maxwell’s Equations with Random Inputs
Authors: Zhiwei Fang
Jichun Li
Tao Tang
Tao Zhou
Publication date: 07-03-2019
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-00936-z

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