A stability study of a new explicit numerical scheme for a system of differential equations with a large skew-symmetric component

Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time step, but require more computations. When the differential equations are dominated by a skew-symmetric component, the problem is not stiffness in the sense that thesize of the eigenvalues are unequal, rather that the real eigenvalues are dominated by imaginary eigenvalues. A skew-dominated system of this type may be seen in magnetohydrodynamics and in control problems. We present and compare analytical results for stable time step limits for several explicit methods including the super-time-stepping method of Alexiades, Amiez, and Gremaud which is an explicit Runge-Kutta method for parabolic partial differential equations and a new method modeled on a predictor-corrector scheme with multiplicative operator splitting. This new explicit method, presented in regular and super-time-stepping form, increases stability without forcing the step size to zero.