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- ItemTime splitting error in DSMC schemes for the inelastic Boltzmann equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Rjasanow, Sergej; Wagner, WolfgangThe paper is concerned with the numerical treatment of the uniformly heated inelastic Boltzmann equation by the direct simulation Monte Carlo (DSMC) method. This technique is presently the most widely used numerical method in kinetic theory. We consider three modifications of the DSMC method and study them with respect to their efficiency and convergence properties. Convergence is investigated both with respect to the number of particles and to the time step. The main issue of interest is the time step discretization error due to various splitting strategies. A scheme based on the Strang-splitting strategy is shown to be of second order with respect to time step, while there is only first order for the commonly used Euler-splitting scheme. On the other hand, a no-splitting scheme based on appropriate Markov jump processes does not produce any time step error. It is established in numerical examples that the no-splitting scheme is about two orders of magnitude more efficient than the Euler-splitting scheme. The Strang-splitting scheme reaches almost the same level of efficiency compared to the no-splitting scheme, since the deterministic time step error vanishes sufficiently fast.
- ItemStochastic models in kinetic theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Wagner, WolfgangThe paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (mono-molecular processes), or via binary interactions (bi-molecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent
- ItemNumerical study of the systematic error in Monte Carlo schemes for semiconductors(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Muscato, Orazio; Di Stefano, Vincenza; Wagner, WolfgangThe paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially one-dimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step.
- ItemA stochastic weighted particle method for coagulation-advection problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Patterson, Robert I.A.; Wagner, WolfgangA spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two Direct Simulation Algorithms which develop instabilities.
- ItemSome properties of the kinetic equation for electron transport in semiconductors(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Wagner, WolfgangThe paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.
- ItemStochastic weighted particle methods for population balance equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Patterson, Robert I.A.; Kraft, Markus; Wagner, WolfgangA class of stochastic algorithms for the numerical treatment of population balance equations is introduced. The algorithms are based on systems of weighted particles, in which coagulation events are modelled by a weight transfer that keeps the number of computational particles constant. The weighting mechanisms are designed in such a way that physical processes changing individual particles (such as growth, or other surface reactions) can be conveniently treated by the algorithms. Numerical experiments are performed for complex laminar premixed flame systems. Two members of the class of stochastic weighted particle methods are compared to each other and to a direct simulation algorithm. One weighted algorithm is shown to be consistently better than the other with respect to the statistical noise generated. Finally, run times to achieve fixed error tolerances for a real flame system are measured and the better weighted algorithm is found to be up to three times faster than the direct simulation algorithm.
- ItemA class of probabilistic models for the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Wagner, WolfgangA class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle systems satisfy the Schrödinger equation. The proofs are based on the theory of piecewise deterministic Markov processes.
- ItemCell size error in stochastic particle methods for coagulation equations with advection(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Patterson, Robert I.A.; Wagner, WolfgangThe paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed.
- ItemA kinetic equation for the distribution of interaction clusters in rarefied gases(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Patterson, Robert I.A.; Simonella, Sergio; Wagner, WolfgangWe consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.
- ItemRandom and deterministic fragmentation models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Wagner, WolfgangRandom and deterministic fragmentation models are considered. Their relationship is studied by deriving different forms of the kinetic fragmentation equation from the corresponding stochastic models. Results related to the problem of non-conservation of mass (phase transition into dust) are discussed. Illustrative examples are given and some open problems are mentioned.