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# Convergence of an explicit iterative leap-frog discontinuous Galerkin method for time-domain Maxwell’s equations in anisotropic materials

- Adérito Araújo
^{1}Email authorView ORCID ID profile, - Sílvia Barbeiro
^{1}and - Maryam Khaksar Ghalati
^{2}

**8**:9

https://doi.org/10.1186/s13362-018-0051-6

© The Author(s) 2018

**Received:**19 February 2018**Accepted:**8 September 2018**Published:**20 September 2018

## Abstract

We propose an explicit iterative leap-frog discontinuous Galerkin method for time-domain Maxwell’s equations in anisotropic materials and derive its convergence properties. The *a priori* error estimates are illustrated by numerical means in some experiments. Motivated by a real application which encompasses modeling electromagnetic wave’s propagation through the eye’s structures, we simulate our model in a 2D domain aiming to represent a simple example of light scattering in the outer nuclear layer of the retina.

## Keywords

- Maxwell’s equations
- Explicit iterative leap-frog discontinuous Galerkin method
- Convergence
- Light scattering

## 1 Introduction

Optical Coherence Tomography (OCT) is an increasingly popular noninvasive technique that has been successfully used as a diagnostic tool in ophthalmology in the past decades. This method allows the assessment of the human retina *in vivo* and has been shown to provide functional information. By analysing data acquired through OCT, several retinal pathologies, such as diabetic retinopathy, or macular edema, can be detected in their early stages, before noticeable morphologic alterations on the retina [16]. As OCT standard techniques only provide structural information [14], it is necessary to expand OCT data analysis to account for both structural and functional information. OCT provides also the possibility of evaluating different elements in measuring the retinal nerve fiber layer (RNFL), namely the tendency of RNFL thinning in glaucoma and other diseases that involve optic nerve atrophy. Waveguides with induced anisotropy may worth to be considered for modeling biological waveguides [10].

*E*and

*H*, respectively electric field and magnetic field. Here we shall consider the time domain Maxwell’s equations in the transverse electric mode, as in [8], where the only non-vanishing components of the electromagnetic fields are \(E_{x}\), \(E_{y}\) and \(H_{z}\). Using the following notation for the vector and scalar curl operators

*ϵ*, and the magnetic permeability of the medium,

*μ*, are varying in space, being

*μ*a scalar function and

*ϵ*an anisotropic tensor

*ϵ*is symmetric and uniformly positive definite for almost every \((x,y) \in\Omega\), and that it is uniformly bounded with a strictly positive lower bound,

*i.e.*, there are constants \(\underline{\epsilon}>0\) and \(\overline{\epsilon}> 0\) such that, for almost every \((x,y)\in\Omega\), \(\underline{\epsilon}|\xi|^{2}\le\xi^{T}\epsilon(x,y)\xi\le\overline{\epsilon}|\xi|^{2}, \forall\xi\in\mathbb{R}^{2}\). We also assume that there are constants \(\underline{\mu}>0\) and \(\overline{\mu}> 0\) such that, for almost every \((x,y) \in\Omega\), \(\underline{\mu}\le\mu(x,y) \le\overline{\mu}\).

*c*is the speed with which a wave travels along the direction of the unit normal, defined, using the effective permittivity \(\epsilon _{\mathrm{eff}}=\det(\epsilon)/(n^{T} \epsilon n)\) (see [8]), by \(c =1/\sqrt{\mu\epsilon_{\mathrm{eff}}}\).

