428 research outputs found

    ADI schemes for heat equations with irregular boundaries and interfaces in 3D with applications

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    In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical tests for both the heat equation and a reaction-diffusion equation. For the Stefan problem, which is a free boundary problem of the heat equation, a level set method is incorporated into the ADI method to capture the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented

    Simulating Self-gravitating Hydrodynamic Flows

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    An efficient algorithm for solving Poisson's equation in two and three spatial dimensions is discussed. The algorithm, which is described in detail, is based on the integral form of Poisson's equation and utilizes spherical coordinates and an expansion into spherical harmonics. The solver can be applied to and works well for all problems for which the use of spherical coordinates is appropriate. We also briefly discuss the implementation of the algorithm into hydrodynamic codes which are based on a conservative finite--difference scheme.Comment: 15 pages, compressed uu-encoded postscript file (232kB), to appear in Computer Physics Communications, special issue Computational Hydrodynamics in Astrophysic

    Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods

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    The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple, high-order implicit-time methods for parabolic PDE and implicit-explicit methods for related nonlinear PDE. We apply the IBSE method to solve the Poisson, heat, Burgers', and Fitzhugh-Nagumo equations, and demonstrate fourth-order pointwise convergence for Dirichlet problems and third-order pointwise convergence for Neumann problems

    A Class of Stable, Globally Noniterative, Nonoverlapping Domain Decomposition Algorithms for the Simulation of Parabolic Evolutionary Systems.

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    Parabolic systems are governed by time dependent partial differential equations. To obtain a high simulation quality that captures important features of a parabolic system requires solving the governing equation to an adequate accuracy, which necessitates a large sampling size in the spatial and temporal dimensions, and hence a large amount of simulation data and high computing cost. Domain decomposition is an effective method of divide-and-conquer paradigm that divides the problem domain into several subdomains, reducing the original problem into several smaller interdependent problems which can be solved in parallel. In this dissertation, we propose a class of stabilized explicit-implicit time marching (SEITM) domain decomposition algorithms for parabolic equations. Explicit-implicit time marching (EITM) algorithms are globally non-iterative nonoverlapping domain decomposition methods, which, when compared with Schwartz algorithm based parabolic solvers, are both computationally and communicationally efficient for each time step simulation but suffer from small time step size restrictions due to conditional stability. The proposed stabilization techniques in the SEITM algorithms retain the time-stepwise efficiency in computation and communication of the EITM algorithms but free the algorithms from small time step size restrictions, rendering SEITM algorithms excellent candidates for large scale parallel simulation problems. Three algorithms of the SEITM class are presented in this dissertation, which are mathematically analyzed and experimentally tested to show excellent numerical stability, computation and communication efficiencies, and high parallel speedup and scalability

    Higher-order in time “quasi-unconditionally stable” ADI solvers for the compressible Navier–Stokes equations in 2D and 3D curvilinear domains

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    This paper introduces alternating-direction implicit (ADI) solvers of higher order of time-accuracy (orders two to six) for the compressible Navier–Stokes equations in two- and three-dimensional curvilinear domains. The higher-order accuracy in time results from 1) An application of the backward differentiation formulae time-stepping algorithm (BDF) in conjunction with 2) A BDF-like extrapolation technique for certain components of the nonlinear terms (which makes use of nonlinear solves unnecessary), as well as 3) A novel application of the Douglas–Gunn splitting (which greatly facilitates handling of boundary conditions while preserving higher-order accuracy in time). As suggested by our theoretical analysis of the algorithms for a variety of special cases, an extensive set of numerical experiments clearly indicate that all of the BDF-based ADI algorithms proposed in this paper are “quasi-unconditionally stable” in the following sense: each algorithm is stable for all couples (h,Δt)of spatial and temporal mesh sizes in a problem-dependent rectangular neighborhood of the form (0,M_h)×(0,M_t). In other words, for each fixed value of Δt below a certain threshold, the Navier–Stokes solvers presented in this paper are stable for arbitrarily small spatial mesh-sizes. The second-order formulation has further been rigorously shown to be unconditionally stable for linear hyperbolic and parabolic equations in two-dimensional space. Although implicit ADI solvers for the Navier–Stokes equations with nominal second-order of temporal accuracy have been proposed in the past, the algorithms presented in this paper are the first ADI-based Navier–Stokes solvers for which second-order or better accuracy has been verified in practice under non-trivial (non-periodic) boundary conditions

    Simulations of Surfactant Spreading

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    Thin liquid films driven by surface tension gradients are studied in diverse applications, including the spreading of a droplet and fluid flow in the lung. The nonlinear partial differential equations that govern thin films are difficult to solve analytically, and must be approached through numerical simulations. We describe the development of a numerical solver designed to solve a variety of thin film problems in two dimensions. Validation of the solver includes grid refinement studies and comparison to previous results for thin film problems. In addition, we apply the solver to a model of surfactant spreading and make comparisons with theoretical and experimental results

    An investigation of alternating-direction implicit finite-difference time-domain (ADI-FDTD) method in numerical electromagnetics

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    In this thesis, the alternating-direction implicit method (ADI) is investigated in conjunction with the finite difference time-domain method (FDTD) to allow crossing of the Courant-Friedrich-Levy (CFL) stability criterion while maintaining stability in the FDTD algorithm. The main reason for this is to be able to use a larger numerical time step than that governed by the CFL criterion. The desired effect is a significant reduction in numerical run-times. Although the ADI-FDTD method has been used in the literature, most analysis and application have been performed on simple three-dimensional cavities.This work makes original contribution in two aspects. Firstly, a new modified alternating-direction implicit method for a three-dimensional FDTD algorithm has been successfully developed and implemented in this research. This new method allows correct modelling of a realistic physical structure such as a microstrip patch with the ADI scheme without causing instability even when the CFL criterion is not observed. However, due to the inherent property of this modified ADI-FDTD method, a decreasing reflection coefficient is observed using this scheme.The second and more important contribution this research makes in the field of numerical electromagnetics is the development of a new method of simulating realistic complex structures such as geometries comprising copper patch antennas on a dielectric substrate. With this new method, for the first time, the ADl-FDTD algorithm remains stable while still in violation of the CFL criterion, even when complex structures are being modelled.However, there is a trade-off between accuracy and computational speed in ADI-FDTD and modified ADI-FDTD methods. The larger the numerical time step, the shorter is the simulation run-time but an increase in numerical time step causes a degradation in accuracy of numerical results. Comparison between speed and accuracy is shown in this thesis and it has to be mentioned here that these values are very much dependent on the structure being modelled
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