Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations

DOI:10.1051/m2an/2012002 www.esaim-m2an.org

TEMPORAL CONVERGENCE OF A LOCALLY IMPLICIT DISCONTINUOUS GALERKIN METHOD FOR MAXWELL’S EQUATIONS

Ludovic Moya

Abstract.In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Partic- ularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is nota prioriclear due to the component splitting which can introduce order reduction

Mathematics Subject Classification. 65M12, 65M60, 78M10.

Received June 8, 2011. Revised December 30, 2011.

Published online March 27, 2012.

1. Introduction

Nowadays, many different types of methods exist for the numerical resolution of time-domain Maxwell’s equations modeling electromagnetic wave propagation. The most prominent method among physicists and engineers is still the finite difference time-domain (FDTD) method based on Yee’s scheme [31]. This popularity is mainly due to its simplicity and efficiency in discretising simple domain problems. However, its inability to effectively handle complex geometries has prompted to search for alternatives methods. Also one of the main features of numerical methods based on finite element meshes like finite element time-domain (FETD) [17], finite volume time-domain (FVTD) [24] or discontinuous Galerkin time-domain (DGTD) [5,10,14,15] methods is the possibility of using locally refined and non-conformal space grids to easily deal with complex geometries.

In recent years there has been an increasing interest in the DGTD method. The latter is particularly well suited to the design ofhp-adaptive strategies (i.e. where the characteristic mesh sizehand the interpolation degreep change locally wherever it is needed) [8]. Thus the DGTD method can achieve a high order of accuracy and is used in many applications [4,15].

In the same time the choice of the temporal integration method is a crucial step for the global efficiency of the numerical method. We distinguish two major families for the temporal integration: implicit and explicit methods.

An implicit integration method to numerically solve a time-dependent PDE leads in general to unconditional

Keywords and phrases. Temporal convergence, discontinuous Galerkin method, time-domain Maxwell equations, component splitting, order reduction.

∗In memory of Prof. dr. Jan Verwer, who passed away unexpectedly before completion of this paper on 16 February 2011, at his home in Heiloo.

1 INRIA Sophia Antipolis - M´editerann´ee, NACHOS project-team, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France.ludovic.moya@inria.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2012

stability. Then the time step can be chosen arbitrarily large. However, implicit methods require the solution of large linear systems resulting in a high computational effort. An explicit integration method results in less computational effort per time step, but readily leads to unduly step size restrictions caused by the smallest grid elements. For examples of implicit and explicit integration methods for the semi-discrete Maxwell’s equations we refer to [3,29] and [1,10], respectively. A possible alternative to overcome step size limitation, induced by local mesh refinements in explicit methods, is to use smaller time steps precisely where the smallest elements are located, given by a local stability criterion. These local time-stepping techniques have been recently studied for second-order wave equations discretized in space by continuous or discontinuous finite element methods in [6]

and more specifically for the time-domain Maxwell’s equations discretized in space by a discontinuous Galerkin method in [11,20,26]. Considering the strengths of implicit and explicit methods, the authors of [7,23,27]

have proposed another alternative: locally implicit time integration methods. More precisely, the smallest grid elements are treated implicitly and the remaining elements explicitly by a technique we call component splitting.

If the ratio of fine to coarse elements is small, the most severe step size restrictions are overcome. The counterpart of this approach is having to solve per time step a linear system. But due to the assumed small fine to coarse cell size ratio, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. Consequently, these methods are particularly well suited if the local refinement is strongly localized. Note that the method from [7,23] has been especially designed for a discontinuous Galerkin discretization, while that from [27] covers the common spatial discretizations like finite-difference and various finite-element discretizations.

In this paper we study the temporal convergence of the locally implicit DGTD method for Maxwell’s equations initially proposed by Piperno in [23]. In particular we examine whether the method retains its second-order ODE convergence towards the true PDE solution under stable simultaneous space-time grid refinement. This question is legitimate because component splitting can cause order reduction.

