DGTD method based on exponential time integrators

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-P'_2_' and DGTD-P'_2_' is respectively equal to 87 and 71.

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

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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.

(:cellnr align='center':) %width=450px% http://www-sop.inria.fr/nachos/pics/news/2017/may/sphere_mesh.png

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Simulations rely on a DGTD-P'_k_' (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-P'_k_' methods using the combined Lawson-LSRK and fully explicit LSRK schemes agree very well (see Figure 2 for a comparsion of the DGTD-P'_1_' solutions).

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Figure 2. Electromagnetic wave radiation from a localized source. Contour lines of the magnitude of the electric field.

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(:cell  align='center':) %width=450px% http://www-sop.inria.fr/nachos/pics/news/2017/may/spherePsrc_YoZ_.png
(:cellnr align='center':) Figure 2. Electromagnetic wave radiation from a localized source. Contour lines of the magnitude of the electric field. 

(:cellnr align='center':) %width=500px% http://www-sop.inria.fr/nachos/pics/news/2017/may/sphere_mesh.png
(:cellnr align='center':) Figure 1. Electromagnetic wave radiation from a localized source. Unstructured tetrahedral mesh and contour lines of the magnitude of the electric field. 

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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 J_z (x,t) = sin⁡(wt) e^(-(x-x_s )^2 ), with frequency 300MHz oriented along the z-axis, is placed in the gap between the two structures. To capture the localized source accurately, the elements around the source point are locally refined.

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(:cellnr align='center':) Electromagnetic wave radiation from a localized source. Unstructured tetrahedral mesh and contour lines of the magnitude of the electric field. 

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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).

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We study a family of exponential-based 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, rising from the modeling of 3D transient multiscale electromagnetic problems. The number of refined elements is assumed far less than that of coarse elements.

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(:title High order DGTD method based on exponential time integrators:)

(:title Exponential-based high order DGTD method for modeling:)
(:title 3D transient multiscale electromagnetic problems:)

(:title Exponential-based high order DGTD method for modeling 3D transient multiscale electromagnetic problems:)