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Title: A Marker Method for the Solution of the Damped Burgers' Equatio

Abstract

A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations.

Authors:
Publication Date:
Research Org.:
Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
934517
Report Number(s):
PPPL-4129
TRN: US0803869
DOE Contract Number:
DE-AC02-76CH03073
Resource Type:
Technical Report
Resource Relation:
Related Information: Submitted to Numerical Methods for Differential Equations
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; PARTIAL DIFFERENTIAL EQUATIONS; SHAPE; PHYSICS; Numerical Methods; Numerical Simulation; Partial Differential Equations

Citation Formats

Jerome L.V. Lewandowski. A Marker Method for the Solution of the Damped Burgers' Equatio. United States: N. p., 2005. Web. doi:10.2172/934517.
Jerome L.V. Lewandowski. A Marker Method for the Solution of the Damped Burgers' Equatio. United States. doi:10.2172/934517.
Jerome L.V. Lewandowski. Tue . "A Marker Method for the Solution of the Damped Burgers' Equatio". United States. doi:10.2172/934517. https://www.osti.gov/servlets/purl/934517.
@article{osti_934517,
title = {A Marker Method for the Solution of the Damped Burgers' Equatio},
author = {Jerome L.V. Lewandowski},
abstractNote = {A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations.},
doi = {10.2172/934517},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Tue Nov 01 00:00:00 EST 2005},
month = {Tue Nov 01 00:00:00 EST 2005}
}

Technical Report:

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  • A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
  • In this paper we consider a random walk algorithm for the solution of Burgers' equation. The algorithm uses the method of fractional steps. The non-linear advection term of the equation is solved by advecting ''fluid'' particles in a velocity field induced by the particles. The diffusion term of the equation is approximated by adding an appropriate random perturbation to the positions of the particles. Though the algorithm is inefficient as a method for solving Burgers' equation, it does model a similar method, the random vortex method, which has been used extensively to solve the incompressible Navier-Stokes equations. The purpose ofmore » this paper is to demonstrate the strong convergence of our random walk method and so provide a model for the proof of convergence for more complex random walk algorithms; for instance, the random vortex method without boundaries.« less
  • This dissertation considers solution methods for damped linear structural systems which are subjected to transient loading. To accommodate a range of possible applications, the damping is assumed to be non-proportional to mass and stiffness and may also be large and/or lumped. Some important characteristic properties of the damped system are presented. The second-order equations of motion are reduced to a first-order set by doubling the size of the problem to facilitate the subsequent analysis and computation. The Rayleigh-Ritz method is generalized for the matrix pencil associated with a damped system. A projection method is discussed as an alternative. Both providemore » a basis for computing partial eigensolutions of a large damped dynamic system. The subspace iteration method is modified to extract eigensolutions of a damped dynamic system. The iteration vectors are arranged in such a way that only real arithmetic is required to describe the complex solution vectors. The algorithm is implemented to solve some typical numerical examples. The Lanczos method also is extended to find eigensolutions of a damped dynamic system. The loss of orthogonality between Lanczos vectors is investigated and two schemes are presented to restore the required orthogonality. The algorithm presented can take full advantage of the symmetry and sparsity of the associated matrices and also involves only real arithmetic during the solution process. The algorithm is implemented and tested with the numerical examples introduced in the subspace solution. 31 refs.« less
  • A scheme for obtaining numerical solutions to the time-dependent incompressible fluid flow equations is presented. The method is designed for use on a computational grid of arbitrary polygons, and can be considered to be the Finite-Volume (FV) generalization of the Marker-and-Cell (MAC) method. A scheme resembling the advection method of van Leer is utilized in a way that yields second-order spatial accuracy and conditional stability with any level of fluid viscosity. The method is fully described in the presentation. Sample results are given for a variety of two-dimensional flow circumstances. These include non-viscous, viscous, steady, and unsteady flows on variousmore » non-orthogonal computational meshes. 10 refs., 7 figs.« less
  • This report is intended to be a ''user manual'' for the Lawrence Livermore Laboratory version of the Eulerian incompressible hydrodynamic computer code ABMAC. The theory of the numerical model is discussed in general terms. The format for data input and data printout is described in detail. A listing and flow chart of the computer code are provided.