^{1}

^{*}

^{2}

^{*}

In this paper, a new kind of energy identities for the Maxwell equations with periodic boundary conditions is proposed and then proved rigorously by the energy methods. By these identities, several modified energy identities of the ADI-FDTD scheme for the two dimensional (2D) Maxwell equations with the periodic boundary conditions are derived. Also by these identities it is proved that 2D-ADI-FDTD is approximately energy conserved and unconditionally stable in the discrete L
^{2} and H
^{1} norms. Experiments are provided and the numerical results confirm the theoretical analysis on stability and energy conservation.

The alternative direction implicit finite difference time domain (ADI-FDTD) methods, proposed in [^{2} and H^{1} norms. Is there any other structure which can keep energy conservation for Maxwell equations? Is there any other energy identity for ADI-FDTD method? This two interesting questions promote us to find other energy- conservation structure.

In this paper, we focus our attention on structure with periodic boundary conditions and propose energy identities in L^{2} and H^{1} norms of the 2D Maxwell equations with periodic boundary conditions. We derive the energy identities of ADI-FDTD for the 2D Maxwell equations (2D-ADI-FDTD) with periodic boundary conditions by a new energy method. Several modified energy identities of 2D-ADI-FDTD in terms of the discrete L^{2} and H^{1} norms are presented. By these identities it is proved that 2D-ADI-FDTD with the periodic boundary conditions is unconditionally stable and approximately energy conserved under the discrete L^{2} and H^{1} norms. To test the analysis, experiments to solve a simple problem with exact solution are provided. Computational results of the energy and error in terms of the discrete L^{2} and H^{1} norms confirm the analysis on the energy conservation and the unconditional stability.

The remaining parts of the paper are organized as follows. In Section 2, energy identities of the 2D Maxwell equations with periodic conditions in L^{2} and H^{1} norms are first derived. In Section 3, several modified energy identities of the 2D-ADI-FDTD method are derived, the unconditional stability and the approximate energy conservation in the discrete L^{2} and H^{1} norms are then proved. In Section 4, the numerical experiments are presented.

Consider the two-dimensional (2D) Maxwell equations:

in a rectangular domain with electric permittivity ε and magnetic permeability μ, where ε and μ are positive constants;

We assume that the rectangular region Ω is surrounded by periodic boundaries, so the boundary conditions can be written as

We also assume the initial conditions

It can be derived by integration by parts and the periodic boundary conditions (2.2)-(2.3) that the above Maxwell equations have the energy identities:

Lemma 2.1 Let

where and in what follows, ^{2} norm with the weights ε (corresponding electric field) or µ (magnetic field). For example,

Identity (2.5) is called the Poynting Theorem and can be seen in many classical physics books. Besides the above energy identities, we found new ones below.

Theorem 2.2 Let

where u = x or y, and ^{1} norm (the H^{1} norm of f is defined by

^{1}-semi norm of f).

Proof. First, we prove Equation (2.7) with u = x. Differentiating each of the Equations in (2.1) with respect to x leads to

By the integration by parts and the periodic boundary conditions (2.2)-(2.3), we have

where

Multiplying the Equations (2.9) by

From (2.1) and the boundary conditions (2.2)-(2.3) we note that

So,

The alternating direction implicit FDTD method for the 2D Maxwell equations (denoted by 2D-ADI-FDTD) was proposed by (Namiki, 1999). For convenience in analysis of this scheme, next we give some notations. Let

where Δx and Δy are the mesh sizes along x and y directions, ∆t is the time step, I, J and N are positive integers. For a grid function

where u = x, y or t. For

Other norms:

the approximations of

Stage 1:

Stage 2:

For simplicity in notations, we sometimes omit the subscripts of these field values without causing any ambiguity. By the definition of cross product of vectors, the boundary conditions for (2.2)-(2.3) become

where

In this Section we derive modified energy identities of 2D-ADI-FDTD and prove its energy conservation and unconditional stability in the discrete H^{1} norm.