The attention to the development of explicit high-order accurate methods for solving time-domain Maxwell’s equations in complex geometries brings to the use of discontinuous Galerkin (DG) methods with finite difference schemes for time discretization. Those methods for time-dependent Maxwell’s equations are derived considering them either in first-order hyperbolic form (see, e.g. [1, 4, 7]) or alternatively in second-order form as wave equations (see, e.g. [6]). Generally, high-order explicit schemes for time discretizations require restrictive stability conditions. For these reasons, the one-step explicit time integration methods, like leap-frog schemes, are usually considered since they are computationally efficient per update cycle and easy to implement (see, e.g. [4, 5] and the references therein). The treatment of anisotropic materials within a DGTD framework was discussed for instance in [4] (with central fluxes) and in [8] (with upwind fluxes). The stability analysis of DGTD methods for Maxwell’s equations was considered in [4], where the scheme that is defined with the central fluxes leads to a locally implicit time method in the case of SM-ABC, and [9], where the scheme is defined with the upwind fluxes leading to an implicit method. In [1] we extends the results in [4] and [9] to a fully explicit in time method for both cases, central fluxes and upwind fluxes. The error estimates derived therein show that the method is only first order convergent in time when SM-ABC are considered.

In the present work we propose an iterative predictor-corrector method based on the explicit method investigated in [1], resulting a fully explicit method that is second order convergent in time for the SM-ABC case. In the Sect. 2 we prove that the explicit iterative method converges to a second order in time implicit method and we deduce the *a priori* error estimates for the fully discrete scheme. In Sect. 3 we illustrate the theoretical results with some numerical examples and, in Sect. 4, we apply the numerical method to a computational model that aims to simulate the light scattering through the outer nuclear layer of the retina.

This work was developed in the framework of a more general project that aims to develop a computational model to simulate the electromagnetic wave’s propagation through the eye’s structures in order to create a virtual optical coherence tomography scan [13].

## 2 An explicit iterative leap-frog discontinuous Galerkin method

### 2.1 Numerical scheme

Assume that the computational domain Ω is a bounded polygonal set that is partitioned into *K* triangular elements \(T_{k}\) such that \(\overline{\Omega}= \bigcup_{k} T_{k}\). The finite element space is then taken to be \(V_{N}=\{v \in L^{2}(\Omega )^{3}: v|_{T_{k}} \in P_{N}(T_{k})^{3}\}\), where \(P_{N}(T_{k})\) denotes the space of polynomials of degree less than or equal to *N* defined on \(T_{k}\). On each element \(T_{k}\), the solution fields \({q}_{k}(x,y,t)= (E_{x}(x,y,t), E_{y}(x,y,t), H_{z}(x,y,t))^{T}\) are approximated by the piecewise polynomial functions \(\hat{q}_{k}(x,y,t) = (\hat{E}_{xk}(x,y,t),\hat{E}_{yk}(x,y,t), \hat{H}_{zk}(x,y,t))^{T}\).

*F*by a numerical flux \(F^{*}\). Reversing the integration by parts yields

*n*is the outward pointing unit normal vector of the contour.

The approximate fields are allowed to be discontinuous across element boundaries. In this way, we introduce the notation for the jumps of the field values across the interfaces of the elements, \([\hat{E}]=\hat{E}^{-}-\hat{E}^{+}\) and \([\hat{H}]=\hat{H}^{-}-\hat{H}^{+}\), where the superscript “+” denotes the neighboring element and the superscript “−” refers to the local cell. Furthermore we introduce, respectively, the cell-impedances and cell-conductances \(Z^{\pm}=\mu^{\pm}c^{\pm}\) and \(Y^{\pm}= (Z^{\pm} )^{-1}\). At the outer cell boundaries we set \(Z^{+}=Z^{-}\).

To define the fully discrete scheme, we divide the time interval into *M* subintervals by the points \(0=t^{0}< t^{1}<\cdots<t^{M}=T\), where \(t^{m}=m \Delta t\), Δ*t* is the time step size and \(T+ \Delta t/2 \leq T_{f}\). The unknowns related to the electric field are approximated at integer time-stations \(t^{m}\) and are denoted by \(\hat{E}_{k}^{m}=\hat{E}_{k}(\cdot,t^{m})\). The unknowns related to the magnetic field are approximated at half-integer time-stations \(t^{m+1/2}=(m+ \frac{1}{2}) \Delta t\) and are denoted by \(\hat{H}_{k}^{m+1/2}=\hat{H}_{k}(\cdot,t^{m+1/2})\).

