A stochastic Galerkin method for the Euler equations with Roe variable transformation

Abstract

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square-roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables and no auxiliary quantities are needed, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

“…Standard quadrature techniques, often in combination with sparse grid techniques [8] can be used to obtain the statistics of interest. Intrusive methods [9,10,11,12,13,14,15,16] are based on polynomial chaos expansions leading to a systems of equations for the expansion coefficients. This implies that new specific non-deterministic codes must be developed.…”

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

“…4.3, where the same peak and valley values are observed in the deterministic problems with different w. and Γ = 1.4. The same setup is considered in [20,8,21,6]. The gPC-SG method may easily fail in this test due to the appearance of negative density caused by the oscillations [20].…”

“…Moreover, a minimization problem needs be solved at each spatial mesh point and time step, and thus the method is time-consuming, especially for multi-dimensional problems. An approach using the Roe variables was proposed for the Euler equations in [21]. Although effective, its extension to general systems is also very limited, due to the Roe linearization.…”

An intrusive hybrid method for discontinuous two-phase flow under uncertainty

“…In general, obtaining accurate eigenvalue estimates may be computationally costly. However, for the piecewise constant and piecewise linear multiwavelet expansion, we have explicit expressions for the system eigenvalues due to the constant eigenvectors of the inner triple product matrices A given by (12), see [25].…”