1,115 research outputs found
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
Stable High Order Quadrature Rules for Scattered Data and General Weight Functions
Numerical integration is encountered in all fields of numerical analysis and
the engineering sciences. By now, various efficient and accurate quadrature
rules are known; for instance, Gauss-type quadrature rules. In many
applications, however, it might be impractical---if not even impossible---to
obtain data to fit known quadrature rules. Often, experimental measurements are
performed at equidistant or even scattered points in space or time. In this
work, we propose stable high order quadrature rules for experimental data,
which can accurately handle general weight functions.Comment: Accepted at SINU
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