37,971 research outputs found
Numerical Solution of Partial Differential Equations Using Polynomial Particular Solutions
Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning.
The derived polynomial particular solutions are also applied in the localized method of particular solutions to solve large-scale problems. Many numerical experiments have been performed to show the effectiveness of the particular solutions on this algorithm.
As another part of the dissertation, a modified method of particular solutions (MPS) has been used for solving nonlinear Poisson-type problems defined on different geometries. Polyharmonic splines are used as the basis functions so that no shape parameter is needed in the solution process. The MPS is also applied to compute the sizes of critical domains of different shapes for a quenching problem. These sizes are compared with the sizes of critical domains obtained from some other numerical methods. Numerical examples are presented to show the efficiency and accuracy of the method
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
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential
Equation which originated in geometric surface theory, and has been applied in
dynamic meteorology, elasticity, geometric optics, image processing and image
registration. Solutions can be singular, in which case standard numerical
approaches fail. In this article we build a finite difference solver for the
Monge-Ampere equation, which converges even for singular solutions. Regularity
results are used to select a priori between a stable, provably convergent
monotone discretization and an accurate finite difference discretization in
different regions of the computational domain. This allows singular solutions
to be computed using a stable method, and regular solutions to be computed more
accurately. The resulting nonlinear equations are then solved by Newton's
method. Computational results in two and three dimensions validate the claims
of accuracy and solution speed. A computational example is presented which
demonstrates the necessity of the use of the monotone scheme near
singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added
coment
Immersed Boundary Smooth Extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods
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 discontinuous Galerkin method for the Vlasov-Poisson system
A discontinuous Galerkin method for approximating the Vlasov-Poisson system
of equations describing the time evolution of a collisionless plasma is
proposed. The method is mass conservative and, in the case that piecewise
constant functions are used as a basis, the method preserves the positivity of
the electron distribution function and weakly enforces continuity of the
electric field through mesh interfaces and boundary conditions. The performance
of the method is investigated by computing several examples and error estimates
associated system's approximation are stated. In particular, computed results
are benchmarked against established theoretical results for linear advection
and the phenomenon of linear Landau damping for both the Maxwell and Lorentz
distributions. Moreover, two nonlinear problems are considered: nonlinear
Landau damping and a version of the two-stream instability are computed. For
the latter, fine scale details of the resulting long-time BGK-like state are
presented. Conservation laws are examined and various comparisons to theory are
made. The results obtained demonstrate that the discontinuous Galerkin method
is a viable option for integrating the Vlasov-Poisson system.Comment: To appear in Journal for Computational Physics, 2011. 63 pages, 86
figure
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
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