## 510 Mathematik

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- Method of lines (5)
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The lattice Boltzmann method is a modern approach to simulate fluid flow. In its original formulation, it is restricted to regular grids, second-order discretizations, and a unity CFL number. This paper describes our new off-lattice Boltzmann solver NATriuM, an extensible and parallel C++ code to perform lattice Boltzmann simulations on irregular grids. NATriuM also allows high-order spatial discretizations and non-unity CFL numbers to be used. We demonstrate how these features can efficiently decrease the number of grid points required in a simulation and thus reduce the computational time, compared to the standard lattice Boltzmann method. We detail the implementation of a recently proposed semi-Lagrangian lattice Boltzmann method and prove its efficiency in comparisons to other state-of-the-art off-lattice Boltzmann schemes.

Simulating free-surface and pressurised flow is important to many fields of application, especially in network approaches. Modelling equations to describe flow behaviour arising in these problems are often expressed by one-dimensional formulations of the hyperbolic shallow water equations. One established approach to realise their numerical computation is the method of lines based on semi-discretisation in space (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001; Steinebach and Weiner, Appl Numer Math 62:1567–1578, 2012; Steinebach et al., Modeling and numerical simulation of pipe flow problems in water supply systems. In: Martin, Klamroth, et al. (eds) Mathematical optimization of water networks. International series of numerical mathematics, vol 162, pp 3–15. Springer, Basel, 2012). It leads to index-one DAE systems as algebraic constraints are required to realise coupling and boundary conditions of single reaches.Linearly implicit ROW schemes proved to be effective to solve these DAE systems (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001). However, under certain conditions an extended partial explicit time-integration of the shallow water equations could be worthwhile to save computational effort. To restrict implicit solution by ROW schemes to stiff components while using explicit solution by RK methods for remaining terms, we adapt ROW method ROS34PRW (Rang, J Comput Appl Math 262:105–114, 2014) to an AMF and IMEX combining approach (Hundsdorfer and Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. Springer, Berlin, Heidelberg, New York, 2003). Applied to first test problems regarding open channel flow, efficiency is analysed with respect to flow behaviour. Results prove to be advantageous especially concerning dynamical flow.

Solving differential-algebraic equations (DAEs) efficiently is an ongoing topic in applied mathematics. Applications are given with respect to many fields of practical interest, such as multiphysics problems or network simulations. Due to the stiffness properties of DAEs, linearly implicit Runge-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes are an appropriate choice for effecitive numerical time-integration. Compared to fully implicit schemes, they are easy to implement and avoid having to solve non-linear equations by including Jacobian information in their formulation explicity. But, especially when having to solve large coupled systems, computing the Jacobian is costly and proves to be a considerable drawback. Inspired by the works of Steihaug and Wolfbrandt [4], we introduce concepts to realize linearly-implicit Runge-Kutta methods for DAEs in the form of so-called W-methods. These schemes allow for arbitrary approximations to given Jacobian entries and, thus, for versatile strategies to reduce computational effort significantly when solving semi-explicit DAE problems of index-1. An approach extending Roche’s procedure [3] will be presented that enables to derive order conditions of the resulting methods by an algebraic theory using rooted trees, a strategy originally introduced by Butcher regarding Runge-Kutta schemes [1,2]. Besides, suitable sets of coefficients for implementing embedded schemes and their potential of increasing efficincy when solving DAEs will be demonstrated.

