Articlebronze2010

Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed

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43 citations0/0 files storedCommunications in Computational Physics

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Probabilistic and Robust Engineering Design Model Reduction and Neural Networks Numerical methods for differential equations

Abstract

For a simple model of a scalar wave equation with a random wave speed, Gottlieb and Xiu [Commun. Comput. Phys., 3 (2008), pp. 505-518] employed the generalized polynomial chaos (gPC) method and demonstrated that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. Consequently, a consistent method of imposing the boundary conditions is proposed and its convergence is established under the assumption that the expansion coefficients decay fast asymptotically. In this work, we investigate stochastic collocation methods for the same type of scalar wave equation with random wave speed. It will be demonstrated that the rate of convergence depends on the regularity of the solutions; and the regularity is determined by the random wave speed and the initial and boundary data. Numerical examples are presented to support the analysis and also to show the sharpness of the assumptions on the relationship between the random wave speed and the initial and boundary data. An accuracy enhancement technique is investigated following the multi-element collocation method proposed by Foo, Wan and Karniadakis [J. Comput. Phys., 227 (2008), pp. 9572-9595].

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References

1991Stochastic Finite Elements: A Spectral Approach 2005High-Order Collocation Methods for Differential Equations with Random Inputs 2007A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data 2009Fast Numerical Methods for Stochastic Computations: A Review 2007Sparse grid collocation schemes for stochastic natural convection problems 2007Efficient Collocational Approach for Parametric Uncertainty Analysis 2009A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems 2008The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

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2015Analysis and computation of the elastic wave equation with random coefficients 2015A Stochastic Galerkin Method for Hamilton--Jacobi Equations with Uncertainty 2011Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials 2019Data-driven polynomial chaos expansions: A weighted least-square approximation

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