The paper is organized as follows. Section2 presents the problem and the notations. In Section 3we intro- duce the component splitting and the implicit-explicit method for time-domain Maxwell’s equations spatially discretized with a discontinuous Galerkin method. This implicit-explicit integration method is a blend of the explicit one-step second-order Leap-Frog scheme (LF2) and the implicit one-step second-order Crank-Nicolson scheme (CN2). Next we study the temporal convergence order. The different steps of this convergence analysis are the same as in [27] where the author proposes another implicit-explicit integration method for the semi- discrete Maxwell’s equations. The latter method is also a blend of LF2 and CN2 and retains its second-order ODE convergence towards the true PDE solution under stable simultaneous space-time grid refinement [27].

In Section 4 we give some numerical results for 2D Maxwell problems to illustrate the previous convergence analysis. Section5concludes the paper with final remarks and future plans.

2. Problem statement

We consider time-domain Maxwell’s equations

∂tE = curlH −σE−JE, μ ∂tH =−curlE,

(2.1) where E and H denote the electric and magnetic field, respectively. JE is the given source current and , μ andσare coefficients representing dielectric permittivity, magnetic permeability and conductivity, respectively.

After discretization in space by a DG method we obtain the semi-discrete Maxwell system

⎧⎨

⎩

M ∂tE =SH−DE+Mf^E, M^μ ∂tH =−S^TE+M^μf^H,

(2.2) where, for convenience, we use the same notation for the electric and magnetic fieldsE andH as in the space- continuous case. For more details on DG spatial discretization we refer to [7]. The matricesM,M^μ are the DG

mass matrices which contain the values of the dielectric permittivity and magnetic permeability coefficient. The matrixS emanates from the discretization of the curl operator. The matrixD is associated with the dissipative conduction term−σE. ThroughoutDmay be assumed symmetric positive semi-definite. The functionsf^E and f^H are associated with source terms. More preciselyf^E represents the given source current −JE, butf^E and f^H may also contain Dirichlet boundary data.

We can give an equivalent formulation of (2.2) without mass matrix. As in [1] we introduce the Cholesky factorizations

M=LML^TM andM^μ=LM^μL^TM^μ, (2.3) whereLM andLM^μ are triangular matrices. Then with (2.2) we have

⎧⎨

⎩

LML^T_M ∂tE =SH−DE+LML^T_Mf^E, LM^μL^T_Mμ ∂tH =−S^TE+LM^μL^T_Mμf^H.

(2.4)

Introducing ˜E=L^T_ME and ˜H=L^T_M^μH, we get

⎧⎨

⎩

∂tE˜ =L⁻¹_M S (L^T_Mμ)⁻¹ H˜ −L⁻¹_M D(L^T_M)⁻¹ E˜+L^T_M f^E,

∂tH˜ =−L⁻¹_Mμ S^T (L^T_M)⁻¹ E˜+L^T_M^μ f^H.

(2.5)

Next we write

S˜=L⁻¹_M S (L⁻¹_Mμ)^T,D˜ =L⁻¹_M D(L⁻¹_M)^T,

f˜^E=L^T_M f^E, f˜^H=L^T_Mμ f^H, (2.6)

and note that

S˜^T = [L⁻¹_M S (L⁻¹_M^μ)^T]^T =L⁻¹_M^μ [L⁻¹_M S]^T =L⁻¹_M^μ S^T (L⁻¹_M)^T. (2.7) Thus we can write the semi-discrete Maxwell system equivalent to (2.2) as

⎧⎨

⎩

∂tE˜ = ˜SH˜ −D˜E˜+ ˜f^E,

∂tH˜ =−S˜^TE˜+ ˜f^H.

(2.8)

For convenience of notation and presentation we use the same notation in (2.2) and (2.8)i.e.

⎧⎨

⎩

∂tE =SH−DE+f^E,

∂tH =−S^TE+f^H.

(2.9)

We will proceed with (2.9), the meaning ofE,H,S,D,f^E andf^H will always be clear from the context or will be precised. In particular results obtained for (2.9) apply to (2.2) and vice versa. Note thatSemanates from an appropriate DG discretization for the Maxwell problem under consideration and further on we will prove that

S∼ 1

h, forh→0, (2.10)

where the parameter h denotes the maximum diameter of the (non-uniform) grid elements. Throughout the remainder we will assume initial values at time t= 0 and the source functions f^E(t),f^H(t)∈ C²[0, T] for an interval [0, T], so thatE(t),H(t)∈C³[0, T].

