17 research outputs found
A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
We are interested in the approximation of a steady hyperbolic problem. In
some cases, the solution can satisfy an additional conservation relation, at
least when it is smooth. This is the case of an entropy. In this paper, we
show, starting from the discretisation of the original PDE, how to construct a
scheme that is consistent with the original PDE and the additional conservation
relation. Since one interesting example is given by the systems endowed by an
entropy, we provide one explicit solution, and show that the accuracy of the
new scheme is at most degraded by one order. In the case of a discontinuous
Galerkin scheme and a Residual distribution scheme, we show how not to degrade
the accuracy. This improves the recent results obtained in [1, 2, 3, 4] in the
sense that no particular constraints are set on quadrature formula and that a
priori maximum accuracy can still be achieved. We study the behavior of the
method on a non linear scalar problem. However, the method is not restricted to
scalar problems
Summation-by-parts operators for general function spaces: The second derivative
Many applications rely on solving time-dependent partial differential
equations (PDEs) that include second derivatives. Summation-by-parts (SBP)
operators are crucial for developing stable, high-order accurate numerical
methodologies for such problems. Conventionally, SBP operators are tailored to
the assumption that polynomials accurately approximate the solution, and SBP
operators should thus be exact for them. However, this assumption falls short
for a range of problems for which other approximation spaces are better suited.
We recently addressed this issue and developed a theory for first-derivative
SBP operators based on general function spaces, coined function-space SBP
(FSBP) operators. In this paper, we extend the innovation of FSBP operators to
accommodate second derivatives. The developed second-derivative FSBP operators
maintain the desired mimetic properties of existing polynomial SBP operators
while allowing for greater flexibility by being applicable to a broader range
of function spaces. We establish the existence of these operators and detail a
straightforward methodology for constructing them. By exploring various
function spaces, including trigonometric, exponential, and radial basis
functions, we illustrate the versatility of our approach. We showcase the
superior performance of these non-polynomial FSBP operators over traditional
polynomial-based operators for a suite of one- and two-dimensional problems,
encompassing a boundary layer problem and the viscous Burgers' equation. The
work presented here opens up possibilities for using second-derivative SBP
operators based on suitable function spaces, paving the way for a wide range of
applications in the future.Comment: 20 pages, 7 figure
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