• Corpus ID: 203626584

Fast integral equation methods for linear and semilinear heat equations in moving domains

@article{Wang2019FastIE,
  title={Fast integral equation methods for linear and semilinear heat equations in moving domains},
  author={Jun Wang and Leslie Greengard and Shidong Jiang and Shravan K. Veerapaneni},
  journal={ArXiv},
  year={2019},
  volume={abs/1910.00755}
}
We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite difference methods in terms of accuracy, stability and space-time adaptivity. In order to be practical, however, a number of technical capabilites are required: fast algorithms for the evaluation of heat potentials, high-order accurate quadratures for singular and… 
A fast time domain solver for the equilibrium Dyson equation
TLDR
This work proposes a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator of Volterra integro-differential equations, and can reach large propagation times, and is thus wellsuited to explore quantum many-body phenomena at low energy scales.
A High‐Order Integral Equation‐Based Solver for the Time‐Dependent Schrödinger Equation
TLDR
A numerical method is introduced for the solution of the time-dependent Schrodinger equation with a smooth potential, based on its reformulation as a Volterra integral equation, which avoids the need for artificial boundary conditions, admits simple, inexpensive high-order implicit time marching schemes, and naturally includes time- dependent potentials.
Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations
TLDR
This note proposes to use the hierarchical interpolative factorization as the preconditioning for the conjugate gradient iteration of the Crank-Nicholson time stepping method, and offers an efficient and accurate approximate inverse of the linear system.
Parallel Skeletonization for Integral Equations in Evolving Multiply-Connected Domains
TLDR
This paper presents an application that leverages a parallel implementation of skeletonization with updates in a shape optimization regime and retains the locality property afforded by the "proxy point method," and allows for a parallelized implementation where different processors work on different parts of the boundary simultaneously.
A fast, high-order numerical method for the simulation of single-excitation states in quantum optics
TLDR
This work considers the numerical solution of a nonlocal partial differential equation which models the process of collective spontaneous emission in a two-level atomic system containing a single photon, and describes an efficient solver for the case of a Gaussian atomic density.

References

SHOWING 1-10 OF 88 REFERENCES
Fast Adaptive Methods for the Free-Space Heat Equation
  • J. Strain
  • Computer Science
    SIAM J. Sci. Comput.
  • 1994
TLDR
Efficient and accurate new adaptive methods that solve the free-space heat equation with large timesteps that combine the fast Gauss transform with an adaptive refinement scheme that represents the solution as a continuous piecewise polynomial, to a user-specified degree of accuracy are presented.
An Efficient Galerkin Boundary Element Method for the Transient Heat Equation
TLDR
If the parabolic multipole method is used to apply parabolic boundary integral operators, the overall complexity of the scheme is log-linear while preserving the convergence of the Galerkin discretization method.
On the numerical solution of the heat equation I: Fast solvers in free space
High Order Accurate Methods for the Evaluation of Layer Heat Potentials
TLDR
It is shown that standard quadrature schemes suffer from a poorly recognized form of inaccuracy, which is referred to as “geometrically induced stiffness,” but that rules based on product integration of the full heat kernel in time are robust.
Efficient high-order integral equation methods for the heat equation
EFFICIENT HIGH-ORDER INTEGRAL EQUATION METHODS FOR THE HEAT EQUATION by Shaobo Wang Efficient high-order integral equation methods have been developed for solving the boundary value problems of the
A fast method for solving the heat equation by layer potentials
A High-Order Solver for the Heat Equation in 1D domains with Moving Boundaries
TLDR
A fast high-order accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces is described, which is characterized as “fast”; that is, it is work-optimal up to a logarithmic factor.
An adaptive fast solver for the modified Helmholtz equation in two dimensions
...
...