3. The implicit-explicit DGTD method

The implicit-explicit integration considered in this note is issued from [7,23]. As we have previously mentioned it is a blend of the second order LF2 scheme that we write in the three-stage form, emanating from Verlet’s method, see [23]

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Hⁿ⁺¹² −Hⁿ

Δt/2 =−S^TEⁿ+f^H(tn), Eⁿ⁺¹−Eⁿ

Δt =SHⁿ⁺¹² −¹₂D(Eⁿ⁺¹+Eⁿ) +¹₂(f^E(tn+1) +f^E(tn)), Hⁿ⁺¹−Hⁿ⁺¹²

Δt/2 =−S^TEⁿ⁺¹+f^H(tn+1),

(3.1)

and the second order, unconditionally stable CN2 scheme that we also write in the three-stage form

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Hⁿ⁺¹² −Hⁿ

Δt/2 =−S^TEⁿ+f^H(tn), Eⁿ⁺¹−Eⁿ

Δt =SHⁿ⁺¹−¹₂D(Eⁿ⁺¹+Eⁿ) +¹₂(f^E(tn+1) +f^E(tn)), Hⁿ⁺¹−Hⁿ⁺¹²

Δt/2 =−S^TEⁿ⁺¹+f^H(tn+1).

(3.2)

which only differ in the middle stage in the time level forH (see [27]). HereinΔt=tn+1−tn denotes the step size and upper indices time levels, as usual.

3.1. Component splitting

The set of grid elements is assumed to be partitioned into two subsets: one made of the smallest elements that will be treated implicitly using the CN2 method and the other one of the remaining elements that will be treated explicitly with the LF2 method. In line with this splitting the problem unknowns are reordered as

E= Ee

Ei

and H = He

Hi

, (3.3)

where the indicesiande are associated to the elements of the subset treated implicitly and explicitly, respec- tively. Likewise the semi-discrete curl operatorS is split into the block form

S =

Se −Aei

−Aie Si

, (3.4)

for the specific meaning of the block-entries ofS we refer to [7].D is supposed to be split accordingly into D=

De 0 0 Di

, (3.5)

and we write

f^E =

⎛

⎝f_e^E f_i^E

⎞

⎠, f^H=

⎛

⎝f_e^H f_i^H

⎞

⎠. (3.6)

Inserting this splitting into (2.9) we obtain the system of ODEs

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

∂tEe =SeHe−AeiHi−DeEe+f_e^E(t),

∂tEi =SiHi−AieHe−DiEi+f_i^E(t),

∂tHe=−S_e^TEe+A^T_ieEi+f_e^H(t),

∂tHi =−S_i^TEi+A^T_eiEe+f_i^H(t).

(3.7)

3.2. The implicit-explicit time integration method

The implicit-explicit method proposed in [7,23] is a blend of LF2 and CN2 applied to (3.7). It reads

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

He^n+1/2−H_eⁿ

Δt/2 =−S_e^TE_eⁿ+A^T_ieE_iⁿ+f_e^H(tn), Ee^n+1/2−E_eⁿ

Δt/2 =SeHe^n+1/2−AeiH_iⁿ−DeEeⁿ+fe^E(tn),

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

E_iⁿ⁺¹−E_iⁿ Δt =Si

H_iⁿ⁺¹+H_iⁿ 2

−AieHe^n+1/2

−Di

E_iⁿ⁺¹+E_iⁿ 2

+f_i^E(tn+1) +f_i^E(tn)

2 ,

H_iⁿ⁺¹−H_iⁿ

Δt =−S_i^T

E_iⁿ⁺¹+E_iⁿ 2

+A^T_eiE_e^n+1/2+f_i^H(tn+1) +f_i^H(tn)

2 ,

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

E_eⁿ⁺¹−Ee^n+1/2

Δt/2 =SeHe^n+1/2−AeiH_iⁿ⁺¹−DeEeⁿ⁺¹+fe^E(tn+1), H_eⁿ⁺¹−He^n+1/2

Δt/2 =−S_e^TE_eⁿ⁺¹+A^T_ieE_iⁿ⁺¹+f_e^H(tn+1).

