  A time discretization method is called strongly stable, if the norm of its
numerical solution is nonincreasing. It is known that, even for linear
semi-negative problems, many explicit Runge--Kutta (RK) methods fail to
preserve this property. In this paper, we enforce strong stability by modifying
the method with superviscosity, which is a numerical technique commonly used in
spectral methods. We propose two approaches, the modified method and the
filtering method for stabilization. The modified method is achieved by
modifying the semi-negative operator with a high order superviscosity term; the
filtering method is to post-process the solution by solving a diffusive or
dispersive problem with small superviscosity. For linear problems, most
explicit RK methods can be stabilized with either approach without accuracy
degeneration. Furthermore, we prove a sharp bound (up to an equal sign) on
diffusive superviscosity for ensuring strong stability. The bound we derived
for general dispersive-diffusive superviscosity is also verified to be sharp
numerically. For nonlinear problems, a filtering method is investigated for
stabilization. Numerical examples with linear non-normal ordinary differential
equation systems and for discontinuous Galerkin approximation of conservation
laws are performed to validate our analysis and to test the performance.
