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# DGTD method based on exponential time integrators

We study a family of exponential time integration methods for the time-domain Maxwell's equations discretized by a high order discontinuous Galerkin (DG) scheme formulated on locally refined unstructured meshes. The number of refined elements in the mesh is assumed to represent a small percentage of the total number of mesh elements.

By following the Lawson procedure, we first split the underlying tetrahedral mesh into coarse and fine parts. We then split the problem unknowns so that the right hand of the differential equation becomes two operators associated to the fine part and the coarse part of the mesh. By introducing a new variable associated with the exponential of the fine part operator, we can remove the explicit dependence in the differential equation on the fine part operator. Thus the stability of the Lawson exponential time integration method for the fine part is excellent or even unconditional. Therefore, the particular exponential time integration method considered here not only removes the stiffness due to the fine part of the mesh on the allowable time step size, but also reverses the global explicitness when time integrated using an explicit time integration technique (e.g. Low-storage Runge-Kutta, LSRK scheme).

We simulate a composite structure made of a PEC sphere and a small PEC cylinder, which is shown in Figure 1. The computational domain is truncated by a Silver-Müller absorbing boundary condition. A localized radiation source is placed in the gap between the two structures. To capture the radiation of this localized source accurately, the elements around the source point are locally refined (see Figure 2 for contour lines of the radiated field).

 Figure 1. Electromagnetic wave radiation from a localized source. Unstructured tetrahedral mesh.

Figure 2. Electromagnetic wave radiation from a localized source. Contour lines of the magnitude of the electric field in the XoY, YoZ and XoZ planes.

Simulations rely on a DGTD-Pk (k=1,2,3) method, which is based on a central flux, and exploit the Lawson exponential integration combined to a fourth order five stage LSRK scheme. The solutions obtained by the DGTD-Pk methods using the combined Lawson-LSRK and fully explicit LSRK schemes agree very well (see Figure 3 for a comparison of the DGTD-P1 solutions).

Figure 3. Electromagnetic wave radiation from a localized source. Time evolution of the Ex and Ez components for the fully explicit LSRK and combined Lawson-LSRK schemes.

For this particular problem and the underlying locally refined tetrahedral mesh, the ratio between the allowable time step for the Lawson-LSRK scheme to that of the fully explicit LSRK scheme is equal to 132. The corresponding gain in CPU time for the DGTD-P1 and DGTD-P2 methods is respectively equal to 87 and 71.