(3.8)

For the stability analysis of this method we refer to [7]. The proof is based on the conservation of a discrete electromagnetic energy. It is proved that this energy is a positive quadratic form of the numerical unknowns E_eⁿ, E_iⁿ, H_eⁿ and H_iⁿ under a condition on the time step size. Consequently the non-dissipative nature of the method yields to the stability of the method.

3.3. Matrix behavior for h → 0

In this subsection we are interested in the behavior of the matrices in (3.8) forh→0. This is an essential point for convergence analysis because some of these matrices can lie at the origin of order reduction. Let us consider the general case of dimensiond(d= 1, 2 or 3). First we investigate the behavior of the matrices in the formulation with the mass matrices. Thereafter we will be able to deduce the behavior of the matrices in (3.8).

We reintroduce the notation with a tilde for the elements involved in the formulation without mass matrix (see (2.8)) in order to avoid confusion. The specific meaning of the block-entries of the different matrices involved in the formulation with mass matrices can be found in [7]. First we observe that the mass matrices are only composed of volumic terms, hence we have

M, M^μ h^d, forh→0. (3.9)

Thus

LM, L^T_M, LM^μ andL^T_M^μ h^d2, forh→0. (3.10)

The matricesSeandSi are composed of volumic and surfacic terms, hence

SeandSi h^d−1, forh→0. (3.11)

The matricesAei andAierepresent surfacic terms (interface matrices). Hence

AeiandAie h^d−1, forh→0 (3.12)

and from the block form of the matrixS (see (3.4)) we deduce with (3.11) and (3.12) that

S h^d−1, forh→0. (3.13)

From (2.6) we get

S˜=L⁻¹_M S (L⁻¹_M^μ)^T, (3.14)

then with the behaviors above we deduce that forh→0 S˜e, S˜i = O

1 h

, A˜ei, A˜ie = O

1 h

,

(3.15)

and we have the expected behavior (2.10) for ˜S.

3.4. Temporal convergence

In this section we are interested in the PDE convergence of method (3.8). More precisely, we will examine whether the method retains its second-order ODE convergence under stable simultaneous space-time grid refine- mentΔth,h→0 towards the true PDE solution. This is nota priori clear due to the component splitting which can introduce order reduction through error constants which grow withh⁻¹, forh→0.

This section is organized in four subsections. In Section 3.4.1 we will eliminate the intermediate values of (3.8) to get an equivalent one step formula fromtn to tn+1that we will use for our convergence analysis. In Section3.4.2we will introduce the perturbed method obtained by substituting the true PDE solution restricted to the assumed space grid into (3.7), and defects (space-time truncation errors) obtained by substituting this true PDE solution into the equivalent one step formula of our method. In Section 3.4.3 we will define the common one-step recurrence relation for the global error. In Section3.4.4we will point out the order reduction mentioned above. Finally in Section3.4.5 we will see that this order reduction, affecting the local error, may (partly) cancel in the transition from the local to the global error.

3.4.1. Elimination of intermediates values

First we treatHe. From the first and last equations of (3.8) we get Heⁿ⁺¹² =H_eⁿ−Δt

2 S_e^TE_eⁿ+Δt

2 A^T_ieE_iⁿ+Δt

2 f_e^H(tn), Heⁿ⁺¹² =H_eⁿ⁺¹+Δt

2 S_e^TEⁿ⁺¹_e −Δt

2 A^T_ieE_iⁿ⁺¹+Δt

2 f_e^H(tn+1).

(3.16)

Inserting the first equation of (3.16) into the last equation of (3.8) yields H_eⁿ⁺¹=H_eⁿ−Δt

2 S_e^T(E_eⁿ+E_eⁿ⁺¹) +Δt

2 A^T_ie(E_iⁿ+E_iⁿ⁺¹) +Δt

2 (f_e^H(tn) +f_e^H(tn+1)). (3.17)

Next we treatEe. From the second and fifth equations of (3.8) we get Eⁿ⁺e ¹² =E_eⁿ+Δt

2 SeHeⁿ⁺¹² −Δt

2 AeiH_iⁿ−Δt

2 DeE_eⁿ+Δt 2 f_e^E(tn), Eⁿ⁺e ¹² =Eeⁿ⁺¹−Δt

2 SeHeⁿ⁺¹² +Δt

2 AeiH_iⁿ⁺¹+Δt

2 DeEeⁿ⁺¹−Δt

2 fe^E(tn+1).

