As the next stage in the numerical solution procedure of the TDIEs in (2.42)-(2.44), the time axis is divided into equal intervals ∆t defining time instants tn= n∆t.
3.2.1
Theta Method
To approximate the time derivatives, e.g. (2.9), the simplest general scheme is the theta (weighted) method
A(r, tn) − A(r, tn−1)
∆t + θ∇φ(r, tn) + (1 − θ)∇φ(r, tn−1) = E
i(r, t
n−(1−θ)) (3.23)
where specific choices of the parameter θ lead to well-known classical methods. Namely θ = 0 reflects the explicit (forward) Euler method, whereas θ = 1 implies the implicit (backward) Euler method. Besides backward and forward Euler, another first-order dif- ference method frequently used in numerical solution of differential algebraic system of equations, known as the Galerkin method, is deduced by θ = 23. Setting θ = 12 results in the only second order approximation represented by (3.23), the so-called Crank-Nicolson or implicit midpoint (trapezoidal) method, in which the time derivative associated with the vector potential term is approximated by the central finite difference and the time averaging is used for the scalar potential term. In fact, (3.23) combines all recent cases into a unified equation so that one can readily develop a unified code for implementing various time integration schemes. Generally, applying the θ-method leads to
apqmk,n = ∂Ik(τ pq mk,n) ∂τ = Ik(tr) − Ik(tr−1) ∆t (3.24)
where
tr−1 < τmk,npq ≤ tr. (3.25)
Assuming that a linear interpolation is used for approximating the values of the currents at retarded times, (3.10) is calculated for the implicit backward Euler and the explicit forward Euler respectively as
bpqmk,n|θ=1= ∆t tr−1 X t=t0 Ik(t) + δ − δ 2 2 − 1 2 Ik(tr−1) + δ2 2Ik(tr) (3.26) bpqmk,n|θ=0 = ∆t tr−2 X t=t0 Ik(t) + δ − δ 2 2 − 1 2 Ik(tr−2) + δ2 2Ik(tr−1) (3.27) where δ = τ pq mk,n − tr−1 ∆t . (3.28)
Apparently, for the general case of (3.23), bpqmk is equal to θ times (3.26) plus (1 − θ) times (3.27).
Considering the temporal variation of A(r, t) as a quadratic polynomial, a more accu- rate representation may be provided by the second order backward finite difference formula
3A(r, tn) − 4A(r, tn−1) + A(r, tn−2)
2∆t + ∇φ(r, tn) = E
i(r, t n).
This three-point backward asymmetric scheme is also 2nd order accurate in time, O(∆t2),
and results in
apqmk,n = 3Ik(tr) − 4Ik(tr−1) + Ik(tr−2) 2∆t
while the same expression as (3.26) is obtained for bpqmk,n.
Considering the DEFIE (2.11), the second time derivative can be approximated by A(r, tn) − 2A(r, tn−1) + A(r, tn−2)
∆t2 + ∇Φ(r, tn) =
∂Ei(r, tn)
∂t . (3.29)
For this case, (3.11) and (3.12), respectively, turn to
˙apqmk,n = Ik(tr) − 2Ik(tr−1) + Ik(tr−2)
∆t2 (3.30)
˙bpq
mk,n = (1 − δ)Ik(tr−1) + δIk(tr). (3.31)
The Newmark-Beta formulation [70] on the DEFIE A(r, tn) − 2A(r, tn−1) + A(r, tn−2)
∆t2 + θ∇Φ(r, tn)
+(1 − 2θ)∇Φ(r, tn−1) + θ∇Φ(r, tn−2) =
∂Ei(r, tn−1)
gives ˙apqmk,n identical to (3.30) and ˙bpq
mk,n = θ(1 − δ)Ik(tr−3) + [(1 − δ) + θ(3δ − 2)]Ik(tr−2)
+[δ + θ(1 − 3δ)]Ik(tr−1) + θδIk(tr). (3.33)
with judiciously chosen values of θ ≥ 0.2 [71].
Here, we also consider the time-domain MFIE (2.24) approximated by the implicit backward difference method,
J(r, tn) 2 − ˆn × 1 4π Z S0 " J(r′, τ n) − J(r′, τn−1) c∆t × R R2 + J(r ′, τ n) × R R3 # dS′ = ˆn × Hi(r, tn) (3.34)
where the retarded time samples τn = tn− Rc are linearly interpolated. As a result, the
discretization coefficients (3.13) and (3.14) are obtained as follows: cpqmk,n = Ik(tr) − Ik(tr−1)
∆t = a
pq
mk,n|in (3.24)
dpqmk,n = (1 − δ)Ik(tr−1) + δIk(tr) = ˙bpqmk,n.
