Transient Analysis of Series and Parallel RLC Circuits using a Higher-Order Hybrid Block Backward Differentiation Formula with Off-Step Points

Switching and fault events excite transient currents and voltages in RLC circuits, and the governing second-order equation becomes stiff when the damping is strong. This paper applies a hybrid block backward differentiation formula (HSBBDF), with two off-step points, a stability-control parameter and step-size ratios r = 1, 2 and 5/8, to the source-free response of series and parallel RLC circuits in the under-damped, critically damped and over-damped regimes. The properties of the scheme were re-examined. It is of order six, which a constant-step convergence test on the six circuit problems supports (observed rates 5.7 to 6.3 in 17 of the 18 estimates), it is zero-stable for the three step-size ratios, and for r = 1 it is A(α)-stable with α ≈ 88.7°. A variable-step implementation with a Milne-type error estimate was run at tolerances 10⁻², 10⁻⁴ and 10⁻⁶ and compared with the variable-step variable-order 2-point diagonal block BDF (VSVO 2DBBDF) and MATLAB’s ode15s, using the results published for those methods, and with the variable-order BDF code of SciPy run on the same platform. HSBBDF gave a smaller maximum mixed error than VSVO 2DBBDF in 17 of the 18 cases and than ode15s in 16 of 18, with 1.6 to 5.4 times fewer steps. Its Python prototype was, however, slower in wall-clock time than SciPy’s BDF, and its global error exceeded the requested tolerance in some runs.