898 research outputs found
Solving PDEs in Python
This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible NavierâStokes equations, and systems of nonlinear advectionâdiffusionâreaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs. This book is open access under a CC BY license
A Fixed-point Method for Computing Steady-state 2D Laser-Fluid Interactions
This research introduces a fixed-point numerical approach for solving the steady-state Navier-Stokes (NS) equations on a finite two-dimensional (2D) domain. The steady-state interaction between a high energy laser beam and its surrounding fluid medium is important to researchers in the field of high energy laser beam propagation. The solutions to the steady-state Navier-Stokes equations provide a model for uncovering the steady-state behavior of the fluid medium, which is useful for the modeling of thermal blooming in laser beam propagation. Numerical solutions remain the only tenable option for solving the NS equations, wherein numerical speed and fidelity beget the utility of any such algorithm. The timing and accuracy results from the novel fixed-point algorithm are compared to a standard Newton solver, where the fixed-point algorithm implements a series of discrete Poisson solvers through successive fixed-point iterations of fluid velocity (u,v), pressure (p), and temperature (T) in a Boussinesq fluid model. The fixed-point scheme consistently proves superior in computational cost by converging after O(N2 log N2 ) flops compared to the O(N6) flops in the Newton Solver for a discrete N x N grid. We provide a proof for the convergence of small amplitude solutions, and discuss the relationship between fluid parameters (Re, Ri, Pe) and the existence of solutions as a function of laser intensity in a bifurcation analysis
HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems
Solving PDEs in Python
This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible NavierâStokes equations, and systems of nonlinear advectionâdiffusionâreaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs. This book is open access under a CC BY license
Multidomain Spectral Method for the Helically Reduced Wave Equation
We consider the 2+1 and 3+1 scalar wave equations reduced via a helical
Killing field, respectively referred to as the 2-dimensional and 3-dimensional
helically reduced wave equation (HRWE). The HRWE serves as the fundamental
model for the mixed-type PDE arising in the periodic standing wave (PSW)
approximation to binary inspiral. We present a method for solving the equation
based on domain decomposition and spectral approximation. Beyond describing
such a numerical method for solving strictly linear HRWE, we also present
results for a nonlinear scalar model of binary inspiral. The PSW approximation
has already been theoretically and numerically studied in the context of the
post-Minkowskian gravitational field, with numerical simulations carried out
via the "eigenspectral method." Despite its name, the eigenspectral technique
does feature a finite-difference component, and is lower-order accurate. We
intend to apply the numerical method described here to the theoretically
well-developed post-Minkowski PSW formalism with the twin goals of spectral
accuracy and the coordinate flexibility afforded by global spectral
interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes
revisions based on referee reports and has two extra figure
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