Large systems of equations in a discrete electromagnetism: formulations and numerical algorithms

The description of discrete electromagnetic fields is achieved with the Maxwell grid equations, a set of matrix equations representing a spatial discretisation of Maxwell´s equations in integral form on a dual mesh pair. Based on this matrix formalism the calculation of electromagnetic fields requires efficient techniques for the solution of commonly large, sparse systems of linear or nonlinear algebraic, differential-algebraic and ordinary differential equations. Field formulations and modern numerical-solution algorithms especially for the solution of static and quasistatic electric and magnetic fields are presented. Numerical schemes for fast-varying transient and time-harmonic current-driven or resonator electromagnetic fields are added. Details such as regularisation techniques, the modelling of motion-induced eddy currents, possible field excitation sources and the algorithmic treatment of ferromagnetic material properties are addressed. The numerical schemes include error-controlled adaptive time-domain algorithms featuring advanced extrapolation methods, linearisation methods for the nonlinear algebraic systems of equations with the possible inclusion of ferromagnetic hysteresis modelling and iterative solution methods for large sparse linear systems of equations.