3.2.2
Time Interpolation Methods
In the TDIE, owing to the presence of delayed terms, knowledge of past solution is required not only at nodal points rather mostly somewhere in between. Unless otherwise stated, the triangular (hat) functions are used to represent the temporal evolution, resulting in unit weights for t = tr and linear interpolation to zero for t = tr± ∆t. In order to illustrate
that the use of smoother interpolators does not necessarily enhance, but, on the contrary, shrink the extent of the stable region, the higher order interpolating functions ensuring a qth order approximation over temporal subdomains are employed as well. Considering successive orders of the Lagrange interpolants, the q + 1 points interpolant is equivalent to qth order piecewise polynomial expanded over (q + 1)∆t intervals, namely [(r − q)∆t, r∆t].
The value of the current at time instance τn thus depends on the one ahead (tr) and the q
earlier values of the discrete current coefficients behind tr. Therefore, the overall effect of
the shifted versions provides an qth order accurate interpolation over the generic rth time
interval, between the samples tr−1 = (r − 1)∆t and tr = r∆t. The result is a continuous
qth order function, with a piecewise (q − 1)th order derivative that is continuous at the
integer multiples of ∆t, except when the function to be interpolated is exactly smoother than the qth order. The present work checks the time-shifted Lagrange interpolants of
orders q = 1, 2, 3, and 4 [64]. For the sake of completeness, the cubic spline interpolation and other alternatively proposed interpolants, namely the cosine squared function [19], the optimized exponential function with all-order continuous derivatives [19], and the non- differentiable sinusoidal dome interpolant [23] are also employed to describe time evolution over the temporal subdomains. However, no explicit averaging technique is used here to filter out the intrinsic high frequency oscillations of the results as posed by many previous works [1, 18, 7].
3.2.3
Delay Differential Equation (DDE) Context
Retarded potential integral equations, namely the time-domain EFIE (2.10) and MFIE (2.24), can be stated as a general form of a DDE problem,
∂
∂ty(t) = f (t, y(t), y(t − τ(t, y(t)))), 0 ≤ t
y(t) = 0, t ≤ 0.
Contrary to the numerical ordinary differential equation (ODE) methods that furnish ap- proximate values of the solution only at nodal points, implementation of any numerical method for the solution of DDEs, e.g. (2.10) and (2.24), requires the knowledge of the approximate solution at somewhere between many past intermediate points tr other than
the nodal points tn. Therefore, in general, the DDE method is based on continuous ex-
tension of numerical ODE schemes. To provide step by step a continuous approximation of the solution, a posteriori interpolation of the solution values given by the underlying discrete ODE method can be utilized. Thus, the success of the resulting DDE method in terms of accuracy and stability depends on the particular choice of the discrete method as well as of the interpolant providing the continuous extension. It can be illustrated that owing to the presence of delayed terms some desirable accuracy and stability properties of the underlying ODE method can be destroyed when the method is applied to a DDE [34]. Therefore, the integration of a DDE can rarely be based on the plain application of some classical ODE codes, rather it requires the use of specifically designed methods considering the presence of the delayed terms.
For continuous extension of the ODE method, i.e., the continuity condition of the interpolation procedure, the order of the interpolation q ≥ 1 should be less or equal than that of the integration method [34]. In addition, it has been proven that the global order p of the piecewise discrete collocation method is preserved for any choice of the mesh by using a uniform interpolant of order q = p − 1 [34]. In fact, for the integration of a DDE only a few set of Runge-Kutta (RK) methods are stable. In the class of one-stage RK methods of order 1, the only one that is ANf-stable is the backward Euler method together
with linear interpolation [34]. One step collocation at one Gaussian point (the Galerkin method) is AN-stable also with linear interpolation. In the class of two-stage RK methods of order 2 the only one that is PN-stable is Lobatto IIIC method [34]. It is also shown that the DDE
∂
∂ty(t) = λ(t)y(t) + µ(t)y(t − τ), t0 ≤ t
y(t) = φ(t), t ≤ t0
is PN-stable for all delay τ and initial function φ(t) in constant time step size n∆t if |yn|n≥0≤ max|φ(t)| and AN-stable if |yn+1| ≤ |yn|.
So far, no A-stable discrete RK method of order 3 is known and it has been proven that no A-stable method can exceed order 4 [34]. In particular, no three-stage discrete RK method of order p ≥ 3 exists which is A-stable. Hence, it is hard to find efficient high order schemes. For example, the promising schemes with averaging over the scalar potential, e.g., the Crank-Nicolson and the proposed scheme in [23], intrigued the author to check the stability of higher order integrators such as the finite difference of fourth order accuracy with three time levels, formulating (2.9) as
A(r, tn) − A(r, tn−2) 2∆t + 1 6[∇φ(r, tn)+4∇φ(r, tn−1)+∇φ(r, tn−2)] = E i(r, t n−1).
It does not, however, provide with any order of the interpolants a reliable scheme for practical purposes.