The SM-ABC are discretised as in [1, 5], that is, for both upwind and central fluxes, consider \(\alpha=1\) for the numerical flux at the outer boundary and \([\tilde{E}_{x}]=\tilde{E}_{x}^{-}\), \([\tilde{E}_{y}]=\tilde{E}_{y}^{-}\) and \([\tilde{H}_{z}]=\tilde{H}_{z}^{-}\). We want to remark that for the case \(\alpha=0\), the inner iterations change the solution only in the triangles which have boundary edges, accordingly to the discretization of the SM-ABC.

This iterative scheme may be viewed as a predictor-corrector type method of the form P(C)^{niter}, where P represents the predictor step (the iteration \(n=0\)), C the correction steps (iterations \(n=1,2,\ldots \)), and niter the number of correction steps needed until the stopping criteria is verified. If we consider no inner iterations (\(\operatorname{niter}=0\)) we obtain the explicit method considered in [1].

### 2.2 Convergence result

*h*be the maximum element diameter, that is, \(h_{k}=\sup_{P_{1},P_{2} \in T_{K}} \|P_{1}-P_{2}\|\), \(h=\max_{T_{k} \in\mathcal {T}_{h}}\{h_{k}\}\). We assume that the mesh is regular in the sense that there exists a constant \(\tau>0\) such that for all \(T_{k} \in\mathcal {T}_{h}\), \(\frac {h_{k}}{\tau_{k}} \leq\tau\), where \(\tau_{k}\) denotes the maximum diameter of a ball inscribed in \(T_{k}\). It may be proved (see [12]) that, for any \(u \in P_{N}(T_{k})\), the following trace inequality holds

*N*but dependent on the shape-regularity

*τ*.

We prove the following theorem by finding upper bounds for \(\delta_{n} \hat{E}_{x_{k}}^{m+1}\), \(\delta_{n} \hat{E}_{y_{k}}^{m+1}\) and \(\delta_{n} \hat{H}_{z_{k}}^{m+3/2}\).

### Theorem 1

*The solution of the iterative predictor*-

*corrector scheme*(5)

*–*(7)

*converges to the solution of the method*(8)

*–*(10)

*provided that the stability condition of the underlying explicit method*(

*i*.

*e*., (5)

*–*(7)

*taking only the iteration*\(n=0\))

*is satisfied*,

*that is*(

*see*[1])

*with*

*where*\(C_{\tau}\)

*satisfies the trace inequality*(12), \(C_{\mathrm{inv}}\)

*is a positive constant independent of*\(h_{k}\)

*and*

*N*,

*and*\(Z_{k}\)

*and*\(Y_{k}\)

*denote respectively the cell*-

*impedance*

*Z*

*and the cell*-

*conductance*

*Y*

*inside the triangle*\(T_{k} \in\mathcal {T}_{h}\).

### Proof

For simplicity of the prove, we consider that the mesh \(\mathcal {T}_{h}\) is conforming.

The stability condition (13) ensures that \(\|\delta_{0}\hat{E}^{m+1}\|_{L^{2}(\Omega)}\) and \(\|\delta _{0}\hat{H}_{z}^{m+3/2}\|_{L^{2}(\Omega)}\) are bounded for all \(m=0,1,\ldots,M-1\).

Let us denote by \(F^{\mathrm{int}}\) the set of internal edges and \(F^{\mathrm{ext}}\) the set of edges that belong to the boundary *δ*Ω. Let \(v_{k}\) be the set of indices of the neighbouring elements of \(T_{k}\). For each \(i\in v_{k}\), we consider the internal edge \(f_{ik} = T_{i}\cap T_{k}\), and we denote by \(n_{ik}\) the unit normal oriented from \(T_{i}\) towards \(T_{k}\). For each boundary edge \(f_{k} = T_{k} \cap\delta\Omega\), \(n_{k}\) is taken to be the unitary outer normal vector to \(f_{k}\).

