On Stability, Monotonicity, and Construction of Difference Schemes I: Theory

In this paper, we focus on the application and illustration of the approach developed in part I. This approach is found to be useful in the construction of stable and monotone central difference schemes for hyperbolic systems. A new modification of the central Lax–Friedrichs scheme is developed to be of second-order accuracy. The stability of several versions of the developed central scheme is proved. Necessary conditions for the variational monotonicity of the scheme are found. A monotone piecewise cubic interpolation is used in the central schemes to give an accurate approximation for the model in question. The monotonicity parameter introduced in part I is found to be important. That parameter, along with the Courant–Friedrichs–Lewy (CFL) number, plays a major role in the criteria for monotonicity and stability of the central schemes. The modified scheme is tested on several conservation laws taking a CFL number equal or close to unity, and the scheme is found to be accurate and robust.