Explicit Time Marching Schemes for Solving the Magnetic Field Volume Integral Equation

Abstract

A method for constructing explicit marching-on-intime (MOT) schemes to solve the time domain magnetic field volume integral equation (TD-MFVIE) on inhomogeneous dielectric scatterers is proposed. The TD-MFVIE is cast in the form of an ordinary differential equation (ODE) and the unknown magnetic field is expanded using curl conforming spatial basis functions. Inserting this expansion into the TD-MFVIE and spatially testing the resulting equation yield an ODE system with a Gram matrix. This system is integrated in time for the unknown time-dependent expansion coefficients using a linear multistep method. The Gram matrix is sparse and well-conditioned for Galerkin testing and consists of only four diagonal blocks for point testing. The resulting explicit MOT schemes, which call for the solution of this matrix system at every time step, are more efficient than their implicit counterparts, which call for inversion of a fuller matrix system at lower frequencies. Numerical results compare the efficiency, accuracy, and stability of the explicit MOT schemes and their implicit counterparts for low-frequency excitations. The results show that the explicit MOT scheme with point testing is significantly faster than the other three solvers without sacrificing from accuracy.

A Time Domain Volume Integral Equation Solver to Analyze Electromagnetic Scattering from Nonlinear Dielectric Objects

“…The fully-discretized TD-EFVIE (20) relates unknowns I E j and I D j to the time derivative of the unknown İE j , and is integrated in time using a P E(CE) m scheme to yield the unknown [39,[47][48][49]. This scheme uses the discretized constitutive relation (32) and the discretized Padé approximant (39) to update I E j and I D j . The steps of the P E(CE) m are provided as follows.…”

“…That said, both FDTD and TD-FEM suffer from several well-known drawbacks of the differential equation solvers: They require the computation domain to be truncated using absorbing boundary conditions or perfectly matched layers, their accuracy is limited by numerical phase dispersion, and their time step size is often restricted by the Courant-Friedrichs-Lewy (CFL) condition [1,9]. Time domain volume integral equation (TD-VIE) solvers [27][28][29][30][31][32][33][34][35][36][37][38][39] do not suffer from these drawbacks of FDTD and TD-FEM. This is because they rely on a formulation where the scattered electromagnetic field is represented as a spatio-temporal convolution between the Green function of the background medium and current/field induced in the geometry.…”

“…Inserting these expansions into TD-EFVIE and testing the resulting equation using SWG functions yield a system of ordinary differential equations (ODEs) in timedependent expansion coefficients of the SWG basis functions. A P E(CE) m scheme is used to integrate this system of ODEs in time for the unknown electric field expansion coefficients [38,39,43,[47][48][49][50][51][52][53][54][55][56][57]. Similarly, expansions of the electric field and the electric flux are inserted into the nonlinear constitutive relation and its "inverse" obtained using the Padé approximant [58][59][60].…”

An Explicit Time Marching Scheme for Efficient Solution of the Magnetic Field Integral Equation at Low Frequencies

“…Substituting (3) in (2) and spatially testing with u(rjq) and v(rjq), j = 1, ..., Nn, q = 1, ..., Np yield a time-dependent semi-discrete system of ODEs. This system has to be sampled at times t = h∆t to carry out the time integration using a PE(CE) m -type scheme [8], [20], [22]. Consequently one has to use temporal interpolation on…”

“…A PE(CE) m scheme is used to integrate the ODE system (5) to yield I h , h = 1, ..., Nt [8], [20], [22]. Steps of this scheme are briefly summarized as follows:…”