*n*, and replacing \(u_{k}\), \(v_{k}\) and \(w_{k}\) by, respectively, \(\delta_{n} \hat{E}_{x_{k}}^{m+1}\), \(\delta_{n} \hat{E}_{y_{k}}^{m+1}\) and \(\delta_{n} \hat{H}_{z_{k}}^{m+3/2}\) and summing over all elements \(T_{k} \in\mathcal {T}_{h}\), we obtain

*n*tends to the infinity, of the iterative predictor-corrector scheme (5)–(7) converges to the unique solution of the method (8)–(10). □

*ϵ*and

*μ*, on the properties of the space discretization namely

*N*and

*h*and on the time step Δ

*t*,

*m*, it is possible to have an estimate for the number of inner iterations needed to satisfy the stopping criterion defined by tol, considering the approximation

The next theorem establishes that the implicit method is second order convergent in time and arbitrary high order in space. So, with the previous result, we may conclude that, if a sufficient number of inner iterations is performed, i.e, if niter is sufficiently large, the iterative predictor-corrector scheme has, in practice, similar convergence properties.

### Theorem 2

*Let us consider the implicit leap*-

*frog DG method*(8)

*–*(10)

*complemented with the discrete boundary conditions defined in Sect*. 2.1

*and suppose that the solution of the Maxwell’s equations*(1)

*complemented by*(3)

*has the following regularity*: \(E_{x}, E_{y}, H_{z} \in L^{\infty}(0,T_{f};H^{s+1}(\Omega))\), \(\frac{\partial E_{x}}{\partial t}, \frac{\partial E_{y}}{\partial t}, \frac{\partial H_{z}}{\partial t} \in L^{2} (0,T_{f};H^{s+1}(\Omega) \cap L^{\infty}(\partial \Omega))\)

*and*\(\frac{\partial^{2} E_{x}}{\partial t^{2}}\), \(\frac {\partial^{2} E_{y}}{\partial t^{2}}\), \(\frac{\partial^{2} H_{z}}{\partial t^{2}}\) ∈ \(L^{2}(0,T_{f};H^{1}(\Omega))\), \(s \geq0\).

*If the time step*Δ

*t*

*satisfies*

*where*\(C_{\mathrm{inv}}\)

*and*\(C_{\tau}\)

*are the positive constants defined in the previous theorem*,

*then*

*holds*,

*where*

*C*

*is a generic positive constant independent of*Δ

*t*

*and the mesh size*

*h*.

### Proof

Follows the steps of the proof of Theorem 4.2 in [1]. □

## 3 Numerical results and discussion

*ϵ*is given by (2), with \(\epsilon _{xx}=4x^{2}+y^{2}+1\), \(\epsilon_{yy}=x^{2}+1\) and \(\epsilon_{xy}=\epsilon _{yx}=\sqrt{x^{2}+y^{2}}\). The source terms are defined in such way that the problem has the exact solution

^{1}. These results correspond to upwind fluxes. The experiments using central fluxes show analogous results in terms of order of convergence in time.

In all the experiments we performed we have observed that second order convergence in time is achieved with only one correction step of the method. So, in practice, it is more efficient to consider only one inner iteration instead of doing enough inner iterations until a certain prescribed tolerance tol is achieved.

## 4 Modeling scattered electromagnetic wave’s propagation through eye’s structures

This work is part of a research project which aims to develop a cellular model of the human retina able to simulate different retinal/cellular conditions and how these changes are translated to an Optical Coherence Tomography scan [13]. Simulating the full complexity of the retina, in particular the variation of the size and shape of each structure, distance between them and the respective refractive indexes, requires a rigorous approach that can be achieved by solving Maxwell’s equations. As the interest is to acquire the backscattered light intensity, we start this section by the scattered field formulation. Then we build up a two dimensional model which tries to represent a single nucleus of the outer nuclear layer (ONL) of the retina. The performance of our method is examined by simulating the light scattering in this 2D domain. The evolution of the scattering field intensity in time is obtained using the predictor-corrector DG method.

