TY - JOUR

T1 - A Finite Difference Method for Dispersive Linear Waves with Applications to Simulating Microwave Pulses in Water

AU - Luke, Jonathan H.C.

N1 - Funding Information:
The author is grateful to the two anonymous referees for the careful reviews and insightful comments. This work was supported by a grant (F49620-96-1-0039) from the Air Force Office of Scientific Research, Air Force Material Command, USAF. Computing equipment obtained through grants (DMS-9305665 and DMS-9407196) from the National Science Foundation was utilized. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation therein. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government.

PY - 1999/1/1

Y1 - 1999/1/1

N2 - The EPd method, a finite difference method for highly dispersive linear wave equations, is introduced and analyzed. Motivated by the problem of simulating the propagation of microwave pulses through water, the method attempts to relieve the computational burden of resolving fast processes, such as dipole relaxation or oscillation, occurring in a material with dynamic structure. This method, based on a novel differencing scheme for the time step, is considered primarily for problems in one spatial dimension with constant coefficients. It is defined in terms of the solution of an initial value problem for a system of ordinary differential equations that, in an implementation of the method, need be solved only once in a preprocessing step. For certain wave equations of interest (nondispersive systems, the telegrapher's equation, and the Debye model for dielectric media) explicit formulas for the method are presented. The dispersion relation of the method exhibits a high degree of low-wavenumber asymptotic agreement with the dispersion relation of the model to which it is applied. Comparisons with a finite difference time-domain approach and an approach based on Strang splitting demonstrate the potential of the method to substantially reduce the cost of simulating linear waves in dispersive materials. A generalization of the EPd method for problems with variable coefficients appears to retain many of the advantages seen for constant coefficients.

AB - The EPd method, a finite difference method for highly dispersive linear wave equations, is introduced and analyzed. Motivated by the problem of simulating the propagation of microwave pulses through water, the method attempts to relieve the computational burden of resolving fast processes, such as dipole relaxation or oscillation, occurring in a material with dynamic structure. This method, based on a novel differencing scheme for the time step, is considered primarily for problems in one spatial dimension with constant coefficients. It is defined in terms of the solution of an initial value problem for a system of ordinary differential equations that, in an implementation of the method, need be solved only once in a preprocessing step. For certain wave equations of interest (nondispersive systems, the telegrapher's equation, and the Debye model for dielectric media) explicit formulas for the method are presented. The dispersion relation of the method exhibits a high degree of low-wavenumber asymptotic agreement with the dispersion relation of the model to which it is applied. Comparisons with a finite difference time-domain approach and an approach based on Strang splitting demonstrate the potential of the method to substantially reduce the cost of simulating linear waves in dispersive materials. A generalization of the EPd method for problems with variable coefficients appears to retain many of the advantages seen for constant coefficients.

KW - Debye media

KW - Dispersion relation

KW - Dispersive linear waves

KW - Electromagnetics waves

KW - FDTD

KW - Finite difference method

KW - Stiff hyperbolic systems

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U2 - 10.1006/jcph.1998.6117

DO - 10.1006/jcph.1998.6117

M3 - Article

AN - SCOPUS:0347040118

SN - 0021-9991

VL - 148

SP - 199

EP - 226

JO - Journal of Computational Physics

JF - Journal of Computational Physics

IS - 1

ER -