(3.18)

Inserting the first equation of (3.18) and half of each expression of (3.16) for Heⁿ⁺¹² into the fifth equation of (3.8) yields

E_eⁿ⁺¹=E_eⁿ+Δt

2 Se(H_eⁿ+H_eⁿ⁺¹)−Δt

2 Aei(H_iⁿ+H_iⁿ⁺¹)−Δt

2 De(E_eⁿ+E_eⁿ⁺¹) +Δt

2 (f_e^E(tn) +f_e^E(tn+1)) +Δt²

4 Se(−S_e^TE_eⁿ+A^T_ieE_iⁿ) +Δt²

4 Se(S_e^TEⁿ⁺¹_e −A^T_ieE_iⁿ⁺¹) +Δt²

4 Se(f_e^H(tn) +f_e^H(tn+1)).

(3.19) Now we considerHi. Inserting half of each expression of (3.18) forEeⁿ⁺¹² in the fourth equation of (3.8) gives

H_iⁿ⁺¹=H_iⁿ−Δt

2 S_i^T(E_iⁿ+E_iⁿ⁺¹) +Δt

2 A^T_ei(E_eⁿ+E_eⁿ⁺¹) +Δt

2 (f_i^H(tn) +f_i^H(tn+1)) +Δt²

4 A^T_eiAei(H_iⁿ⁺¹−H_iⁿ) +Δt²

4 A^T_eiDe(E_eⁿ⁺¹−E_eⁿ) +Δt²

4 A^T_ei(f_e^E(tn) +f_e^E(tn+1)). (3.20) Finally we treatEi. Inserting half of each expression of (3.16) forHeⁿ⁺¹² in the third equation of (3.8) yields

E_iⁿ⁺¹=E_iⁿ+Δt

2 Si(H_iⁿ+H_iⁿ⁺¹)−Δt

2 Aie(H_eⁿ+H_eⁿ⁺¹)−Δt

2 Di(E_iⁿ+E_iⁿ⁺¹) +Δt

2 (f_i^E(tn) +f_i^E(tn+1))

−Δt²

4 Aie(−S_e^TE_eⁿ+A^T_ieE_iⁿ)−Δt²

4 Aie(S_e^TEⁿ⁺¹_e −A^T_ieE_iⁿ⁺¹)−Δt²

4 Aie(f_e^H(tn) +f_e^H(tn+1)). (3.21) The equivalent method of (3.8) with its intermediate values eliminated thus reads

E_eⁿ⁺¹ =E_eⁿ+Δt

2 Se(H_eⁿ+H_eⁿ⁺¹)−Δt

2 Aei(H_iⁿ+H_iⁿ⁺¹)−Δt

2 De(E_eⁿ+E_eⁿ⁺¹) +Δt

2 (f_e^E(tn) +f_e^E(tn+1)) +Δt²

4 Se[(−S_e^TE_eⁿ+A^T_ieEⁿ_i)−(S^T_eE_eⁿ⁺¹−A^T_ieE_iⁿ⁺¹)] +Δt²

4 Se(f_e^H(tn) +f_e^H(tn+1)), E_iⁿ⁺¹ =E_iⁿ+Δt

2 Si(H_iⁿ+H_iⁿ⁺¹)−Δt

2 Aie(H_eⁿ+H_eⁿ⁺¹)−Δt

2 Di(E_iⁿ+E_iⁿ⁺¹) +Δt

2 (f_i^E(tn) +f_i^E(tn+1))

−Δt²

4 Aie[(−S_e^TE_eⁿ+A^T_ieE_iⁿ)−(S_e^TE_eⁿ⁺¹−A^T_ieE_iⁿ⁺¹)]−Δt²

4 Aie(f_e^H(tn) +f_e^H(tn+1)), H_eⁿ⁺¹=H_eⁿ−Δt

2 S_e^T(E_eⁿ+E_eⁿ⁺¹) +Δt

2 A^T_ie(E_iⁿ+Eⁿ⁺¹_i ) +Δt

2 (f_e^H(tn) +f_e^H(tn+1)), H_iⁿ⁺¹=H_iⁿ−Δt

2 S_i^T(E_iⁿ+E_iⁿ⁺¹) +Δt

2 A^T_ei(Eeⁿ+Eⁿ⁺¹e ) +Δt

2 (f_i^H(tn) +f_i^H(tn+1)) +Δt²

4 A^T_eiAei(H_iⁿ⁺¹−H_iⁿ) +Δt²

4 A^T_ei[De(E_eⁿ⁺¹−E_eⁿ) + (f_e^E(tn) +f_e^E(tn+1))].

(3.22)

