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Wed, 07 Feb 2018 14:27:16 +0100Wed, 07 Feb 2018 14:27:16 +0100Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234
We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal design of zonal pricing electricity markets. This together with nonlinearities due to economic modeling yields extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof, whereas the second is a tailored generalized Benders decomposition. We prove for both methods that they yield global optimal solutions. Finally, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation.Thomas Kleinert; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234Wed, 07 Feb 2018 14:27:16 +0100Robust Optimal Discrete Arc Sizing for Tree-Shaped Potential Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233
We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing's optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.Martin Robinius; Lars Schewe; Martin Schmidt; Detlef Stolten; Johannes Thürauf; Lara Welderpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233Thu, 01 Feb 2018 15:28:47 +0100Optimal Model Switching for Gas Flow in Pipe Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232
We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.Fabian Rüffler; Volker Mehrmann; Falk Hantearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232Mon, 29 Jan 2018 14:11:31 +0100A Bayesian approach for parameter identification in gas networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231
The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the
friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.Soheil Hajian; Michael Hintermüller; Claudia Schillings; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231Mon, 29 Jan 2018 14:11:31 +0100DYNAMIC BOUNDARY CONTROL GAMES WITH NETWORKS OF STRINGS
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229
Consider a star-shaped network
of strings. Each string is governed by the wave equation.
At each boundary node of the network there is
a player that performs Dirichlet boundary control action
and in this way influences the system state.
At the central node, the states are coupled
by algebraic conditions in such a way that the energy is conserved.
We consider the corresponding antagonistic game
where each player minimizes a certain quadratic objective function
that is given by the sum of a control cost and
a tracking term for the final state.
We prove that under suitable assumptions
a unique Nash equilibrium exists
and give an explicit representation
of the equilibrium strategies.Martin Gugat; Sonja Steffensenarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229Fri, 19 Jan 2018 11:16:58 +0100Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227
Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.M. Hassan Farshbaf-Shaker; Rene Henrion; Dietmar Hömbergarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227Tue, 09 Jan 2018 11:32:43 +0100Subdifferential characterization of probability functions under Gaussian distribution
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226
Probability functions figure prominently in optimization problems
of engineering. They may be nonsmooth even if all input data are smooth.
This fact motivates the consideration of subdifferentials for such typically just
continuous functions. The aim of this paper is to provide subdifferential formulae
of such functions in the case of Gaussian distributions for possibly
infinite-dimensional decision variables and nonsmooth (locally Lipschitzian)
input data. These formulae are based on the spheric-radial decomposition of
Gaussian random vectors on the one hand and on a cone of directions of
moderate growth on the other. By successively adding additional hypotheses,
conditions are satisfied under which the probability function is locally Lipschitzian
or even differentiable.Abderrahim Hantoute; Rene Henrion; Pedro Perez-Arosarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226Tue, 09 Jan 2018 11:32:42 +0100On the identification of the friction coefficient in a semilinear system for gas transport through a network
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225
An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions is established. Finally, numerical experiments validate the theoretical findings.
Michael Hintermüller; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225Tue, 28 Nov 2017 10:57:52 +0100Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224
The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.Volker Mehrmann; Martin Schmidt; Jeroen J. Stolwijkpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224Mon, 27 Nov 2017 12:56:12 +0100Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223
In this paper we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203–215, 2017]
to a broader class of two-step methods that allow the construction of super-
convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of ex-
plicit methods to efficiently solve systems of ordinary differential equations with
both stiff and non-stiff parts included in the source term. To construct super-
convergent IMEX-Peer methods with favourable stability properties, we derive
necessary and sufficient conditions on the coefficient matrices and apply an
extrapolation approach based on already computed stage values. Optimised
super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are
given as result of a search algorithm carefully designed to balance the size of
the stability regions and the extrapolation errors. Numerical experiments and
a comparison to other IMEX-Peer methods are included.Moritz Schneider; Jens Lang; Willem Hundsdorferpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223Sun, 26 Nov 2017 20:46:40 +0100Uniqueness of Market Equilibria on Networks with Transport Costs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222
We study market equilibria for perfect competition in capacitated transport networks. Using piecewise linear and symmetric transport costs, we prove uniqueness of the resulting equilibria under mild assumptions.Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222Wed, 22 Nov 2017 11:30:24 +0100A Global Optimization Approach for Instationary Gas Transport in Pipeline Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221
We consider the optimization of a gas network that is described by a coupled system of parabolic partial differential equations. Energy and mass balance at network junctions and active elements, like compressors or valves, are modeled as additional algebraic equations. The resulting optimal control problem is discretized in space and time by a particular finite volume method, which can be shown to be well-posed under rather general assumptions. This first-discretize-then-optimize procedure yields a mixed-integer nonlinear problem (MINLP) that can be solved to global optimality.
