Courant condition

The finite difference time step, Δt, can be made very small. However, there are limits on the upper value of Δt. This upper limit is given by the equation below.

where c is the speed of light in the medium in which the computation is being conducted.

The equation above is known as the Courant criterion or Courant condition. Violation of the Courant condition will result in a numerical instability that is quickly manifested. Therefore, if the Courant condition is violated, an error message will be issued in EMA3D and the program terminated

As the equation indicates, the time step can be made as small as desired without violating the Courant condition. However, a small time step implies more time iterations to attain a desired response for a given time window, and therefore an increase in computation time. In addition, an extremely small time step may impart numerical errors due to limitations in computer dynamic range.

It is therefore advantageous, from a computational time perspective, to make the time step as large as possible without violating the Courant condition. However, if the time step is too close, within 10% to 20% of the upper limit, then numerical instabilities may arise. These instabilities are exacerbated by thin wires with large diameters (large meaning over 20% of the largest cell increment in the cross-sectional direction) and also the presence of dielectric materials.

To avoid the manifestation of such instabilities, it is best to set the time step no closer than 20% of the maximum value mandated by the Courant condition. If numerical instabilities still arise, then lowering of the time step value further may be required. Some experimentation may be necessary in some cases.

The time step value in EMA3D is the same at all mesh locations. Therefore, it must satisfy the Courant condition at all mesh locations within the finite difference problem space. This comprises both material parameters, through the speed of propagation, and mesh increments (both constant and variable). The problem space may be searched to find the overall minimum value at each mesh location. However, it is usually easier to use the overall minimum spatial increments and the speed of light in a vacuum. This assures that the Courant condition is not violated.