The numerical solution of the Cauchy problem for a partial differential equation of the type \[ {\partial \over \partial t} u(x,t) + {\partial \over \partial x} f(u(x,t)) =0, \quad t \geq 0\;\;-\infty < x < \infty \] is investigated. Using the method of lines the numerical problem leads to solving a system of ordinary differential equations of the form \[ {d\over dt} U(t) = F(U(t)) (t\geq 0), \;U(0)=u_0. \] To solve this system via some Runge-Kutta (RK) method we obtain approximate solution \(u_n\) of \(U(n\Delta t)\) in some time \(n\Delta t\), where \(\Delta t>0\) denotes time step. For all these methods it is desirable, that this process is total-variation-diminishing (TVD). Some conditions for this property of special types of Runge-Kutta methods were done in previous paper of C.-W. Shu and S. Osher [J. Comput. Phys. 77, No. 2, 439–471 (1988; Zbl 0653.65072)] or S. Gottlieb and C.-W. Shu [Math. Comput. 67, No. 221, 73–85 (1998; Zbl 0897.65058)].
The authors solve the problem of extension these conditions for more general types of RK methods and they want to give some restrictions on the step-size which are necessary and sufficient for the TVD property. The authors define a step-size-coefficient for monotonicity, which give the restriction for time step for the RK method to rest monotonic. This coefficient corresponds to the so called Courant-Friedrichs-Levy (CFL) coefficient in the context of discretization of hyperbolic partial differential equations. The authors prove that if some \(c\) is less or equal the quantity \(R(A,b)\) which characterizes the RK method and was introduced by J. F. B. M. Kraaijevanger [BIT 31, No. 3, 482–528 (1991; Zbl 0763.65059)], then this \(c\) is a step-size-coefficient for monotonicity with respect to all vector spaces \(\mathcal V\) and seminorms \(\| .\| \) on \(\mathcal V\). This theorem allows the authors to prove the TVD property for more general Runge-Kutta methods.