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Before applying multigrid methods to a project, mathematicians, scientists, and engineers need to answer questions related to the quality of convergence, whether a development will pay out, whether multigrid will work for a particular application, and what the numerical properties are. Practical Fourier Analysis for Multigrid Methods uses a detailed and systematic description of local Fourier k-grid (k=1,2,3) analysis for general systems of partial differential equations to provide a framework that answers these questions.
This volume contains software that confirms written statements about convergence and efficiency of algorithms and is easily adapted to new applications. Providing theoretical background and the linkage between theory and practice, the text and software quickly combine learning by reading and learning by doing. The book enables understanding of basic principles of multigrid and local Fourier analysis, and also describes the theory important to those who need to delve deeper into the details of the subject.
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.
Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies
(2018)
Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995)] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012)], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.
Pseudopotential-based Lattice Boltzmann models provide an efficient means to simulate multiphase flows. Although thoroughly analysed and extended, they still suffer from various numerical deficiencies, such as stability issues at large density and viscosity ratios as well as spurious velocities in interfacial regions. In this work, we show how to optimize the model numerically with regard to the latter. This optimized scheme is analysed in static as well as oscillating droplet simulations and is compared to different higher-order isotropic schemes. We found that the optimized scheme substantially lowers spurious velocities and competes well with comparable schemes in both test cases.
The lattice Boltzmann method is a simulation technique in computational fluid dynamics. In its standard formulation, it is restricted to regular computation grids, second-order spatial accuracy, and a unity Courant-Friedrichs-Lewy (CFL) number. This paper advances the standard lattice Boltzmann method by introducing a semi-Lagrangian streaming step. The proposed method allows significantly larger time steps, unstructured grids, and higher-order accurate representations of the solution to be used. The appealing properties of the approach are demonstrated in simulations of a two-dimensional Taylor-Green vortex, doubly periodic shear layers, and a three-dimensional Taylor-Green vortex.
Results Obtained with a Semi-lagrangian Mass-Integrating Transport Algorithm by Using the GME Grid
(2008)
The defect of mass by actual schemes within the GME has motivated the implementation of a conservative semi-Lagrangian scheme on an icosahedral mesh. This scheme for the GME-mesh is unique. The properties are demonstrated by applying the new algorithm to typical test cases as they can be derived from the shallow water test suite. The results using coarse grids show: this scheme is extremely attractive for a climate version of the GME. Even for periods longer than those as specified in the SWE test suite, a defect of mass in the range of accuracy of the machine is obtained. The relative error results show that the order of accuracy of the proposed algorithm is two.
