1,276 research outputs found
Calibration-Based ALE Model Order Reduction for Hyperbolic Problems with Self-Similar Travelling Discontinuities
Full text linkWe propose a novel Model Order Reduction framework that is able to handle solutions
of hyperbolic problems characterized by multiple travelling discontinuities. By means of
an optimization based approach, we introduce suitable calibration maps that allow us to
transform the original solution manifold into a lower dimensional one. The novelty of the
methodology is represented by the fact that the optimization process does not require the
knowledge of the discontinuities location. The optimization can be carried out simply by
choosing some reference control points, thus avoiding the use of some implicit shock tracking
techniques, which would translate into an increased computational effort during the offline
phase. In the online phase, we rely on a non-intrusive approach, where the coefficients of
the projection of the reduced order solution onto the reduced space are recovered by means
of an Artificial Neural Network. To validate the methodology, we present numerical results
for the 1D Sod shock tube problem, for the 2D double Mach reflection problem, also in the
parametric case, and for the triple point problem
Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs
Full text linkIn this work, we propose viable and efficient strategies for stabilized parametrized advection dominated problems, with random inputs. In particular, we investigate the combination of wRB (weighted reduced basis) method for stochastic parametrized problems with stabilized reduced basis method, which is the integration of classical stabilization methods (SUPG, in our case) in the Offline--Online structure of the RB method. Moreover, we introduce a reduction method that selectively enables online stabilization; this leads to a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We present numerical test cases to assess the performance of the proposed methods in steady and unsteady problems related to heat transfer phenomena
Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs
Full text linkIn this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown
A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity
Full text linkThe deferred correction (DeC) is an iterative procedure, characterized by increasing the accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in the Runge-Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts
On improving the efficiency of ADER methods
Full text linkThe (modern) arbitrary derivative (ADER) approach is a popular technique for
the numerical solution of differential problems based on iteratively solving an
implicit discretization of their weak formulation. In this work, focusing on an
ODE context, we investigate several strategies to improve this approach. Our
initial emphasis is on the order of accuracy of the method in connection with
the polynomial discretization of the weak formulation. We demonstrate that
precise choices lead to higher-order convergences in comparison to the existing
literature. Then, we put ADER methods into a Deferred Correction (DeC)
formalism. This allows to determine the optimal number of iterations, which is
equal to the formal order of accuracy of the method, and to introduce efficient
-adaptive modifications. These are defined by matching the order of accuracy
achieved and the degree of the polynomial reconstruction at each iteration. We
provide analytical and numerical results, including the stability analysis of
the new modified methods, the investigation of the computational efficiency, an
application to adaptivity and an application to hyperbolic PDEs with a Spectral
Difference (SD) space discretization
DeC and ADER: Similarities, Differences and a Unified Framework
Full text linkIn this paper, we demonstrate that the explicit ADER approach as it is used inter alia in
Zanotti et al. (Comput Fluids 118:204–224, 2015) can be seen as a special interpretation of the
deferred correction (DeC) method as introduced in Dutt et al. (BIT Numer Math 40(2):241–
266, 2000). By using this fact, we are able to embed ADER in a theoretical background of
time integration schemes and prove the relation between the accuracy order and the number
of iterations which are needed to reach the desired order. Next, we extend our investigation to
stiff ODEs, treating these source terms implicitly. Some differences in the interpretation and
implementation can be found. Using DeC yields typically a much simpler implementation,
while ADER benefits from a higher accuracy, at least for our numerical simulations. Then,
we also focus on the PDE case and present common space-time discretizations using DeC
and ADER in closed forms. Finally, in the numerical section we investigate A-stability for
the ADER approach—this is done for the first time up to our knowledge—for different order
using several basis functions and compare them with the DeC ansatz. Then, we compare the
performance of ADER and DeC for stiff and non-stiff ODEs and verify our analysis focusing
on two basic hyperbolic problems
Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes
Full text linkIn [1] is proposed a simplified DeC method, that, when combined with the
residual distribution (RD) framework, allows to construct a high order,
explicit FE scheme with continuous approximation avoiding the inversion of the
mass matrix for hyperbolic problems. In this paper, we close some open gaps in
the context of deferred correction (DeC) and their application within the RD
framework. First, we demonstrate the connection between the DeC schemes and the
RK methods. With this knowledge, DeC can be rewritten as a convex combination
of explicit Euler steps, showing the connection to the strong stability
preserving (SSP) framework. Then, we can apply the relaxation approach
introduced in [2] and construct entropy conservative/dissipative DeC (RDeC)
methods, using the entropy correction function proposed in [3].
[1] R. Abgrall. High order schemes for hyperbolic problems using globally
continuous approximation and avoiding mass matrices. Journal of Scientific
Computing, 73(2):461--494, Dec 2017.
[2] D. Ketcheson. Relaxation Runge--Kutta methods: Conservation and stability
for inner-product norms. SIAM Journal on Numerical Analysis, 57(6):2850--2870,
2019.
[3] R. Abgrall. A general framework to construct schemes satisfying
additional conservation relations. application to entropy conservative and
entropy dissipative schemes. Journal of Computational Physics, 372:640--666,
2018
Issues with Positivity-Preserving Patankar-type Schemes
Full text linkPatankar-type schemes are linearly implicit time integration methods designed
to be unconditionally positivity-preserving. However, there are only little
results on their stability or robustness. We suggest two approaches to analyze
the performance and robustness of these methods. In particular, we demonstrate
problematic behaviors of these methods that, even on very simple linear
problems, can lead to undesired oscillations and order reduction for vanishing
initial condition. Finally, we demonstrate in numerical simulations that our
theoretical results for linear problems apply analogously to nonlinear stiff
problems
Correction to: When terminology hinders research: the colloquialisms of transitions of control in automated driving (Cognition, Technology & Work, (2022), 10.1007/s10111-022-00705-3)
Full text linkIn the original article, author affiliation published with error. The correct affiliations are: Davide Maggi—Institute for Transport Studies, Leeds, UK. Richard Romano—Institute for Transport Studies, Leeds, UK. Oliver Carsten—Institute for Transport Studies, Leeds, UK. Joost C. F. De Winter—Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Delft, The Netherlands. The original article has been corrected.Human-Robot Interactio
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