Matthew F. Causley

Higher order A-stable schemes for the wave equation using a successive convolution approach

In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL

Method of lines transpose: An implicit solution to the wave equation

^T

Method of Lines Transpose: High Order L-Stable

) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL

{\mathcal O}(N)

^T

Schemes for Parabolic Equations Using Successive Convolution

formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient

On the Time-Domain Response of Havriliak-Negami Dielectrics

O(N)

A particle-in-cell method for the simulation of plasmas based on an unconditionally stable field solver

convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers...

Method of Lines Transpose: An Efficient Unconditionally Stable Solver for Wave Propagation

We present a new method for solving the wave equation implicitly. Our approach is to discretize the wave equation in time, following the method of lines transpose, sometimes referred to as the transverse method of lines, or Rothe’s method. We differ from conventional methods that follow this approach, in that we solve the resulting system of partial differential equations using boundary integral methods. Our algorithm extends to higher spatial dimensions using an alternating direction implicit (ADI) framework. Thus we develop a boundary integral solution, that is competitive with explicit finite difference methods, both in terms of accuracy and speed. However, it provides more flexibility in the treatment of source functions, and complex boundaries. We provide the analytical details of our one-dimensional method herein, along with a proof of the convergence of our schemes, in free space and on a bounded domain. We find that the method is unconditionally stable, and achieves second order accuracy. A caveat of the analysis is the derivation of a unique and novel optimal quadrature method, which can be viewed as a Lax-type correction.

Angled derivative approximation of the hyperbolic heat conduction equations

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Greens function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions.

Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field Models

We apply a combination of asymptotic and numerical methods to study electromagnetic pulse propagation in the Havriliak-Negami permittivity model of fractional relaxation. This dielectric model contains the Cole-Cole and Cole-Davidson models as special cases. We analytically determine the impulse response at short and long distances behind the wavefront, and validate our results with numerical methods for performing inverse Laplace transforms and for directly solving the time-domain Maxwell equations in such dielectrics. We find that the time-domain response of Havriliak-Negami dielectrics is significantly different from that obtained for Debye dielectrics. This makes possible using pulse propagation measurements in TDR setups in order to determine the appropriate dielectric model, and its parameters, for the actual dielectric whose properties are being measured.