For the numerical solution of the MINLP, we consider a relaxation approach allowing to solve the problem globally by a sequence of mixed-integer problems (MIPs) with any required accuracy. The relaxation is based on piecewise linearization of the nonlinear constraints modeling the gas dynamics on the pipelines. Due to the particular discretization of the state equations, only univariate nonlinearities have to be approximated. This substantially facilitates the numerical treatment of the nonlinear constraints. To illustrate the efficiency of the proposed approach, we present numerical tests for typical benchmark problems.Robert Burlacu; Herbert Egger; Martin Groß; Alexander Martin; Marc Pfetsch; Lars Schewe; Mathias Sirvent; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221Tue, 21 Nov 2017 15:12:15 +0100Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220
This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa.Veronika Grimm; Julia Grübel; Lars Schewe; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220Tue, 21 Nov 2017 13:42:30 +0100Optimal Boundary Control of Hyperbolic Balance Laws with State Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219
In this paper we analyze the optimal control of initial-boundary value problems
for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Hereby,
we suppose that the initial and the boundary data switch between different C¹-functions at certain
switching points, where the C¹ -functions and the switching points are considered as the control. For
a class of cost functionals, we prove first order necessary optimality conditions for the corresponding
optimal control problem with state contraints. Furthermore, we use a Moreau-Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality
conditions for the regularized problems and finally prove convergence to the solution of the optimal
control problem with state constraints.Johann Michael Schmitt; Stefan Ulbrichpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219Tue, 21 Nov 2017 13:19:55 +0100Improving branching for disjunctive polyhedral models using approximate convex decompositions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/218
Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.Benjamin Hiller; Tom Waltherreporthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/218Tue, 21 Nov 2017 13:08:05 +0100Numerical optimal control of instationary gas transport with control and state constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/214
We consider the optimal control of a nonlinear hyperbolic system of balance
laws on a one-dimensional network which arises in the context of gas transport in pipeline
systems. State constraints, which are required for the safe operation of the system, are
incorporated by a barrier method. We discuss the well-posedness of the governing system
of partial differential-algebraic equations and investigate the existence of minimizers. For
the numerical solution, we then consider the approximation of the state equation by
mixed finite elements in space and a particular linear implicit time integration scheme
that can be interpreted as a discontinuous Galerkin approximation. We establish well-
posedness of this discretization scheme and prove the existence of minimizers for the
corresponding discretized optimal control problem and discuss its numerical solution
by a projected Gauß-Newton method. The efficient realization of the Jacobian and
Hessian of the quadratic approximations that have to be minimized in every iteration
of the Gauß-Newton method can be obtained via the solution of discretized sensitivity
and adjoint equations. These are obtained by formal differentiation and transposition
of the Galerkin methods employed for the discretization of the state equations. All
approximations obtained after discretization can thus be interpreted as functions on the
continuous level and, since the functional analytic setting is not changed by the Galerkin
discretization, we observe mesh independence of the resulting fully discrete methods. For
illustration of our theoretical results and to demonstrate the efficiency of the proposed
method, we present numerical results for two test problems that model typical situations
that may arise in the daily operation of gas networks.Herbert Egger; Thomas Kugler; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/214Mon, 20 Nov 2017 19:59:45 +0100Joint model of probabilistic/robust (probust) constraints with application to gas network optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/215
Optimization tasks under uncertain conditions abound in many
real-life applications. Whereas solution approaches for probabilistic constraints
are often developed in case the uncertainties can be assumed to follow a
certain probability distribution, robust approaches are usually used in case
solutions are sought that are feasible for all realizations of uncertainties within
some pre-defined uncertainty set. As many applications contain different types
of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focussing on complex uncertain gas
network optimization problems, we show the relevance of this class of problems
for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution.
