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New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

Alina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Maria Lukacova-Medvidova

TL;DR

The paper develops a robust semi-discrete finite-volume framework for hyperbolic PDEs with uncertainties by coupling second-order physical-space reconstruction with fifth-order Ai-WENO-Z interpolation in random space, effectively avoiding Gibbs phenomena. Numerical fluxes use central-upwind schemes and Gauss-Legendre quadrature in the random space, while well-balanced and positivity-preserving treatments are integrated for Saint-Venant equations. The approach is validated on Euler and Saint-Venant systems across 1-D and 2-D spatial cases with varying stochastic dimensions, demonstrating high accuracy in the random variables and stable, non-oscillatory behavior near discontinuities. This method offers a scalable alternative to spectral stochastic methods, with demonstrated gains in robustness and sharp statistical estimates in practical uncertainty quantification tasks, and points to future enhancements in handling high-dimensional randomness and parallel implementation.

Abstract

In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

TL;DR

The paper develops a robust semi-discrete finite-volume framework for hyperbolic PDEs with uncertainties by coupling second-order physical-space reconstruction with fifth-order Ai-WENO-Z interpolation in random space, effectively avoiding Gibbs phenomena. Numerical fluxes use central-upwind schemes and Gauss-Legendre quadrature in the random space, while well-balanced and positivity-preserving treatments are integrated for Saint-Venant equations. The approach is validated on Euler and Saint-Venant systems across 1-D and 2-D spatial cases with varying stochastic dimensions, demonstrating high accuracy in the random variables and stable, non-oscillatory behavior near discontinuities. This method offers a scalable alternative to spectral stochastic methods, with demonstrated gains in robustness and sharp statistical estimates in practical uncertainty quantification tasks, and points to future enhancements in handling high-dimensional randomness and parallel implementation.

Abstract

In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.
Paper Structure (38 sections, 3 theorems, 111 equations, 12 figures, 1 table)

This paper contains 38 sections, 3 theorems, 111 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. One-Dimensional Semi-Discrete Scheme
  3. Numerical Fluxes and Source Terms
  4. Reconstruction of the Point Values
  5. Reconstruction in Physical Space
  6. Interpolation in Random Space
  7. Internal points.
  8. Near the $\xi$-boundary points.
  9. Higher-Dimensional Extensions
  10. Case $d=1$ and $s=2$
  11. Numerical Fluxes and Source Terms
  12. Reconstruction of the Point Values
  13. Reconstruction in Physical Space.
  14. Interpolation in Random Space.
  15. Case $d=2$ and $s=1$
  16. ...and 23 more sections

Key Result

Theorem 6.2

Assume that at a certain time level, the discrete solution is at a "lake-at-rest" steady state, that is, Then, the RHS of the system of ODEs (2.2) vanishes and hence the proposed semi-discrete scheme (2.2), (2.4)--(2.7), (6.5) is WB.

Figures (12)

  • Figure 5.1: Example 1, Test 1: Mean, 95%-quantile, and standard deviation of $\rho$ for $\xi\sim{\cal U}(-1,1)$ and different $\sigma$.
  • Figure 5.2: Example 1, Test 1: Same as in Figure \ref{['fig51']}, but for $\xi\sim{\cal N}(0,1/36)$.
  • Figure 5.3: Example 1, Test 1: Mean, 95%-quantile, and standard deviation of $\rho u$, $p$, and $E$ for $\sigma=0.1$ and $\xi\sim{\cal U}(-1,1)$.
  • Figure 5.4: Example 1, Test 1: Same as in Figure \ref{['fig52']}, but for $\xi\sim{\cal N}(0,1/36)$.
  • Figure 5.5: Example 1, Test 2: Mean, 95%-quantile, and standard deviation of $\rho$, $\rho u$, $p$, and $E$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Remark 6.1
  • Theorem 6.2
  • proof
  • Theorem 6.3
  • Theorem 6.4