3.4.2. The perturbed method and defects for the PDE solution

LetE_e^h(t) denote at timetthe true solution of the PDE problem restricted to the assumed space grid that we have approximated with the semi-discrete system (3.7).E_e^h(tn) thus represents the vector that is approximated by E_eⁿ. Assume the same notation forEi, He and Hi. Substituting E_e^h(t), E_i^h(t), H_e^h(t) andH_i^h(t) into (3.7) reveals the spatial truncation errors

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ d

dtE_e^h(t) =SeH_e^h(t)−AeiH_i^h(t)−DeE_e^h(t) +f_e^E(t) +σ_e^E(t), d

dtE_i^h(t) =SiH_i^h(t)−AieH_e^h(t)−DiE^h_i(t) +f_i^E(t) +σ_i^E(t), d

dtHe^h(t) =−S_e^TEe^h(t) +A^T_ieE_i^h(t) +fe^H(t) +σ^He (t), d

dtH_i^h(t) =−S_i^TE_i^h+A^T_eiE_e^h(t) +f_i^H(t) +σ^H_i (t),

(3.23)

whereσ^E_e(t),σ_i^E(t),σ_e^H(t) andσ^H_i (t) denote the spatial truncation errors.

Substituting Ee^h(t),Ei^h(t),He^h(t) andHi^h(t) into (3.22) reveals the defects for the PDE solution (space-time truncation errors) and gives what we call the perturbed method

E_e^h(tn+1) =E_e^h(tn) +Δt

2 Se(H_e^h(tn) +H_e^h(tn+1))−Δt

2 Aei(H_i^h(tn) +H_i^h(tn+1))

−Δt

2 De(E_e^h(tn) +E_e^h(tn+1)) +Δt

2 (f_e^E(tn) +f_e^E(tn+1)) +Δt²

4 Se[(−S_e^TE_e^h(tn) +A^T_ieE_i^h(tn))−(S_e^TE_e^h(tn+1)−A^T_ieE_i^h(tn+1))]

+Δt²

4 Se(f_e^H(tn) +f_e^H(tn+1)) +Δt δ^E_e,n, E_i^h(tn+1) =E_i^h(tn) +Δt

2 Si(H_i^h(tn) +H_i^h(tn+1))−Δt

2 Aie(H_e^h(tn) +H_e^h(tn+1))

−Δt

2 Di(E_i^h(tn) +E_i^h(tn+1)) +Δt

2 (f_i^E(tn) +f_i^E(tn+1))

−Δt²

4 Aie[(−S_e^TE_e^h(tn) +A^T_ieE_i^h(tn))−(S_e^TE_e^h(tn+1)−A^T_ieE_i^h(tn+1))]

−Δt²

4 Aie(f_e^H(tn) +f_e^H(tn+1)) +Δt δ^E_i,n, He^h(tn+1) =He^h(tn)−Δt

2 Se^T(Ee^h(tn) +Ee^h(tn+1)) +Δt

2 A^Tie(Ei^h(tn) +Ei^h(tn+1)) +Δt

2 (f_e^H(tn) +f_e^H(tn+1)) +Δt δ_e,n^H , H_i^h(tn+1) =H_i^h(tn)−Δt

2 S_i^T(E_i^h(tn) +E_i^h(tn+1)) +Δt

2 A^T_ei(E_e^h(tn) +E_e^h(tn+1)) +Δt

2 (f_i^H(tn) +f_i^H(tn+1)) +Δt²

4 A^TeiAei(Hi^h(tn+1)−Hi^h(tn)) +Δt²

4 A^T_ei[De(E_e^h(tn+1)−E_e^h(tn)) + (f_e^E(tn) +f_e^E(tn+1))] +Δt δ_i,n^H,

(3.24) whereδ_e,n^E , δ_i,n^E ,δ^H_e,nandδ^H_i,n denote the defects for the PDE solution.

