1,105 research outputs found
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
A posteriori analysis of discontinuous galerkin schemes for systems of hyperbolic conservation laws
In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
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