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High-Order Finite-Volume Methods for Hyperbolic PDEs with Uncertainties


Core Concepts
The authors develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The methods are realized in the semi-discrete finite-volume framework and use fifth-order weighted essentially non-oscillatory (WENO) interpolations in the random space combined with second-order piecewise linear reconstruction in the physical space.
Abstract
The paper focuses on developing efficient and robust numerical methods for hyperbolic systems of conservation and balance laws with uncertainties. The authors propose a new approach that combines a semi-discrete finite-volume framework with high-order WENO interpolations in the random space. Key highlights: The methods achieve high-order accuracy without suffering from the Gibbs phenomenon, which can occur with spectral approximations in the random space. The authors implement a second-order finite-volume method in the physical space, coupled with a high-order WENO interpolation in the random space. The numerical fluxes are computed using the Riemann-problem-solver-free central-upwind (CU) fluxes, and the generalized minmod reconstruction is used in the physical space. The fifth-order Gauss-Legendre quadrature and the fifth-order affine-invariant WENO-Z (Ai-WENO-Z) interpolation are employed in the random space. The authors also develop a one-sided Ai-WENO-Z interpolation near the boundaries in the random space, where no boundary conditions are imposed. The proposed methods are tested on numerical examples for the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations, demonstrating their high-order accuracy, robustness, and ability to preserve well-balanced and positivity-preserving properties.
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Deeper Inquiries

How can the proposed methods be extended to handle more general types of uncertainties, such as non-Gaussian distributions or correlated random variables

The proposed methods can be extended to handle more general types of uncertainties by adapting the interpolation techniques in the random space to accommodate non-Gaussian distributions or correlated random variables. For non-Gaussian distributions, the interpolation schemes can be modified to capture the characteristics of the specific distribution, such as skewness and kurtosis. This may involve using different interpolation weights or modifying the reconstruction algorithms to better represent the non-Gaussian nature of the uncertainties. In the case of correlated random variables, the interpolation in the random space can be adjusted to account for the dependencies between the variables. This could involve developing new interpolation methods that consider the joint probability distribution of the correlated variables and incorporate this information into the reconstruction process. By incorporating these modifications, the high-order numerical methods can effectively handle a wider range of uncertainty types beyond simple Gaussian distributions.

What are the potential challenges and limitations of the WENO-based approach compared to other stochastic methods, such as generalized polynomial chaos (gPC) expansions, when dealing with high-dimensional random spaces

While the WENO-based approach offers advantages in terms of high-order accuracy and non-oscillatory behavior, there are potential challenges and limitations compared to other stochastic methods like generalized polynomial chaos (gPC) expansions, especially in high-dimensional random spaces. One challenge is the computational cost associated with the high-order WENO interpolations, especially in higher dimensions where the number of random variables increases. The complexity of the interpolation schemes and the computational resources required to implement them may pose challenges in terms of efficiency and scalability. Additionally, the WENO-based approach may face limitations in capturing complex probability distributions or handling highly correlated random variables. The interpolation techniques used in WENO methods may struggle to accurately represent the intricate relationships between variables in high-dimensional spaces, leading to potential inaccuracies in the numerical solutions. Furthermore, the stability and convergence properties of WENO-based methods in high-dimensional stochastic problems need to be carefully analyzed and validated, as the complexity of the random space can impact the overall numerical performance of the approach.

Can the ideas presented in this work be applied to other types of partial differential equations, such as elliptic or parabolic problems with uncertainties

The ideas presented in this work can be applied to other types of partial differential equations with uncertainties, such as elliptic or parabolic problems, by adapting the numerical methods to suit the specific characteristics of these equations. For elliptic problems, where the solutions are smooth and typically involve boundary value conditions, the high-order numerical methods can be tailored to ensure accurate representation of uncertainties in the boundary conditions or coefficients of the equations. The interpolation techniques in the random space can be adjusted to handle the steady-state nature of elliptic problems and provide reliable solutions in the presence of uncertainties. Similarly, for parabolic problems involving time-dependent phenomena, the numerical methods can be extended to incorporate uncertainties in the initial conditions or time-dependent coefficients of the equations. The time integration schemes can be modified to account for the stochastic nature of the problem and ensure robust solutions over time. Overall, the concepts of high-order numerical methods and stochastic interpolation presented in this work can be adapted and applied to a variety of partial differential equations with uncertainties, providing a versatile framework for addressing stochasticity in different types of PDEs.
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