### 4.1 Optical coherence tomography

### 4.2 The scattered field formulation

We can exploit the linearity of the Maxwell’s equations in order to separate the electromagnetic fields (*E*, *H*) into incident fields (\(E^{i}\), \(H^{i}\)) and scattered components (\(E^{s}\), \(H^{s}\)), *i.e.*, \(E = E^{s} + E^{i}\) and \(H = H^{s} + H^{i}\).

### 4.3 Light scattering in the outer nuclear layer

We use our numerical model to simulate light scattering in the ONL. This layer has a special relevance among the retina’s layers as it consistently presents the characteristics of diabetic macular edema [2, 3]. The ONL is mostly populated by the cells bodies of light sensitive photoreceptor cells (rods and cons). Thus, we postulate that the main contribution to light scattering in this layer comes from the nucleus [15], as it is the biggest organelle in the soma and presents a high refractive index difference to the surrounding medium. As such, the ONL can be modelled as a population of spherical nuclei in an homogenous medium. As a proof of concept we present a simple simulation in a two dimensional square domain which contains circles that aims to represent, respectively, a single nucleus and three nuclei in the ONL. The permittivity inside the circles and in the background domain has different values.

Let us consider equations (15)–(17), in \(\Omega=(-1,1)^{2}\), complemented with SM-ABC and null initial conditions. The absorbing boundary conditions are chosen for the model as they avoid undesirable reflections that invade the computational domain. In the first experiment we will consider the case of just one circle: \({\mathcal {C}} = \{(x,y)\in\Omega: x^{2}+y^{2}<0.25\}\). In the second example we will consider the case of three circles: \({\mathcal {C}}_{1} = \{(x,y)\in\Omega: x^{2}+(y-0.5)^{2}<0.01\}\); \({\mathcal {C}}_{2} = \{(x,y)\in\Omega: x^{2}+y^{2}<0.01\}\); \({\mathcal {C}}_{3} = \{(x,y)\in\Omega: x^{2}+(y+0.5)^{2}<0.01\}\). In the experiments the relative permittivity and permeability and magnetic permeability are considered as constants, \(\epsilon^{i}=1\) and \(\mu^{i}=\mu=1\). The electric permittivity is considered as a diagonal matrix with \(\epsilon_{xx}(x,y)=\epsilon_{yy}(x,y)=1.2\) for \((x,y)\) inside the circles and \(\epsilon_{xx}(x,y)=\epsilon_{yy}(x,y)=1\) otherwise. For the incident wave we consider the planar wave \(E_{y}^{i}(x,t) = \cos(10(x-t))\).

^{1}), considering \(\alpha=0\) (central flux) and the approximation polynomial degree \(N=4\). The time step was chosen to be \(\Delta t=0.002\) and the final simulation time is \(T = 0.8\). The meshes are illustrated in Fig. 5. The evolution in time of the scattered field intensity (21) is plotted in Fig. 6. These results show that the scatterers are clearly identified. With this model, we can simulate more complex cellular structures only by changing the electric permittivity tensor

*ϵ*.

## 5 Conclusions

We presented an iterative explicit leap-frog DG method for time dependent Maxwell’s equations in anisotropic media, considering SM-ABC. The numerical scheme is fully explicit and converges to a second order in time implicit method. The results of a set of numerical experiments support the theoretical results and show that the second order of convergency is achieved for any number of inner iterations \(n\ge1\). Moreover we developed a 2D model which simulates the light scattering by a single nucleus in the outer nuclear layer of the retina. This work was elaborated in the framework of a more general project with a real application (see [3, 13]).

## Declarations

### Acknowledgements

Not applicable.

### Availability of data and materials

Please contact the corresponding author for data requests.

### Funding

The design, analysis, interpretation of data and writing of the manuscript have been supported by: Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020; by the Portuguese Government through the BD grant SFRH/BD/51860/2012; and by FCT/MCTES through the project reference UID/Multi/04044/2013.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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