Charge Density-Scalar Potential Formulation for Adaptive Time-Integration of Nonlinear Electroquasistatic Problems

An adaptive parabolic-elliptic time-integration method based on a singly diagonally implicit Runge-Kutta (SDIRK) algorithm is described for the finite element (FE) solution of nonlinear electroquasistatic (EQS) problems. The method uses the nodal charges as dynamic variables in addition to the electric scalar potential, thereby achieving better stability and performance than methods based on the scalar potential only. No Newton iteration is required in a time step because the Jacobian is incorporated into the time integration formula; only one linear equation with multiple right hand sides has to be solved. The global time-integration error is controlled by limiting the local error at each time step selection. The efficiency of the formulation and time stepping algorithm is illustrated by solving a typical nonlinear benchmark problem.