Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach

Abstract

In this paper, we consider Burgers' equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especially on stability, which is proven for CPR methods and the systems arising from the PC approach. Due to the usage of split-forms, the major challenge is to construct entropy stable numerical fluxes. For the first time, such numerical fluxes are constructed for all systems resulting from the PC approach for Burgers' equation. In numerical tests, we verify our results and show also the advantage of the given ansatz using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation equipped with an initial shock, demonstrates quite fascinating observations. The behaviour of the numerical solutions from several methods (finite volume, finite difference, CPR) differ significantly from each other. Through careful investigations, we conclude that the reason for this is the high sensitivity of the system to varying dissipation. Furthermore, it should be stressed that the system is not strictly hyperbolic with genuinely nonlinear or linearly degenerate fields.

References in corpus (6)

• Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations
• Shallow water equations: Split-form, entropy stable, well-balanced, and positivity preserving numerical methods
• A hyperbolicity-preserving stochastic Galerkin approximation for uncertain hyperbolic systems of equations
• Application of Modal Filtering to a Spectral Difference Method
• A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations
• A posteriori error analysis for random scalar conservation laws using the Stochastic Galerkin method

Cited by in corpus (16)

• High order entropy preserving ADER scheme
• Smooth and compactly supported viscous sub-cell shock capturing for Discontinuous Galerkin methods
• Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points
• Summation-by-parts operators for general function spaces
• Efficient iterative arbitrary high order methods: an adaptive bridge between low and high order
• Intrusive acceleration strategies for Uncertainty Quantification for hyperbolic systems of conservation laws
• Towards Stable Radial Basis Function Methods for Linear Advection Problems
• High Order Edge Sensors with ${\ell }^1$ Regularization for Enhanced Discontinuous Galerkin Methods
• Stabilizing Radial Basis Function Methods for Conservation Laws Using Weakly Enforced Boundary Conditions
• Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
• Multiresolution-analysis for stochastic hyperbolic conservation laws
• Summation-by-parts operators for general function spaces: The second derivative
• Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods
• Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
• Stochastic Discontinuous Galerkin Methods for Robust Deterministic Control of Convection Diffusion Equations with Uncertain Coefficients
• On the Connection between Residual Distribution Schemes and Flux Reconstruction