In this paper an overview on modelling techniques and numerical methods applied to problems in water network simulation is given. The considered applications cover river alarm systems (Rentrop and Steinebach, Surv Math Ind 6:245–265, 1997), water level forecast methods (Steinebach and Wilke, J CIWEM 14(1):39–44, 2000) up to sewer and water supply networks (Steinebach et al., Mathematical Optimization of Water Networks Martin. Springer, Basel, 2012).
The hyperbolic modelling equations are derived from mass and momentum conservation laws. A typical example are the well known Saint-Venant equations. For their numerical solution a conservative semi-discretisation in space by finite differences is proposed. A new well-balanced space discretisation scheme is presented which improves the local Lax-Friedrichs approach applied so far. Higher order discretisations are achieved by WENO methods (Kurganov and Levy, SIAM J Sci Comput 22(4):1461–1488, 2000).
Together with appropriate boundary and coupling conditions this method of lines approach leads to an index-one DAE system. Efficient solution of the DAE system is the topic of Jax and Steinebach (ROW methods adapted to network simulation for fluid flow, in preparation).

A new type of Rosenbrock-Wanner (ROW) methods for solving semi-explicit DAEs of index-1 is introduced. The scheme considers arbitrary approximations to Jacobian entries resulting for the differential part and thus corresponds to a first attempt of applying W methods to DAEs. Besides, it is a generalized class covering many ROW-type methods known from literature. Order conditions are derived by a consistent approach that combines theories of ROW methods with exact Jacobian for DAEs (Roche, 1988) and W methods with arbitrary Jacobian for ODEs (Steihaug and Wolfbrandt, 1979). In this context, rooted trees based on Butcher’s theory that include a new type of vertices are used to describe non-exact differentials of the numerical solution. Resulting conditions up to order four are given explicitly, including new conditions for realizing schemes of higher order. Numerical tests emphasize the relevance of satisfying these conditions when solving DAEs together with approximations to Jacobian entries of the differential part.

Since being introduced in the sixties and seventies, semi-implicit RosenbrockWanner (ROW) methods have become an important tool for the timeintegration of ODE and DAE problems. Over the years, these methods have been further developed in order to save computational effort by regarding approximations with respect to the given Jacobian [5], reduce effects of order reduction by introducing additional conditions [2, 4] or use advantages of partial explicit integration by considering underlying Runge-Kutta formulations [1]. As a consequence, there is a large number of different ROW-type schemes with characteristic properties for solving various problem formulations given in literature today.

Hydrological processes, like the water flow or transport of soluble substances in rivers, are usually described by partial differential equations (PDEs). The numerical solution of these equations offer various fields of important applications, e.g., the prediction of water stages during floods or low water periods (optimal load capacity of the ships) and the prediction of the fate of pollution plumes in case of an accident. The aim is to develop numerical schemes that are applicable to mostly all those simulations and well-tested with respect to their stability and efficiency. The method of lines (MOL)-approach was found to be a suitable numerical technique to solve the whole class of the discussed problems. The subject of this paper is the comparison of different numerical schemes for the space discretizations in the MOL-approach, in particular the ENO-discretizations which are found to be very useful even when sharp gradients occur.

River alarm systems are designed for the forecasting of water stages during floods or low flow conditions or the prediction of the transport of pollution plumes. The basic model equations are introduced and a Method of Lines approach for their numerical solution is discussed. The approach includes adaptive space-mesh strategies and a Rosenbrock-Wanner scheme for the time integration. It fits into a PC environment and fulfills the requirements on an implementation within river alarm systems.

Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.

The numerical solution of implicit ordinary differential equations arising in vehicle dynamic
(1988)

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature β and the interaction strength K. The rates of convergence results are obtained as (β,K) converges along appropriate sequences (βn,Kn) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.

In the present paper we obtain rates of convergence for limit theorems via Stein's Method of exchangeable pairs in the context of the Curie–Weiss–Potts model and we consider only the case of non-zero external field h∈ R q. Our interest is in the limit distribution of the empirical vector of the spin variables and we obtain bounds for multivariate normal approximation.

The simulation of fluid flows is of importance to many fields of application, especially in industry and infrastructure. The modelling equations applied describe a coupled system of non-linear, hyperbolic partial differential equations given by one-dimensional shallow water equations that enable the consistent implementation of free surface flows in open channels as well as pressurised flows in closed pipes. The numerical realisation of these equations is complicated and challenging to date due to their characteristic properties that are able to cause discontinuous solutions.

