A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations

Abstract

Uncertainty Quantification through stochastic spectral methods is rising in popularity. We derive a modification of the classical stochastic Galerkin method, that ensures the hyperbolicity of the underlying hyperbolic system of partial differential equations. The modification is done using a suitable "slope" limiter, based on similar ideas in the context of kinetic moment models. We apply the resulting modified stochastic Galerkin method to the compressible Euler equations and the $M_1$ model of radiative transfer. Our numerical results show that it can compete with other UQ methods like the intrusive polynomial moment method while being computationally inexpensive and easy to implement.

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• Cross-mode Stabilized Stochastic Shallow Water Systems Using Stochastic Finite Element Methods
• New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties
• Hyperbolicity-Preserving and Well-Balanced Stochastic Galerkin Method for Shallow Water Equations
• Spline-Based Stochastic Collocation Methods for Uncertainty Quantification in Nonlinear Hyperbolic PDEs
• Parameter identification in uncertain scalar conservation laws discretized with the discontinuous stochastic Galerkin Scheme