Monotonicity-preserving splitting schemes for solving balance laws

In this paper, some monotonicity-preserving (MP) and positivity-preserving (PP) splitting methods for solving the balance laws of the reaction and diffusion source terms are investigated. To capture the solution with high accuracy and resolution, the original equation with reaction source termis separated through the splitting method into two sub-problems including the homogeneous conservation law and a simple ordinary differential equation (ODE). The resulting splitting methods preserve monotonicity and positivity property for a normal CFL condition. A trenchant numerical analysis made it clear that the computing time of the proposed methods decreases when the so-called MP process for the homogeneous conservation law is imposed. Moreover, the proposed methods are successful in recapturing the solution of the problem with high-resolution in the case of both smooth and non-smooth initial profiles. To show the efficiency of proposed methods and to verify the order of convergence and capability of these methods, several numerical experiments are performed through some prototype examples.