Finally, we study the problem of controlling a transient system that is governed
by the wave equation. The task consists in determining controls such that a
certain robustness measure remains below some given upper bound, with high
probability.Dennis Adelhütte; Denis Aßmann; Tatiana Gonzàlez Grandòn; Martin Gugat; Holger Heitsch; Frauke Liers; René Henrion; Sabrina Nitsche; Rüdiger Schultz; Michael Stingl; David Wintergerstpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/215Mon, 20 Nov 2017 19:59:45 +0100Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/213
We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. Herbert Egger; Anke Böttcherpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/213Mon, 20 Nov 2017 19:03:43 +0100Super-convergence and post-processing for mixed finite element approximations of the wave equation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/212
We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes. Herbert Egger; Bogdan Radupreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/212Mon, 20 Nov 2017 19:03:42 +0100A mixed variational discretization for non-isothermal compressible flow in pipelines
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/211
We consider the non-isothermal flow of a compressible fluid through pipes. Starting from the full set of Euler equations, we propose a variational characterization of solutions that encodes the conservation of mass, energy, and entropy in a very direct manner. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. Three different spaces are used for approximation of density, mass flux, and temperature, and we consider a mixed finite element method as one possible choice of suitable approximation spaces. We also investigate the subsequent discretization in time by a problem adapted implicit time stepping scheme for which exact conservation of mass as well as a slight dissipation of energy and increase of entropy are proven which are due to the numerical dissipation of the implicit time discretization. The main arguments of our analysis are rather general and allow us to extend the approach with minor modification to more general boundary conditions and flow models taking into account friction, viscosity, heat conduction, and heat exchange with the surrounding medium.
Herbert Eggerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/211Mon, 20 Nov 2017 18:58:29 +0100On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/210
We consider the identification of nonlinear diffusion coefficients of the form a(t,u) or a(u) in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail. Herbert Egger; Jan-Frederik Pietschmann; Matthias Schlottbompreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/210Mon, 20 Nov 2017 18:53:45 +0100Decomposable Robust Two-Stage Optimization: An Application to Gas Network Operations Under Uncertainty
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/209
We study gas network problems with compressors and control valves under uncertainty that can be formulated as two-stage robust optimization problems. Uncertain data are present in the physical parameters of the pipes as well as in the overall demand. We show how to exploit the special decomposable structure of the problem in order to reformulate the two-stage robust problem as a standard single-stage optimization problem. Since this structure is present in similar problems on e.g., water or direct current electricity networks, we investigate the consequences of the decomposable structure in an abstract setting: The right-hand side of the single-stage problem can be precomputed by solving a series of optimization problems and multiple elements of the right-hand side can be combined into one optimization task. In order to apply our results to gas network problems, we extend piecewise relaxations and preprocessing techniques to incorporate uncertain input data. The practical feasibility and effectiveness of our approach is demonstrated with benchmarks on realistic gas network instances. We observe large speedups due to the described aggregation method together with the developed preprocessing strategies. Furthermore, we are able to solve even comparably large gas network instances quickly for the price of slightly more conservative solutions.Denis Aßmann; Frauke Liers; Michael Stinglpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/209Mon, 20 Nov 2017 18:53:45 +0100Enhancement of flow measurements using fluid dynamic constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/208
Novel experimental modalities acquire spatially resolved velocity measurements for steady state and transient flows which are of interest for engineering and biological applications. One of the drawbacks of such high resolution velocity data is their susceptibility to measurement errors. In this paper, we propose a novel filtering strategy that allows enhancement of noisy measurements to obtain reconstruction of smooth divergence free velocity and corresponding pressure fields, which together approximately comply to a prescribed flow model. The main step in our approach consists of the appropriate use of the velocity measurements in the design of a linearized flow model which can be shown to be well-posed and consistent with the true velocity and pressure fields up to measurement and modeling errors. The reconstruction procedure is formulated
as a linear quadratic optimal control problem and the resulting filter has analyzable smoothing and approximation properties. We also discuss briefly the discretization of our approach by finite element methods and comment on the efficient solution of the linear optimality system by iterative solvers. The capability of the proposed method to significantly reduce data noise is demonstrated by numerical tests in which we also compare to other methods like smoothing and solenoidal filtering.Herbert Eggerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/208Mon, 20 Nov 2017 18:46:54 +0100An asymptotic preserving mixed finite element method for wave propagation in pipelines
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/207
We consider a parameter dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation we then consider a mixed finite element method in space together with a Runge-Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general. Herbert Egger; Thomas Kuglerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/207Mon, 20 Nov 2017 17:51:36 +0100A class of Galerkin schemes for time-dependent radiative transfer
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/206
The numerical solution of time-dependent radiative transfer problems is
challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to preserve basic properties like exponential stability and decay to equilibrium also on the discrete level. We present the basic a-priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure like first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed PN-finite element time stepping scheme
is demonstrated by computational results.Herbert Eggerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/206Mon, 20 Nov 2017 17:51:35 +0100