Eliminating all source term contributionsf_e^E,f_i^E,f_e^H andf_i^H (3.23) yields δ_e,n^E =δE_e^h+Δt

4 Se

d

dt(H_e^h(tn+1)−H_e^h(tn)) +1

2(σ^E_e(tn) +σ^E_e(tn+1)) +Δt

4 Se(σ^H_e (tn) +σ^H_e (tn+1)), δ_i,n^E =δ_E^h

i −Δt 4 Aie

d

dt(H_e^h(tn+1)−H_e^h(tn)) +1

2(σ_i^E(tn) +σ^E_i (tn+1))−Δt

4 Aie(σ_e^H(tn) +σ_e^H(tn+1)), δ_e,n^H =δH_e^h+1

2(σ^H_e(tn) +σ_e^H(tn+1)) δ_i,n^H =δ_Hi^h−Δt

4 A^T_eiSe(H_e^h(tn+1)−H_e^h(tn)) +Δt 4 A^T_eid

dt(E_e^h(tn+1)−E_e^h(tn)) +1

2(σ^H_i (tn) +σ^H_i (tn+1)) +Δt

4 A^T_ei(σ^E_e(tn) +σ_e^E(tn+1)),

(3.25)

whereδ_Ee^h denotes the implicit trapezoidal (CN) rule defect (see [27]) for variableE_e^h(similarly forE_i^h,H_e^hand H_i^h),i.e.

δE_e^h(t) = Ee^h(t+Δt)−Ee^h(t)

Δt −1

2 d

dt(E_e^h(t+Δt) +E_e^h(t)). (3.26) 3.4.3. The error scheme

Letε^E_e,n=E_e^h(tn)−Eⁿ_e denote the global error (similarly we introduceε^E_i,n,ε^H_e,nandε^H_i,n). Substracting (3.22) and (3.24) we obtain the error scheme

ε^E_e,n+1 =ε^Ee,n+Δt

2 Se(ε^He,n+ε^H_e,n+1)−Δt

2 Aei(ε^H_i,n+ε^H_i,n+1)−Δt

2 De(ε^Ee,n+ε^E_e,n+1) +Δt²

4 Se[(−S_e^Tε^E_e,n+A^T_ieε^E_i,n)−(S_e^Tε^E_e,n+1−A^T_ieε^E_i,n+1)] +Δt δ^E_e,n, ε^E_i,n+1 =ε^E_i,n+Δt

2 Si(ε^H_i,n+ε^H_i,n+1)−Δt

2 Aie(ε^H_e,n+ε^H_e,n+1)−Δt

2 Di(ε^E_i,n+ε^E_i,n+1)

−Δt²

4 Aie[(−S_e^Tε^E_e,n+A^T_ieε^E_i,n)−(S_e^Tε^E_e,n+1−A^T_ieε^E_i,n+1)] +Δt δ_i,n^E , ε^H_e,n+1 =ε^H_e,n−Δt

2 S_e^T(ε^E_e,n+ε^E_e,n+1) +Δt

2 A^T_ie(ε^E_i,n+ε^E_i,n+1) +Δt δ_e,n^H , ε^H_i,n+1 =ε^H_i,n−Δt

2 S^T_i (ε^E_i,n+ε^E_i,n+1) +Δt

2 A^T_ei(ε^E_e,n+ε^E_e,n+1) +Δt²

4 A^T_eiAei(ε^H_i,n+1−ε^H_i,n) +Δt²

4 A^T_eiDe(ε^E_e,n+1−ε^E_e,n) +Δt δ_i,n^H.

(3.27)

Let

εn =

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝ ε^Ee,n

ε^E_i,n ε^He,n

ε^H_i,n

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠

and δn=

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎝ δe,n^E

δ_i,n^E δe,n^H

δ_i,n^H

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎠

, (3.28)

then from (3.27) we can write the global error in a more compact form (one-step recurrence relation)

εn+1=Rεn+Δtρn, R=R_L⁻¹RR, ρn=R⁻¹_L δn, (3.29)