We present an a posteriori error analysis for one-dimensional random hyperbolic systems of conservation laws. For the discretization of the random space we consider the Non-Intrusive Spectral Projection method, the spatio-temporal discretization uses the Runge--Kutta Discontinuous Galerkin method. We derive an a posteriori error estimator using smooth reconstructions of the numerical solution, which combined with the relative entropy stability framework yields computable error bounds for the space-stochastic discretization error. Moreover, we show that the estimator admits a splitting into a stochastic and deterministic part.
Methods for quantifying the effects of uncertainties in hyperbolic problems can be divided into intrusive and non-intrusive techniques. Non-intrusive methods allow the usage of a given deterministic solver in a black-box manner, while being embarrassingly parallel. However, avoiding intrusive modifications of a given solver takes away the ability to use several inherently intrusive numerical acceleration tools. Moreover, intrusive methods are expected to reach a given accuracy with a smaller number of unknowns compared to non-intrusive techniques. This effect is amplified in settings with high…
In this article we study intrusive uncertainty quantification schemes for systems of conservation laws with uncertainty. Standard intrusive methods lead to oscillatory solutions which sometimes even cause the loss of hyperbolicity. We consider the stochastic Galerkin scheme, in which we filter the coefficients of the polynomial expansion in order to reduce oscillations. We further apply the multi-element approach and ensure the preservation of hyperbolic solutions through the hyperbolicity limiter. In addition to that, we study the intrusive polynomial moment method, which guarantees hyperboli…
The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution ha…
We present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in [52]. The technique is based on the reformulation of DG scheme as a finite volume (FV) like method through the definition of some specific numerical fluxes referred to as reconstructed fluxes. High-order DG numerical scheme combined with this new local subcell correction technique ensures positivity preservation of the solution, along with a low oscillatory and sharp shocks representation. The main idea of this correction procedure is to …
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online…
In this paper we blend high-order Compact Approximate Taylor (CAT) numerical methods with the a posteriori Multi-dimensional Optimal Order Detection (MOOD) paradigm to solve hyperbolic systems of conservation laws. The resulting methods are highly accurate for smooth solutions, essentially non-oscillatory for discontinuous ones, and almost fail-safe positivity preserving. Some numerical results for scalar conservation laws and systems are presented to show the appropriate behavior of CAT-MOOD methods.
A posteriori error analysis for evolution nonlinear Schrödinger equations up to the critical exponent
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；IACM–FORTH（应用数学研究所-希腊技术研究院）；Dept. of Math. & Applied Mathematics, Univ. of Crete（克里特大学数学与应用数学系）
A Posteriori Subcell Limiting of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws
University of Trento（特伦托大学）；CNRS and Institut de Mathématiques de Toulouse (IMT) Université Paul-Sabatier（法国国家科学研究中心与图卢兹数学研究所（IMT）保罗-萨巴蒂耶大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Surrogate-to-code verification of a non-intrusive POD-GPR machine-learning emulator of peak thermomechanical fields, with application to a carbon-carbon aircraft brake disc
organization= Independent Researcher , city= Paris , country= France
Residual-based a posteriori error estimates for a conforming finite element discretization of the Navier-Stokes/Darcy coupled problem
Université d’Abomey-Calavi (UAC)（阿波美-卡拉维大学）；African Institute for Mathematical Sciences (AIMS)（非洲数学科学研究所）；Institut de Mathématiques et de Sciences Physiques (IMSP)（数学与物理科学研究所）
We present an a posteriori shock-capturing finite volume method algorithm called GP-MOOD that solves a compressible hyperbolic conservative system at high-order solution accuracy (e.g., third-, fifth-, and seventh-order) in multiple spatial dimensions. The GP-MOOD method combines two methodologies, the polynomial-free spatial reconstruction methods of GP (Gaussian Process) and the a posteriori detection algorithms of MOOD (Multidimensional Optimal Order Detection). The spatial approximation of our GP-MOOD method uses GP's unlimited spatial reconstruction that builds upon our previous studies o…
In this paper, we propose a novel approximation strategy for time-dependent hyperbolic systems of conservation laws for the Euler system of gas dynamics that aims to represent the dynamics of strong interacting discontinuities. The goal of our method is to allow an approximation with a high-order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients, in particular across shocks. Following the Multidimensional Optimal Order Detection (MOOD) (Clain et al., Journal of Computational Physics 2011; Diot et al., Computer…
Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of th…
An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations
Well-balanced high order schemes on non-uniform grids and entropy residuals
Dipartimento di Scienza e Alta Tecnologia Università dell’Insubria（因苏布里亚大学科学与技术系）；Dipartimento di Matematica “G. Peano” Università di Torino（都灵大学G.佩阿诺数学系）
Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: quasi-conservative formulation with subcell finite volume corrections
