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

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

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows

A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines approach combining discontinuous Galerkin spatial discretization with single- or multi-step methods in time. The construction of the estimators requires a reconstruction in time for which we present a very general framework first for odes and then apply the approach to conservation laws. The reconstruction does not depend on the actual method used for evolving the solution in time. Most importantly it covers in addition to implicit methods also the wide range of explicit methods typically used to solve conservation laws. For the spatial discretization, we allow for standard choices of numerical fluxes. We use reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. We study under which conditions on the numerical flux the estimate is of optimal order pre-shock. While the estimator we derive is computable and valid post-shock for fixed meshsize, it will blow up as the meshsize tends to zero. This is due to a breakdown of the relative entropy framework when discontinuities develop. We conclude with some numerical benchmarking to test the robustness of the derived estimator