Stochastic Finite Volume Approximation with Clustering in the Parameter Space for the Forward Uncertainty Quantification of Differential Equations with Random Parameters

The uncertainty quantification (UQ) for mathematical models with random parameters is important for many science and engineering problems.

Forward UQ quantifies the impact of random parameters on the output of system.

In the current study, we propose a new stochastic finite volume (SFV) scheme by combining SFV with clustering algorithm in the parameter space such that each cluster can be regarded as a finite volume with implicit boundaries.

The advantage of SFV is that no specific form of the random variable is required such that discontinuous solutions and sharp interfaces can be accurately approximated.

Compared to classic SFV based on structured grids, the new SFV-cluster scheme extends SFV to parameter spaces of higher dimensions.

For demonstration and validation, we present the construction of SFV schemes for the Kraichnan-Orszag three-mode problem and the Buckley-Leverett equation.

The error analysis of SFV is extended and the new algorithm is validated by numerical experiments.