For many practical problems an efficient solution of the one-dimensional shallow-water equations (Saint–Venant equations) is important, especially when large networks of rivers, channels or pipes are considered. In order to test and develop numerical methods four test problems are formulated. These tests include the well-known dam break and hydraulic jump problems and two steady state problems with varying channel bottom, channel width and friction.
The space discretization of the partial differential equations is based on a finite volume approach with central WENO interpolation and local Lax–Friedrich fluxes (Kurganov and Levy, 2000) [7]. For time-integration new linearly-implicit two-step peer methods of orders three and four are developed. These methods are especially adapted to the usage within the method of lines framework. They show a good performance compared to the well-established methods like ode15s, radau5 or rodasp.

In this chapter the simulation of a water supply system on a mesoscale abstraction level is considered. The water network consists of storage tanks, pipes, pumps and valves. It is operated by the characteristics of the water supplier, the consumer and the pumps. For all network elements the modeling equations are given. They include mass and momentum conservation for pressurized pipe flow. For their numerical solution the method of lines is proposed. The discretization in space is based on a finite volume approach together with a local Lax-Friedrich splitting and central WENO reconstruction. Boundary and coupling conditions are implemented as algebraic equations. This leads to a system of differential-algebraic equations in time which is solved by a special Rosenbrock method. The paper ends with some typical simulation results of the network.

For many practical problems an efficient solution of the one-dimensional shallow water equations (Saint-Venant equations) is important, especially when large networks of rivers, channels or pipes are considered. In order to test and develop numerical methods four test problems are formulated. These tests include the well known dam break and hydraulic jump problems and two steady state problems with varying channel bottom, channel width and friction.

A Method of Lines Flux-Difference Splitting Finite Volume Approach for 1D and 2D River Flow Problems
(2001)

The reliable forecasting of water levels is a very important issue. The modelling approach for water level forecasting at the Middle and Lower River Rhine is based on hydrodynamic river flow models coupled with precipitation-runoff models. The hydrodynamic model is defined by a numerical solution of the one-dimensional (1d) shallow water equations. If flood plains or flood risk maps are important, a two-dimensional (2d) model is required.
In this paper the usage of the Alcrudo-Garcia-Navarro scheme (Alcrudo and Garcia-Navarro, 1993) for 1d and 2d problems within the method of lines framework is described. The scheme is slightly modified, in order to allow a more accurate solution of problems with strong variations in the bottom topography. The problem of drying and re-wetting of mesh-cells is not yet fully sufficiently solved. Numerical results for some test problems and an application to a natural river will be presented.

Integrated modelling approaches for whole river catchments require the coupling of different types of models. As an example, river flow and forecast models in one- and two-space dimensions are discussed. Usually, these models are based on the hyperbolic shallow water equations and require special discretizations like ENO or Godunov-type methods.
The basic coupling mechanisms like coupling via source terms, via boundary conditions, via state variables and simulator coupling are introduced by examples. Their properties with respect to performance and accuracy requirements and implementation issues are presented.
If coupling conditions are considered, additional algebraic equations arise. By the method of lines approach it is possible to translate the partial differential equations and the algebraic equations into a large system of differential algebraic equations (DAEs). The DAEs can efficiently be solved if the special structure of the Jacobian of the coupled model components is taken into account.

The system for operational water level forecast and prediction of (fortunately not daily) pollutant transport for the river Rhine is in daily use. This model is based on the Saint-Venant or one-dimensional shallow water equations.
The model is augmented by additional terms and equations to model the effect of dead zones and the transport of soluble components.
The next step is to move towards two-dimensional models. An important problem that arises is that the domain of the fluid is not fixed by the given data, but depends on the water level and is therefore part of the solution of the model. Even worse, depending on the topography of the river bed, even the topology of the fluid domain may change, as islands may appear at low water and get flooded at high water situations.