Theorem 3.1 Let

where for

Proof. First we prove (3.1). Applying

Multiplying both sides of the equations, (3.3)-(3.4) by

Applying summation by parts, we see that

where we have used that

So, if summing each of the Equalities (3.9)-(3.11) over their subscripts, adding the updated equations, multiplying both sides by ΔxΔy, and using the two identities, (3.12) and (3.13), together with the norms defined in Subsection 2.2, we arrive at

Similar argument is applied to the second Stage (3.6)-(3.8), we have

Combination of (3.14) and (3.15) leads to the identity (3.1). Identity (3.2) is similarly derived by repeating the above argument from the operated Equations (2.14)-(2.19) by

In the above proof, if taking

Theorem 3.2 Let

Combining the results in Theorems 3.1 and 3.2 we have

Theorem 3.3 If the discrete H^{1} semi-norm and H^{1} norm of the solution of 2D-ADI-FDTD are denoted respectively by

then, the following energy identities for 2D-ADI-FDTD hold

Remark 3.4 It is easy to see that the identities in Theorems 3.1, 3.2 and 3.3converge to those in Lemma 2.1 and Theorem 2.2 as the discrete step sizes approach zero. This means that2D-ADI-FDTD is approximately energy-conserved and unconditionally stable in the modified discrete form of the L^{2} and H^{1} norms.

In this section we solve a model problem by 2D-ADI-FDTD, and then test the analysis of the stability and energy conservation in Section 3 by comparing the numerical solution with the exact solution of the model. The model considered is the Maxwell equations (2.1) with

lution is:

It is easy to compute the norms of this solution are

To show the accuracy of 2D-ADI-FDTD, we define the errors:

where_{2}, R-ErL_{2}, ErH_{1} and R-ErH_{1}, i.e.

where log is the logarithmic function.

the CFL number

^{2} and H^{1} norm.

In this subsection we check the energy conservation of 2D-ADI-FDTD by computing the modified energy norms derived in Section 3 for the solution to the scheme. Denote these modified energy norms by

In

Δt | R-ErL_{2} | ErL_{2} | Rate | R-ErH_{1} | ErH_{1} | Rate |
---|---|---|---|---|---|---|

4h | 6.0284e−2 | 8.5254e−2 | 6.0287e−2 | 7.6675e−1 | ||

2h | 1.6264e−2 | 2.3001e−2 | 1.8901 | 1.6265e−2 | 2.0595e−1 | 1.8901 |

h | 5.1571e−3 | 7.2932e−3 | 1.6571 | 5.1571e−3 | 6.5229e−2 | 1.3182 |

Δx = Δy | R-ErL_{2} | ErL_{2} | Rate | R-ErH_{1} | ErH_{1} | Rate |
---|---|---|---|---|---|---|

2h | 5.0019e−3 | 8.3182e−3 | 5.0019e−3 | 7.4333e−3 | ||

h | 1.4981e−3 | 2.1186e−3 | 1.7393 | 1.4981e−3 | 1.8942e−3 | 1.7393 |

0.5h | 4.0200e−4 | 5.6851e−4 | 1.8979 | 4.0200e−4 | 5.0834e−4 | 1.8978 |

Fields\Norms | ||||
---|---|---|---|---|

8.9367 | 8.9367 | 1.4226 | 12.7183 | |

8.9367 | 8.9367 | 1.4226 | 12.7183 | |

8.9367 | 8.9367 | 1.4226 | 12.7183 | |

8.9367 | 8.9367 | 1.4226 | 12.7183 | |

3.2685e−13 | 3.2685e−13 | 5.2403e−14 | 4.6718e−13 | |

3.2685e−13 | 3.2685e−13 | 5.2403e−14 | 4.6718e−13 |

In this paper, the modified energy identities of the 2D-ADI-FDTD scheme with the periodic boundary conditions in the discrete L^{2} and H^{1} norms are established which show that this scheme is approximately energy conserved in terms of the two energy norms. By the deriving methods for the energy identities, new kind of energy identities of the Maxwell equations are proposed and proved by the new energy method. Numerical experiments are provided and confirm the analysis of 2D-ADI-FDTD.