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Wed, 30 Oct 2024 11:07:12 +0100Wed, 30 Oct 2024 11:07:12 +0100Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/566
Conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations have been obtained. An example demonstrating the application of the obtained results has been considered. Isothermal models of gas networks have been proposed as applications.Maria Filipkovskaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/566Wed, 30 Oct 2024 11:07:12 +0100Modeling and optimization of gas mixtures on networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/565
This paper presents a model for the mixture of gases on
networks in the stationary case. The model is based on an equation
of state for the mixture, the stationary isothermal Euler equations and
coupling conditions for the flow and mixture. The equation of state or
pressure law is based on the change of the speed of sound in a mixture of
gases. We use this model to solve stationary gas flow problems to global
optimality on large networks and present computational results.Pascal Börner; Marc E. Pfetsch; Stefan Ulbrichconferenceobjecthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/565Sat, 19 Oct 2024 21:16:45 +0200Robust chance-constrained optimization with discrete distributions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/564
Typically, probability distributions that generate uncertain parameters are uncertain themselves or even unknown. As a remedy, distributional robustness determines optimized decisions that are protected in a robust fashion against all probability distributions in some appropriately chosen ambiguity set. In this work, we consider robust joint chance-constrained optimization problems and focus on discrete probability distributions. Many methods for this kind of problems study convex or even linear constraint functions. In contrast, we introduce a practically efficient scenario-based bundle method without convexity assumptions on the constraint functions. We start by deriving an approximation problem to the original robust chance-constrained version by using smoothing and penalization techniques that build on our former work on chance-constrained optimization. Our convergence results with respect to the smoothing approximation and well-known results for penalty approximations suggest replacing the original problem with the approximation problem for large smoothing and penalty parameters. Our scenario-based bundle method starts by solving the approximation problem with a bundle method, and then uses the bundle solution to decide which scenarios to include in a scenario-expanded formulation. This formulation is a standard nonlinear optimization problem. In our numerical experiments we demonstrate the efficiency of our approach on real-world gas transport problems with uncertain demands. Comparing our results to the classical robust reformulations for ambiguity sets consisting of confidence intervals and Wasserstein balls, we observe that the scenario-based bundle method often outperforms solving the classical reformulation directly.Daniela Bernhard; Frauke Liers; Michael Stinglpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/564Tue, 15 Oct 2024 20:53:16 +0200Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/562
We study a state estimation problem for a 2x2 linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term.Jan Giesselmann; Teresa Kunkelpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/562Sun, 13 Oct 2024 20:29:05 +0200Strategic Retailers in the Energy Sector
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/561
We propose a general framework which allows to analyze the strategic interaction of retail companies, where customers choose retail contracts over a longer period of time based on price and non-price characteristics of retail contracts. We allow for many, possibly asymmetric retailers which can offer fixed price tariffs or dynamic prices, as typically observed in energy
markets. Our framework considers uncertainties and allows for price-responsive consumption choices of customers. We analytically characterize all resulting market equilibria for the general asymmetric setting. Based on those results we then propose a solution algorithm which is capable to determine all resulting equilibria. For the case of symmetric retailers we provide analytical comparisons of the different tariff structures. To show the applicability of our framework and our algorithm to real-world instances, we calibrate it to data of the German retail electricity market.
Our results show, that firms profits remain unchanged but consumer surplus and welfare increase when switching from fixed price tariffs to real-time pricing. This effect is more pronounced under higher wholesale price fluctuations. Finally we also propose a surrogate, reduced order model, which is shown to be equivalently capable to quantify the welfare difference of the different tariffs.Ann-Kathrin Wiertz; Andrea Walther; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/561Tue, 01 Oct 2024 12:44:28 +0200Discrete Single-Parameter Optimal Auction Design
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/560
We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over "finite" supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility.
Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a linear combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules.
Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction.Yiannis Giannakopoulos; Johannes Hahnpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/560Tue, 30 Jul 2024 17:03:18 +0200Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/559
The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided.Martin Gugat; Meizhi Qian; Jan Sokolowskiarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/559Mon, 01 Jul 2024 16:41:43 +0200Modeling and Optimal Control of the Flow of a Gas Mixture
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/558
We consider the Euler equations for a pipeline flow of a mixture of two gases. An important application is hydrogen blending. Existence and uniqueness of semi-global solutions is shown and possible boundary conditions are analyzed.
Secondly, we consider classes of associated optimal control problems and show existence of solutions.Christian Langepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/558Tue, 18 Jun 2024 23:18:20 +0200Location Problem for Compressor Stations in Pipeline Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/556
In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related
with the gas dynamics that governs the network flow. That results in non-convex mixed integer stochastic optimization problems with probabilistic constraints.
Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, the problem of finding the optimal location for the
control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered.
In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting,
an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result
for the optimal compressor location is presented.
Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. Results for the existence of optimal locations on a graph in
both, the deterministic and the probabilistic setting, are presented and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings.Martin Gugat; Michael Schuster; Jan Sokolowskipreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/556Tue, 18 Jun 2024 09:13:02 +0200On the Convergence of Optimization Problems with Kernel Density Estimated Probabilistic Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/557
Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields, even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluate corresponding probabilities.
In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity.
The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems.Michael Schusterpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/557Tue, 18 Jun 2024 09:12:27 +0200Stabilizing GNEP-Based Model Predictive Control: Quasi-GNEPs and End Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/555
We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP), in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop if stabilizing end constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing end constraints. The theoretical results are complemented by numerical experiments.Falk M. Hante; Martin Schmidt; Antonia Topalovicpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/555Thu, 13 Jun 2024 14:26:12 +0200Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/551
We consider mixed-integer nonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex. We propose a robust optimization method that requires no certain structure of the adversarial problem, but only approximate worst-case evaluations. The method integrates a bundle method, for continuous subproblems, into an outer approximation approach. We prove that our algorithm converges and finds an approximately robust optimal solution and propose robust gas transport as a suitable application.Martina Kuchlbauerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/551Thu, 16 May 2024 23:40:42 +0200Adjustable Robust Nonlinear Network Design under Demand Uncertainties
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/550
We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations.Johannes Thürauf; Julia Grübel; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/550Sat, 06 Apr 2024 22:41:05 +0200Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/549
High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.Hendrik Wilka; Jens Langpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/549Thu, 04 Apr 2024 19:13:31 +0200A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/547
Method-of-lines discretizations are demanding test problems for stiff inte-
gration methods. However, for PDE problems with known analytic solution
the presence of space discretization errors or the need to use codes to compute
reference solutions may limit the validity of numerical test results. To over-
come these drawbacks we present in this short note a simple test problem with
boundary control, a situation where one-step methods may suffer from order
reduction. We derive exact formulas for the solution of an optimal boundary
control problem governed by a one-dimensional discrete heat equation and an
objective function that measures the distance of the final state from the target
and the control costs. This analytical setting is used to compare the numeri-
cally observed convergence orders for selected implicit Runge-Kutta and Peer
two-step methods of classical order four which are suitable for optimal control
problems.Jens Lang; Bernhard A. Schmittarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/547Mon, 25 Mar 2024 12:09:02 +0100Physics-Informed Neural Networks: A Case Study for Gas Transport Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/546
Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.Erik Laurin Strelow; Alf Gerisch; Jens Lang; Marc E. Pfetscharticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/546Mon, 25 Mar 2024 12:05:36 +0100Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/545
This paper is concerned with the construction and convergence analysis
of novel implicit Peer triplets of two-step nature with four stages for nonlinear
ODE constrained optimal control problems. We combine the property of superconvergence
of some standard Peer method for inner grid points with carefully
designed starting and end methods to achieve order four for the state variables
and order three for the adjoint variables in a first-discretize-then-optimize approach
together with A-stability. The notion triplets emphasizes that these
three different Peer methods have to satisfy additional matching conditions.
