A new approach to the electromagnetic Darwin scheme

In the Darwin scheme of electromagnetic simulation, the electric field is separated into solenoidal and irrotational parts. The solenoidal part of the displacement current is omitted to eliminate propagation of electromagnetic waves and remove the time step limit due to the light speed. The resulting equation for the solenoidal electric field includes the time derivative of the solenoidal electric current which is not straightforward to obtain. One method dealing with this is the Streamlined Darwin Formulation proposed by Gibbons and Hewett [Journal of Computational Physics 120, 231-247, 1995]. The system of linear equations produced in this method, however, is difficult to solve. The present paper describes a different approach based on the curl of the Darwin equation for the solenoidal electric field. Applying the curl automatically removes the contribution of the irrotational electric current. The resulting system of linear equations is easily solved by a standard iterative solver. As an example, the method is applied to simulate a two-dimensional inductively coupled plasma system with the driving current wrapped around the plasma chamber.