Four such Peer triplets of practical interest are constructed. Also as a benchmark
method, the well-known backward differentiation formula BDF4, which is
only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint
consistent method of higher order and BDF type with equidistant nodes. Within
the class of Peer triplets, we found a diagonally implicit A(84)-stable method
with nodes symmetric in [0,1] to a common center that performs equally well.
Numerical tests with three well established optimal control problems confirm
the theoretical findings also concerning A-stability.Jens Lang; Bernhard A. Schmittarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/545Thu, 21 Mar 2024 21:02:23 +0100Gas Network Modeling: An Overview (Extended English Version)
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/543
With this overview we want to provide a compilation of different models for
the description of gas flow in networks in order to facilitate the introduction
to the topic. Special attention is paid to the hierarchical structure inherent
to the modeling, and the detailed description of individual components such
as valves and compressors. Also included are network model classes based
on purely algebraic relations, and energy-based port-Hamiltonian models. A
short overview of basic numerical methods and concepts for the treatment
of hyperbolic balance equations is also given. We do not claim completeness
and refer in many places to the existing literature.Pia Domschke; Jan Giesselmann; Jens Lang; Tobias Breiten; Volker Mehrmann; Riccardo Morandin; Benjamin Hiller; Caren Tischendorfpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/543Thu, 21 Mar 2024 20:55:41 +0100Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/548
It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems. Gradients
of the objective function are most efficiently computed by approximate
adjoint variables. High accuracy with moderate computing time can be achieved
by such time integration methods that satisfy a sufficiently large number of adjoint
order conditions and supply gradients with higher orders of consistency. In
this paper, we upgrade our former implicit two-step Peer triplets constructed in
[Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods
use several stages of the same high stage order, a decisive advantage is their lack
of order reduction as for semi-discretized PDE problems with boundary control.
Additional order conditions for the control and certain positivity requirements
now intensify the demands on the Peer triplet. We discuss the construction of
4-stage methods with order pairs (4,3) and (3,3) in detail and provide three
Peer triplets of practical interest. We prove convergence for s-stage methods,
for instance, order s for the state variables even if the adjoint method and the
control satisfy the conditions for order s-1, only. Numerical tests show the
expected order of convergence for the new Peer triplets.Jens Lang; Bernhard A. Schmittpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/548Thu, 21 Mar 2024 20:53:30 +0100Solving Linear Generalized Nash Games Using an Active Signature Method
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/542
We propose a method to solve linear generalized Nash equilibrium problems (LGNEPs). For this purpose, a reformulation of the LGNEPs as piecewise linear problems is considered. This requires the calculation of all vertices for a special kind of unbounded convex polyhedra. Then the active signature method for constrained abs-linear problems can be used to determine the Nash equilibria. We analyse the computational effort for the resulting solution procedure. This includes also the verification of suitable optimality conditions. Finally, we present and analyse numerical results for some test problems.Gertrud Graser; Timo Kreimeier; Andrea Waltherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/542Thu, 29 Feb 2024 15:15:52 +0100On the convergence of the regularized entropy-based moment method for kinetic equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/536
The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.Graham Alldredge; Martin Frank; Jan Giesselmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/536Tue, 13 Feb 2024 17:44:42 +0100A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/540
We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ.Jan Giesselmann; Niklas Kolbepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/540Tue, 13 Feb 2024 17:44:07 +0100A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/541
We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.Jan Giesselmann; Kiwoong Kwonpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/541Tue, 13 Feb 2024 17:43:18 +0100Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/539
In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension.
Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates.
We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].
Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.Jan Giesselmann; Sam Krupapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/539Tue, 13 Feb 2024 17:42:30 +0100Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/538
We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem.Herbert Egger; Jan Giesselmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/538Tue, 13 Feb 2024 17:41:44 +0100